If anyone wants to understand how everything really works, one thing that is vital is the Inverse Square Law. This is a simple, yet extremely wide-reaching rule which describes the relationships between energy, information and, distance. The Inverse Square Law simply states that, if an object is twice as far away, it's light or gravity will be reduced to one-quarter because four is the square of two (2 x 2 = 4). If an object is three times as far away, it's light or gravity will be reduced to one-ninth because 3 x 3 = 9.
TABLE OF CONTENTS
1) THE INVERSE SQUARE LAW AND THE NUMBER OF ELECTRIC CHARGES
2) LIGHT, GRAVITY AND, THE INVERSE SQUARE LAW
3) GALILEO'S PARADOX AND NEWTON
4) GRANULARITY AND THE INVERSE SQUARE LAW
5) THE EVEN NUMBER BIAS
6) PROGRESS AND THE INVERSE SQUARE LAW
7) DO WE REALLY NEED CALCULUS"?
8) THE FIRST ATOMS AND THE INVERSE SQUARE LAW
9) THE AVERAGE DISTANCE OF A PLANET FROM THE SUN
I see the Inverse Square Law as even more useful and versatile than it is usually regarded. Hopefully, I can add some more to this simple rule of the universe and you will come away today with a renewed appreciation of it.
1) THE INVERSE SQUARE LAW AND THE NUMBER OF ELECTRIC CHARGES
It has been known for centuries that the propagation into space of forces like light and gravity operate by what is known as the Inverse Square Law. This means that the intensity of the light or gravity decreases by the square of the distance as we move further away from the source. Put simply, if something is twice as far away it will appear as only one-quarter of the former apparent size or brightness, or it's gravity will be only one-quarter. This is because square means multiplied by itself, and 2 x 2 = 4. If the object were three times as far away, it would be one-ninth.
It has also long been known that there are two electric charges in the universe, negative and positive, that are opposite and equal. In my cosmology theory, the entire universe is composed of these electric charges. Where there is the usual alternating pattern of negative and positive charges, because opposite charges attract while like charges repel, we have empty space. Where there is a concentration of like charges, held together by energy against the mutual repulsion of like charges, we have matter and the extra energy required is what gives us the well-known Mass-Energy Equivalence.
What I want to point out today is the relationship between the two. My observation is that the reason for the Inverse Square Law in our universe is that it is composed of the two electric charges.
An electromagnetic wave gets weaker as it moves out into space because it's energy is spread over more of the electric charges of space. But energy is really the same thing as information, we cannot apply energy to something without also adding information to it, and we cannot add information to something without applying energy to it.
The rate at which the intensity of the wave will fade, as it moves out into space, is thus proportional to the amount of information (which is also energy) that it must match in the electric charges composing space. The negative and positive electric charges means that there are two possibilities for each charge in space. If there were three possibilities for each electric charge in space, instead of the two of negative and positive, we would have the "Inverse Cube Law" because there would be more probability for each charge in space. Light or gravity would diminish more rapidly, at the cube instead of the square, of the distance.
If there were three electric charges in the universe, rather than two, it would mean that there would be more information that would have to be included. If we looked at an object that was then moved a certain distance away, it would have to diminish in size and brightness proportionally more than in a universe with only two electric charges because there would be proportionally more information at that distance which would have to be added into the field of vision, which is of a fixed size.
No matter what and how many different electric charges the universe might have been made of, the inverse law regarding the relative strength of an electromagnetic wave at a distance will have, as it's exponent, the minimum number of dimensions necessary to convey the information of how the electric charges interact. This is equal to the number of different electric charges, which in our universe is two.
If there were three equal charges that had to form a regular pattern in space, there would have to be an "Inverse Cube Law". It would take more energy to create an electromagnetic wave, and the strength of the wave would fall off faster with distance. Gravity would also be stronger, but would fall off faster with distance.
Suppose that the universe was somehow composed of only one electric charge, instead of our two. Theoretically, if a wave could be sent out into space, it would never diminish because there would be no information at all in the electric charges that the wave was crossing. This is because all information must be in the permutations of electric charges, and there would be no meaningful permutations because there would only be one kind of electric charge.
The Inverse Square Law that we have in our universe with two electric charges has nothing to do with the number of spatial dimensions. But for a wave to diminish with distance, there must be at least the number of dimensions as there are electric charges of which space is composed. If there were to be more electric charges in space than there are dimensions involved, a wave would not diminish with distance. (This is the principle behind a laser). An illustration of a wave not diminishing if the number of dimensions is less than the number of electric charges is a line of dominoes, with each falling one knocking over the one in front of it. As long as the dominoes are correctly lined up, the energy of the wave never diminishes.
A wave on the water, which is in two-dimensions, and an electromagnetic wave, which is in three dimensions, will both diminish with distance at the same rate. But this is not true of the line of dominoes, which is one-dimensional. This shows that there must be at least an equal number of dimensions to the two electric charges of which the universe is composed for a wave to diminish according to the Inverse Square Law, but that the same Inverse Square Law will apply when there are more dimensions than this.
The reason for this is that for the diminishing in intensity with distance, according to the Inverse Square Law, the wave or force must occupy enough dimensions to illustrate how the electric charges interact with one another. In our universe, with two electric charges, that requires two dimensions. there is a checkerboard of negative and positive charges, with each charges surrounded by four opposite charges, because opposite charges attract while like charges repel.
With two electric charges and two dimensions of space, the same Inverse Square Law applies regardless of how many more dimensions there might be. A wave propagating outward from a one-dimensional line in two-dimensional space, or from a two-dimensional sheet in three-dimensional space, will not diminish with distance because there will be no spreading with distance, but a wave from a one-dimensional antenna line or a point in three-dimensional space will diminish with distance, according to the Inverse Square Law.
From any given point in space, there are two opposite directions in each dimension. This can only be because there are two electric charges composing that space, so that there must be one direction for each charge. If there were three equal charges in space that had to be arranged to make up space, there would have to be three directions in each dimension from a given point, even though it is difficult for us to imagine this. The applicable Inverse Law exponent will also be the number of directions in space, from a given point along the same dimension.
Because there are two electric charges, which alternate to form each dimension of space, there must be two separate "information routes" in a dimension, which we see as two opposite directions along a straight line from any given point on the line. It is because light, emanating from one point, must go along two "information routes", or directions, in each dimension of space, diminishes in brightness and objects diminish in apparent size, according to the Inverse Square Law.
Suppose that there is a point in space between two of the electric charges comprising space. Since space consists of alternating negative and positive charges, because opposite charges attract, there would be a negative charge on one side of the point and a positive charge on the other side. This means that there would be two different "information sets" in that dimension from that point. One one side, the information set would start with a negative charge, and would be - + - + - +... On the other side, the information set would start with a positive charge, and would be + - + - + -... Because two electric charges means that there has to be two different "information sets" in a given dimension from any given point in space, that means that there must be two opposite directions in space in the given dimension from the point.
If there were three electric charges in the universe, there would have to be a third such "information set" and thus a third direction from any given point in the same dimension. Light and gravity would diminish by the Inverse Cube Law, rather than our Inverse Square Law, because it would have to "divide" itself along this third information set or route also.
There is a formula for the number of directions that we can go from any given point in space, separated from one another by a right angle. Possible Directions = Number Of Electric Charges Composing Space x Number Of Dimensions Of Space. This means that, in our three dimensional space, there are six directions from a given point in space, separated by right angles. We might express these six directions as: up, down, left, right, backwards and, forwards.
Thus we see how day and night, for one example, ultimately result from the fact that the universe is composed of two electric charges. If light, from a given source, is coming from one direction, then there must be another opposite direction that it is not coming from. If we lived in a universe with three electric charges in space, we could expect that day and night would be more complex.
If the universe was composed of only one electric charge, any waves or movement or movement of energy really would not make any sense. For something to move, by application of energy, it must involve a change of information, because information is energy, or else it makes no sense. There could thus be no movement, no information and, no energy in a one-charge universe. There could be no application of outside energy to anything. It would make no sense to move anything from one place to another because with only one electric charge, there could be no permutations of electric charge to store the information which is energy.
If anything has the energy to move from any point, there must be more than one possible direction in which it can move because movement must necessarily involve more information than there is in empty space. For that to be possible, space must be composed of more than one electric charge.
This proves, as described in my cosmology theory, that space is composed of alternating negative and positive electric charges, and also that space and distance and information and energy are really the same thing. This is the cosmology behind the Inverse Square Law.
2) LIGHT, GRAVITY AND, THE INVERSE SQUARE LAW
The gravitational effect of massive objects are stronger than is to be expected, if we used the same logic as with the rules governing light. The larger the object, the more out of proportion is the gravity. The gravity of a massive object includes it's orbital energy and gravitational attractions to further gravitational nodes to which that object is connected, and which extend throughout the entire universe.
Consider that, from earth, the sun and moon appear as about the same angular size in the sky. The moon is 2.4 times as dense as the sun. If gravity operated in the same way as light, with regard to the Inverse Square Law, then the force of the moon's gravity on earth would be 2.4 times that of the sun.
Yet, the reality is that the gravitational force of the sun on the earth is 169 times that of the moon.
Now, suppose that there was an observer on the moon. The earth would appear in the sky as having four times the angular diameter of the sun. The earth is also more dense than the sun, 3.37 times as dense. Using the same logic as with light, this should mean that the earth's gravitational effect on the moon should be 13.48 times that of the sun.
Yet, this is not the case. The sun's gravitational effect on the moon is actually 2.08 times that of the earth. The sun is 400 times as far from the moon as the earth. using the Inverse Square Law, 400 squared is 160,000. but the sun is 333,000 times the mass of the earth, and 333,000 / 160,000 = 2.08.
Very clearly, although both gravity and light operate by the same Inverse Square Law there are great differences between the two which require special explanation. My conclusion is that gravity is cumulative throughout the universe, while light is not.
The sun is 27 million times the mass of the moon. The cube root of 27 million is 300. The moon is 2.4 times the density of the sun, and the two are about the same angular size in the sky. 300 / 2.4 = 125, yet we know that the sun's gravitational force on the earth is 169 times that of the moon.
The reason that the sun's gravitational effect on earth, relative to the moon, is somewhat more than 125 is that the earth is a concentrated point as seen from the sun, while it has an angular diameter of about 2 degrees as seen from the moon. This makes the sun's gravitational pull on the earth relatively more concentrated than that of the moon, because it is less dispersed.
(Note-One way that we can test these figures is by calculating the tidal effects of the moon and the sun on the earth's oceans. We see that the sun exerts a gravitational force on the earth that is 169 times that of the moon. This is because the sun has 27 million times the mass of the moon (Remember that the sun is 333,000 times the mass of the earth, while the earth is 81 times the mass of the moon). The sun and moon appear as about the same size in the sky. The sun has 400 times the diameter of the moon, but is 400 times as distant. The earth is 3.37 times the density of the sun, while the moon is .606 earth's density, lacking a heavy iron core. The tidal effect of the sun on the earth is known to be only about 40% that of the moon, we can see this figure by dividing 169/400 because being closer to the earth means that there is more difference in the moon's gravity at the top of the ocean than there is at the bottom.
Another test is that we know the earth's mass to be 81 times that of the moon. From the moon, the gravity of the sun is 2.08 that of the earth. From the earth, the sun's gravity is 169 times that of the moon. Notice that 169 / 2.08 = 81.
Here are some basic figures which might be helpful here, and which I use in the calculations:
The earth is 49 times the volume and 81 times the mass of the moon, so that it is about 1.6 times the density.
The earth is 3.37 times the density of the sun, the moon is 2.04 times the density of the sun.
The sun and moon appear from earth as about the same size in the sky, but the sun is about 400 x as far away.
The earth is about 4 times as wide as the sun, as seen from the moon.
The mass of the sun is 333,000 times that of the earth and 27 million times that of the moon).
MATHEMATICAL CHECK OF ABOVE CALCULATIONS
But if the application of the Inverse Square Law to light and gravity cannot be directly compared, then there must be some formula of the relationship between the two. The formula that I come up with is as follows:
The square root of the apparent size difference between two distant objects X the density difference between the two objects X the difference in observed gravitational force between the two objects = the square root of (the mass difference divided by the ratio of distance of the two objects)
Applying this formula to the difference between the earth and sun, as seen from the moon: the apparent size difference is 16, the earth appearing as four times as wide and four times as high as the sun, and the square root of this is 4. The density difference is the earth is 3.37 times as dense as the sun. The gravity difference is that the sun has 2.08 times the gravity of the earth, as measured from the moon. This gives us 28.04.
This is very close to the square root of the mass difference between the sun and the earth, which is 333.000, divided by the distance difference between the sun and the earth, which is 400. This gives us the square root of 832.5, which is 28.85.
28.04 is close to 28.85. We do not get an exact answer because, of course, the figures are average figures and the distance between the earth and the moon and the sun is not constant.
Notice that, to the left of the equals sign, the larger figure must come first. The first two figures are the earth relative to the sun, but the gravity ratio is the sun relative to the earth. The same holds true for the right side of the equation, both are the sun relative to the earth.
I see the Inverse Square Law as even more useful and versatile than it is usually regarded. Hopefully, I can add some more to this simple rule of the universe and you will come away today with a renewed appreciation of it.
1) THE INVERSE SQUARE LAW AND THE NUMBER OF ELECTRIC CHARGES
It has been known for centuries that the propagation into space of forces like light and gravity operate by what is known as the Inverse Square Law. This means that the intensity of the light or gravity decreases by the square of the distance as we move further away from the source. Put simply, if something is twice as far away it will appear as only one-quarter of the former apparent size or brightness, or it's gravity will be only one-quarter. This is because square means multiplied by itself, and 2 x 2 = 4. If the object were three times as far away, it would be one-ninth.
It has also long been known that there are two electric charges in the universe, negative and positive, that are opposite and equal. In my cosmology theory, the entire universe is composed of these electric charges. Where there is the usual alternating pattern of negative and positive charges, because opposite charges attract while like charges repel, we have empty space. Where there is a concentration of like charges, held together by energy against the mutual repulsion of like charges, we have matter and the extra energy required is what gives us the well-known Mass-Energy Equivalence.
What I want to point out today is the relationship between the two. My observation is that the reason for the Inverse Square Law in our universe is that it is composed of the two electric charges.
An electromagnetic wave gets weaker as it moves out into space because it's energy is spread over more of the electric charges of space. But energy is really the same thing as information, we cannot apply energy to something without also adding information to it, and we cannot add information to something without applying energy to it.
The rate at which the intensity of the wave will fade, as it moves out into space, is thus proportional to the amount of information (which is also energy) that it must match in the electric charges composing space. The negative and positive electric charges means that there are two possibilities for each charge in space. If there were three possibilities for each electric charge in space, instead of the two of negative and positive, we would have the "Inverse Cube Law" because there would be more probability for each charge in space. Light or gravity would diminish more rapidly, at the cube instead of the square, of the distance.
If there were three electric charges in the universe, rather than two, it would mean that there would be more information that would have to be included. If we looked at an object that was then moved a certain distance away, it would have to diminish in size and brightness proportionally more than in a universe with only two electric charges because there would be proportionally more information at that distance which would have to be added into the field of vision, which is of a fixed size.
No matter what and how many different electric charges the universe might have been made of, the inverse law regarding the relative strength of an electromagnetic wave at a distance will have, as it's exponent, the minimum number of dimensions necessary to convey the information of how the electric charges interact. This is equal to the number of different electric charges, which in our universe is two.
If there were three equal charges that had to form a regular pattern in space, there would have to be an "Inverse Cube Law". It would take more energy to create an electromagnetic wave, and the strength of the wave would fall off faster with distance. Gravity would also be stronger, but would fall off faster with distance.
Suppose that the universe was somehow composed of only one electric charge, instead of our two. Theoretically, if a wave could be sent out into space, it would never diminish because there would be no information at all in the electric charges that the wave was crossing. This is because all information must be in the permutations of electric charges, and there would be no meaningful permutations because there would only be one kind of electric charge.
The Inverse Square Law that we have in our universe with two electric charges has nothing to do with the number of spatial dimensions. But for a wave to diminish with distance, there must be at least the number of dimensions as there are electric charges of which space is composed. If there were to be more electric charges in space than there are dimensions involved, a wave would not diminish with distance. (This is the principle behind a laser). An illustration of a wave not diminishing if the number of dimensions is less than the number of electric charges is a line of dominoes, with each falling one knocking over the one in front of it. As long as the dominoes are correctly lined up, the energy of the wave never diminishes.
A wave on the water, which is in two-dimensions, and an electromagnetic wave, which is in three dimensions, will both diminish with distance at the same rate. But this is not true of the line of dominoes, which is one-dimensional. This shows that there must be at least an equal number of dimensions to the two electric charges of which the universe is composed for a wave to diminish according to the Inverse Square Law, but that the same Inverse Square Law will apply when there are more dimensions than this.
The reason for this is that for the diminishing in intensity with distance, according to the Inverse Square Law, the wave or force must occupy enough dimensions to illustrate how the electric charges interact with one another. In our universe, with two electric charges, that requires two dimensions. there is a checkerboard of negative and positive charges, with each charges surrounded by four opposite charges, because opposite charges attract while like charges repel.
With two electric charges and two dimensions of space, the same Inverse Square Law applies regardless of how many more dimensions there might be. A wave propagating outward from a one-dimensional line in two-dimensional space, or from a two-dimensional sheet in three-dimensional space, will not diminish with distance because there will be no spreading with distance, but a wave from a one-dimensional antenna line or a point in three-dimensional space will diminish with distance, according to the Inverse Square Law.
From any given point in space, there are two opposite directions in each dimension. This can only be because there are two electric charges composing that space, so that there must be one direction for each charge. If there were three equal charges in space that had to be arranged to make up space, there would have to be three directions in each dimension from a given point, even though it is difficult for us to imagine this. The applicable Inverse Law exponent will also be the number of directions in space, from a given point along the same dimension.
Because there are two electric charges, which alternate to form each dimension of space, there must be two separate "information routes" in a dimension, which we see as two opposite directions along a straight line from any given point on the line. It is because light, emanating from one point, must go along two "information routes", or directions, in each dimension of space, diminishes in brightness and objects diminish in apparent size, according to the Inverse Square Law.
Suppose that there is a point in space between two of the electric charges comprising space. Since space consists of alternating negative and positive charges, because opposite charges attract, there would be a negative charge on one side of the point and a positive charge on the other side. This means that there would be two different "information sets" in that dimension from that point. One one side, the information set would start with a negative charge, and would be - + - + - +... On the other side, the information set would start with a positive charge, and would be + - + - + -... Because two electric charges means that there has to be two different "information sets" in a given dimension from any given point in space, that means that there must be two opposite directions in space in the given dimension from the point.
If there were three electric charges in the universe, there would have to be a third such "information set" and thus a third direction from any given point in the same dimension. Light and gravity would diminish by the Inverse Cube Law, rather than our Inverse Square Law, because it would have to "divide" itself along this third information set or route also.
There is a formula for the number of directions that we can go from any given point in space, separated from one another by a right angle. Possible Directions = Number Of Electric Charges Composing Space x Number Of Dimensions Of Space. This means that, in our three dimensional space, there are six directions from a given point in space, separated by right angles. We might express these six directions as: up, down, left, right, backwards and, forwards.
Thus we see how day and night, for one example, ultimately result from the fact that the universe is composed of two electric charges. If light, from a given source, is coming from one direction, then there must be another opposite direction that it is not coming from. If we lived in a universe with three electric charges in space, we could expect that day and night would be more complex.
If the universe was composed of only one electric charge, any waves or movement or movement of energy really would not make any sense. For something to move, by application of energy, it must involve a change of information, because information is energy, or else it makes no sense. There could thus be no movement, no information and, no energy in a one-charge universe. There could be no application of outside energy to anything. It would make no sense to move anything from one place to another because with only one electric charge, there could be no permutations of electric charge to store the information which is energy.
If anything has the energy to move from any point, there must be more than one possible direction in which it can move because movement must necessarily involve more information than there is in empty space. For that to be possible, space must be composed of more than one electric charge.
This proves, as described in my cosmology theory, that space is composed of alternating negative and positive electric charges, and also that space and distance and information and energy are really the same thing. This is the cosmology behind the Inverse Square Law.
2) LIGHT, GRAVITY AND, THE INVERSE SQUARE LAW
Has anyone ever noticed that both light, which is a form of electromagnetic radiation, and gravity operate by the Inverse Square Law, but that the two cannot be directly compared? The Inverse Square Law simply means that, if an object is twice as far away, it will have only one-quarter of the brightness, apparent angular diameter or, gravitational force.
But yet light and gravity, with regard to the Inverse Square law, simply do not compare directly. I find that gravity is cumulative, but light is not, and that is why the two operate by different rules within the Inverse Square Law. The force of gravity links to the structure of the entire universe, while the intensity of light does not. This reveals a lot about how the universe operates.The gravitational effect of massive objects are stronger than is to be expected, if we used the same logic as with the rules governing light. The larger the object, the more out of proportion is the gravity. The gravity of a massive object includes it's orbital energy and gravitational attractions to further gravitational nodes to which that object is connected, and which extend throughout the entire universe.
Consider that, from earth, the sun and moon appear as about the same angular size in the sky. The moon is 2.4 times as dense as the sun. If gravity operated in the same way as light, with regard to the Inverse Square Law, then the force of the moon's gravity on earth would be 2.4 times that of the sun.
Yet, the reality is that the gravitational force of the sun on the earth is 169 times that of the moon.
Now, suppose that there was an observer on the moon. The earth would appear in the sky as having four times the angular diameter of the sun. The earth is also more dense than the sun, 3.37 times as dense. Using the same logic as with light, this should mean that the earth's gravitational effect on the moon should be 13.48 times that of the sun.
Yet, this is not the case. The sun's gravitational effect on the moon is actually 2.08 times that of the earth. The sun is 400 times as far from the moon as the earth. using the Inverse Square Law, 400 squared is 160,000. but the sun is 333,000 times the mass of the earth, and 333,000 / 160,000 = 2.08.
Very clearly, although both gravity and light operate by the same Inverse Square Law there are great differences between the two which require special explanation. My conclusion is that gravity is cumulative throughout the universe, while light is not.
CUBE ROOT FOR GRAVITY
What I have found is that the differences in gravity, with regard to the Inverse Square Law, is proportional to the cube root of the difference in mass. This is what makes the behavior of gravity different from that of light, even though both operate by the same Inverse Square Law. When the gravitational effect of a larger object is compared to that of a smaller object, as seen from a third object, the larger object will have a gravitational force out of proportion to the rules of light by an amount equal to the cube root of the relative masses of the larger and the smaller distant objects.
If A x A x A = B, then A is the cube root of B. Cube means three because a cube has three dimensions that are multiplied together to get it's volume. The cube root of 27 is 3 because 3 x 3 x 3 = 27.
The reason that we use the cube root of the mass difference between objects in space of different mass, rather than the direct mass difference itself, is that the massive object is more linked by it's stronger gravity to the branches of the universal gravitational structure. The mass proportional difference between the two distant objects is reduced to it's cube root because outside gravity from the galactic center acts on all three objects. The sun has a gravitational relationship with the center of the galaxy, which is in turn linked to our Local Group of galaxies, which is linked by gravity to the spurs and filaments making up the structure of the entire universe.
Let's have a look at an example, the effect of the sun's gravity on the earth relative to that of the moon.
What I have found is that the differences in gravity, with regard to the Inverse Square Law, is proportional to the cube root of the difference in mass. This is what makes the behavior of gravity different from that of light, even though both operate by the same Inverse Square Law. When the gravitational effect of a larger object is compared to that of a smaller object, as seen from a third object, the larger object will have a gravitational force out of proportion to the rules of light by an amount equal to the cube root of the relative masses of the larger and the smaller distant objects.
If A x A x A = B, then A is the cube root of B. Cube means three because a cube has three dimensions that are multiplied together to get it's volume. The cube root of 27 is 3 because 3 x 3 x 3 = 27.
The reason that we use the cube root of the mass difference between objects in space of different mass, rather than the direct mass difference itself, is that the massive object is more linked by it's stronger gravity to the branches of the universal gravitational structure. The mass proportional difference between the two distant objects is reduced to it's cube root because outside gravity from the galactic center acts on all three objects. The sun has a gravitational relationship with the center of the galaxy, which is in turn linked to our Local Group of galaxies, which is linked by gravity to the spurs and filaments making up the structure of the entire universe.
Let's have a look at an example, the effect of the sun's gravity on the earth relative to that of the moon.
The sun is 27 million times the mass of the moon. The cube root of 27 million is 300. The moon is 2.4 times the density of the sun, and the two are about the same angular size in the sky. 300 / 2.4 = 125, yet we know that the sun's gravitational force on the earth is 169 times that of the moon.
The reason that the sun's gravitational effect on earth, relative to the moon, is somewhat more than 125 is that the earth is a concentrated point as seen from the sun, while it has an angular diameter of about 2 degrees as seen from the moon. This makes the sun's gravitational pull on the earth relatively more concentrated than that of the moon, because it is less dispersed.
Another factor why 169 is more than 125 is that since the moon is also in the sun's gravitational field, and the sun's gravity on the moon is stronger than the earth's gravity, the sun's gravitational force on the earth is also acting through the moon, although pointing toward the sun and not toward the moon.
That is what I mean by gravity being cumulative. The moon's gravitational force on the earth does not act through the sun in the same way because the distance from moon to sun, and back to earth, is so great and the moon's mass is so utterly insignificant, relative to that of the sun.
CONCLUSION ABOUT GRAVITY AND LIGHT
The cumulative gravity of the entire universe is why it does not operate by the same rules as light. This is why the sun and moon appear as about the same size in the sky, the moon is actually 2.4 times the density of the sun, yet the sun's gravitational effect on the earth is 169 times that of the moon. The sun's greater mass gives it a stronger link to outside gravity, the center of the galaxy, and this outside gravity acts through the sun. The directional alignments of the earth, sun and, moon with center of the gravity matter little. The effect of the sun's gravity is proportional to only the cube root of the mass difference because the earth, moon and sun are three objects and all are ultimately under the gravitational effects of the center of the galaxy.
Light, unlike gravity, is not cumulative and so the two operate by different rules, even though they both operate by the Inverse Square Law.
The cumulative gravity of the entire universe is why it does not operate by the same rules as light. This is why the sun and moon appear as about the same size in the sky, the moon is actually 2.4 times the density of the sun, yet the sun's gravitational effect on the earth is 169 times that of the moon. The sun's greater mass gives it a stronger link to outside gravity, the center of the galaxy, and this outside gravity acts through the sun. The directional alignments of the earth, sun and, moon with center of the gravity matter little. The effect of the sun's gravity is proportional to only the cube root of the mass difference because the earth, moon and sun are three objects and all are ultimately under the gravitational effects of the center of the galaxy.
Light, unlike gravity, is not cumulative and so the two operate by different rules, even though they both operate by the Inverse Square Law.
(Note-One way that we can test these figures is by calculating the tidal effects of the moon and the sun on the earth's oceans. We see that the sun exerts a gravitational force on the earth that is 169 times that of the moon. This is because the sun has 27 million times the mass of the moon (Remember that the sun is 333,000 times the mass of the earth, while the earth is 81 times the mass of the moon). The sun and moon appear as about the same size in the sky. The sun has 400 times the diameter of the moon, but is 400 times as distant. The earth is 3.37 times the density of the sun, while the moon is .606 earth's density, lacking a heavy iron core. The tidal effect of the sun on the earth is known to be only about 40% that of the moon, we can see this figure by dividing 169/400 because being closer to the earth means that there is more difference in the moon's gravity at the top of the ocean than there is at the bottom.
Another test is that we know the earth's mass to be 81 times that of the moon. From the moon, the gravity of the sun is 2.08 that of the earth. From the earth, the sun's gravity is 169 times that of the moon. Notice that 169 / 2.08 = 81.
Here are some basic figures which might be helpful here, and which I use in the calculations:
The earth is 49 times the volume and 81 times the mass of the moon, so that it is about 1.6 times the density.
The earth is 3.37 times the density of the sun, the moon is 2.04 times the density of the sun.
The sun and moon appear from earth as about the same size in the sky, but the sun is about 400 x as far away.
The earth is about 4 times as wide as the sun, as seen from the moon.
The mass of the sun is 333,000 times that of the earth and 27 million times that of the moon).
MATHEMATICAL CHECK OF ABOVE CALCULATIONS
But if the application of the Inverse Square Law to light and gravity cannot be directly compared, then there must be some formula of the relationship between the two. The formula that I come up with is as follows:
The square root of the apparent size difference between two distant objects X the density difference between the two objects X the difference in observed gravitational force between the two objects = the square root of (the mass difference divided by the ratio of distance of the two objects)
Applying this formula to the difference between the earth and sun, as seen from the moon: the apparent size difference is 16, the earth appearing as four times as wide and four times as high as the sun, and the square root of this is 4. The density difference is the earth is 3.37 times as dense as the sun. The gravity difference is that the sun has 2.08 times the gravity of the earth, as measured from the moon. This gives us 28.04.
This is very close to the square root of the mass difference between the sun and the earth, which is 333.000, divided by the distance difference between the sun and the earth, which is 400. This gives us the square root of 832.5, which is 28.85.
28.04 is close to 28.85. We do not get an exact answer because, of course, the figures are average figures and the distance between the earth and the moon and the sun is not constant.
Notice that, to the left of the equals sign, the larger figure must come first. The first two figures are the earth relative to the sun, but the gravity ratio is the sun relative to the earth. The same holds true for the right side of the equation, both are the sun relative to the earth.
THE SOLUTION TO THE THREE BODY PROBLEM
There is an age-old problem in physics and astronomy that appears simple but it seems impossible to find a solution to it.
Unlike our sun most stars exist as dual or multiple systems, all in orbit around their mutual center of gravity. If two stars are in a mutual orbit the less-massive one will be further out, in a higher-energy orbit. Gravity is a simple force and this is a rather simple arrangement.
But surprisingly if a third star should join the system things start to get complicated. While it is easy to model the dual system it becomes extremely difficult when the third star joins. That is why it is called the Three Body Problem.
But, if gravity is such a simple force, why should it get so much more complicated when a third body is added? There is an answer and we have seen it before on this blog.
Gravity and light both operate by the Inverse Square Law, but not in the same way. The Inverse Square Law states that gravity is directly proportional to the mass but inversely proportional to the square of the distance. This means that an object at twice the distance will exert one-fourth of the gravitational force, because four is the square of two, and at three times the distance will exert one-ninth of the gravitational force, because nine is the square of three.
Light operates by exactly the same rule. The intensity of light from an object, as well as it's angular diameter, decreases with distance proportional to the square of the distance.
But why would light and gravity both operate by the Inverse Square Law, but not in the same way? When a third, or more, objects are added light remains simple but gravity gets complicated.
There is a simple reason why. Gravity is cumulative but light is not. We could say that light is two dimensional, because it is only reflected or radiated from the surface of an object, while gravity is three-dimensional, because it is exerted by the full depth of an object.
This is why gravity is so simple when there are two stars or objects but so complex when a third is added. Gravity is cumulative.
Suppose that we are in orbit around Star A. Further away is Star B, which is in a mutual orbit around their common center of gravity with Star A. We are in the gravitational domain of Star A. But it is complicated because we are also in the gravitational domain of Star B, not only directly but also indirectly.
What I mean by this is that is that not only is the gravity from Star B directly acting on us but it is also acting through Star A, around which we are in orbit. That is what I mean by gravity being cumulative. When we are in the gravitational domain of an object, whether on it's surface or in it's orbit, we are effectively part of it so that we are also in the gravitational domains of the other objects that share a gravitational domain with the object.
This was first thought of with the earth, the moon and, the sun. The moon and the sun are about the same angular diameter in the sky. The moon is much more dense than the sun. So if gravity operated in the same way as light then the moon should have a greater gravitational effect on earth than the moon. But yet that is not the case.
It is true that the moon has a greater effect on the earth's ocean tides than the sun. But that is only because the sun is four hundred times as distant as the moon. It is not due to mass. Being much closer there is more proportional difference in the moon's gravitational effect at the surface of the ocean, in comparison with the bottom, so that the sun's tidal effect on the earth's oceans is only about 40% that of the moon.
When we consider just Star A and Star B, in a mutual orbit around their common center of gravity, we only have to consider their gravitational influence on each other. But when a third star, Star C, joins the gravitational system now we have to deal with all of the following:
The direct gravitational relationship between Star A and Star B.
The direct gravitational relationship between Star A and Star C.
The direct gravitational relationship between Star B and Star C.
The indirect gravitational relationship between Star A and Star B through Star C. This is because, since Star B is in Star C's gravitational domain, there is not only a direct gravitational attraction between Star A and Star B, but also an indirect attraction through Star C.
The indirect gravitational relationship between Star A and Star C through Star B.
The indirect gravitational relationship between Star B and Star C through Star A.
So by adding a third star to the gravitational system we have increased the number of factors that we need to consider from one to six.
But that is not all. It gets more complicated than that. With a two star system the distance between the two stars tends to remain constant. But with a third star added the distances between the three stars may be constantly in flux, which means that all six gravitational relationships are constantly changing because gravity, according to the Inverse Square Law, is inversely proportional to the square of the distance.
That is why, even though gravity is a simple force, the Three Body Problem is virtually unsolvable because we have to remember that gravity not only operates in a direct straight line but also indirectly through a third gravitational object that is part of the system.
Not to mention if we add a fourth or a fifth object.
3) GALILEO'S PARADOX AND NEWTONI notice that there is a simple solution to the thought puzzle known as Galileo's Paradox, involving the Inverse Square Law. Galileo pointed out that "Every number must have a perfect square, yet few numbers are perfect squares". This simple observation proves that numbers must be infinite since this is clearly true, but cannot be true of any finite set of numbers. If every number in a finite set of numbers had a perfect square, then every number would also have to be a perfect square or else there would not be enough perfect squares to go around. Galileo's observation can only be true if numbers continue to infinity.
A perfect square is simply a number multiplied by itself. The perfect square of 2 is 4, and of 3 is 9. Most square roots are not perfect, meaning that they have a remainder. The square root of 3, for example, is 1.414. There is an article about Galileo's Paradox on www.wikipedia.org .
What I would like to announce today is that Galileo's Paradox is actually solved by Sir Isaac Newton's Law of Inverse Squares. There is a popular story that Newton was born on the day Galileo died, or at least in the same year. Apparently, this is due to confusion caused by the fact that there was still two calendars in use at the time, the Gregorian Calendar and the Julian Calendar.
Newton's Law of Inverse Squares is based on the fact that a circle, or any geometric shape, with twice the diameter will have four times the area. This means that if a light is twice as far away, it will be only 1/4 as bright because at twice the distance it's light must be spread over four times the area. Likewise, an object at twice the distance will appear as only 1/4 the size.
The way that this relates to Galileo's perfect squares is that all of the numbers that are less than a given number must have their perfect squares within the perfect square of the given number. 100 is the perfect square of 10 so that all of the numbers from 1-10 have their perfect squares within 100. Every number contains the perfect squares of it's square root, and numbers lower than it.
But this is the same as stating, in Newton's Inverse Square Law, that a light at ten times the distance will be only 1/100 as bright. Galileo's Paradox is referring to numbers, while Newton's Law is referring to space, but both are essentially the same concept.
The thing that has not been pointed out in Galileo's Paradox is that, as we move higher in numbers, perfect squares become more sparse as described by Newton's Inverse Square Law. Consider that 36 doubled equals 72. There are six perfect squares within 36, those of the numbers 1-6. So, if what I am claiming here is correct, there should be the square root of 6 perfect squares between 37 and 72.
Indeed there is. The square root of 6 is 2.45. There are two square roots between 37 and 72, which are 49 and 64, and 72 is nearly half way from 64 to the next square root beyond 72, which is 81. Put another way, the square root of 6 is 2.45 and there are two perfect squares from 37-72 as well as .45 of the distance from the second of these perfect squares, 64, to the next perfect square, 81.
We live in a spatial universe and I find this Inverse Square Law to be extremely useful.
Should Newton be given belated credit for solving Galileo's Paradox, or should Galileo be given credit for at least partially foreseeing Newton's Inverse Square Law? That depends on one's perspective, as well as possibly on the reader's nationality or ancestry. But we can safely conclude that the puzzle known as Galileo's Paradox is solved by Newton's Law of Inverse Squares.
4) GRANULARITY AND THE INVERSE SQUARE LAW
In the posting "Granularity", on the progress blog, I explained my version of presuming that we are dealing with something infinite, infinitesimal or, homogenous when, in fact, we are dealing with something finite or composed of discreet units, invites error in calculations. The Inverse Square Law is intended for dealing with that which is either infinite or infinitesimal, and not with that which is finite or composed of discrete units. But we can get around this, and make the Inverse Square law even more useful than it is, if we take granularity into account.
One of the simplest examples of my concept of granularity is that of gloves in a drawer. Suppose that you have ten gloves in a drawer, five left and five right and well mixed. Now, suppose that you reach in without looking and pull out two gloves. What are the odds that you have a matching left-right pair?
Your first reaction might be that the odds of having a left-right pair are fifty percent. Yet, this is not correct. When you take the first glove, it leaves nine as a choice for the second glove. Five would make a match with the first glove, and four would not. Therefore the odds of picking out a matching pair are 5/9.
The odds are greater than 50% because we are dealing with a finite entity. It is only if there was an infinite number of gloves in the drawer that the odds would be 50% of pulling out a matching pair.
Another example of granularity concerns the difference between the center of mass and the center of gravity of a planet. Textbook illustrations tend to consider the center of mass and the center of gravity as one and the same. Yet, this cannot be correct.
Suppose that we are in a spacecraft a certain distance from a planet. The center of mass of the planet will always be the same. But the near side of the planet is closer to us than the far side. This means that, in accordance with the Inverse Square Law, it has a greater gravitational influence than the far side of the planet. The result is that the center of mass of a planet is fixed, while the center of gravity varies with distance from the planet. It is only when we are at an infinite distance from the planet that the center of mass and the center of gravity of the planet would be one and the same.
The reason that the center of mass and the center of gravity cannot be the same is that we are dealing with a finite quantity, rather than the infinite or the infinitesimal. The finite quantity that we are dealing with is the proportion of our distance from the planet relative to the diameter of the planet, accounting for whether the mass of the planet was evenly distributed or not. This is what I mean by granularity.
Another example of granularity is the presumed equality of day and night, at least over the course of a year due to seasonal differences. Only if the sun was at an infinite distance from the earth, appearing as an infinitesimal point of light, are day and night likely to be exactly equal. In reality, if the sun were close to the earth there would be a band of the earth's surface along the circumference parallel to the direction to the sun which would be hidden from the sun by the curvature of the earth, this would make the night a little bit longer. On the other hand, the sun has a certain angular width in the sky that would make the day longer.
Once again, we must remember that we are dealing with a finite quantity so that granularity comes into play. Those quantities are the angular width of the sun, relative to the circumference of the earth and, the horizon circle on earth from the perspective of the sun due to the distance of the sun from earth. There could conceivably be an arrangement where these two factors would exactly balance out and night, and day would be exactly equal, but other than that we must take granularity into account if we want really precise figures for night and day.
Today, I would like to add another example of how we must take granularity into account. This example involves the Inverse Square Law.
The Inverse Square Law basically states that, if a light is twice as distant it will only be 1/4 as bright. If the light is three times as distant, it will only be 1/9 as bright. This is because 4 is the square of 2 and 9 is the square of 3. The Inverse Square Law also applies to surface area, gravity, apparent angular size of an object and, strength of electromagnetic radiation or sound.
If we apply the Inverse Square Law to surface area, for example, a square or circle with twice the diameter will have four times the area. If we wanted a square or circle with twice the surface area, we would increase the diameter by the square root of two (1.414). If we wanted three times the surface area, we would increase the diameter by the square root of three (1.732).
(Note-By the way, this applies only to surface area and does not apply to volume since volume is cubed rather than squared. If we wanted a three-dimensional box or a sphere with twice the volume, we would increase the side of the cube or the radius of the sphere by the cube root of two (1.26). If we wanted a cube or sphere with three times the volume, we would increase the side of the cube or the radius of the sphere by the cube root of three (1.442)).
But notice that, while the Inverse Square Law works fine in two dimensions, it does not work in one dimension with a line of numbers. If we consider multiplication as two-dimensional, then adding is one-dimensional.
If the Inverse Square Law worked with addition then all numbers added up to a certain number should sum to four times that of all numbers added up to half that number. Yet, this is not the case. the numbers from 1-4 add up to 10. The numbers from 1-8 add up to 36, rather than 40. The numbers added up to a certain number are four times that up to half that number, except for the lower number itself.
This is simply because we are dealing with finite quantities in the numbers. With the two dimensions of space, we are dealing with a homogenous medium and the Inverse Square Law applies. But this is not the case with finite numbers, and so the Inverse Square Law does not apply. This is an example of granularity.
Another example of how granularity affects the Inverse Square Law that I would like to look at today involves operators in call centers. Suppose that you worked in a business where frequent calls were made to a call center, such as for tech support. You wanted an estimate of how many operators worked in the call center.
The way to go about getting such an estimate would be to keep track of the number of calls made to the call center until you reached an operator that you had talked to before. That would provide an estimate of how many operators worked in the call center because the more operators there were the more calls it would require, on the average, to reach an operator that you had reached previously.
The next thing to do would seem to be to square, multiply by itself, the number of calls that were made to the call center before any of the operators had been reached for the second time. But, if we calculate how many operators there should be for the number of calls required to reach one of the operators for a second time, we see that simple squaring will give us too high of an estimate, and the reason is granularity.
If there are ten operators in the call center, with an equal chance of answering our call, we will on average be able to make four calls before the odds of reaching the same operator twice rises above 1 / 2. For the initial call, the odds are certain that we will be connected to that operator for the first time, let's express this as 0. Since there will be 9 operators out of ten remaining which will make a first time operator for the second call, the odds will be 1 / 10. For the third call the odds will be 2 / 10. For the fourth call the odds will be 3 / 10 presuming, of course, that we haven't gotten the same operator on a call twice already.
These odds only apply if we set the odds before we start making the calls and apply only to getting the same operator twice, not any particular operator.
Since such odds are fractions must be added, we find that 0 + 1 / 10 + 2 / 10 + 3 / 10 = 6 / 10, which is just above 1/2. With four calls, our odds are better than even of getting an operator for a second time.
Notice that we cannot use simple squaring to estimate the number of operators in the call center. If we square the four calls that we will make, on average, before we are connected to one of the operators for the second time, we get 16 when there are only 10 operators.
Since such odds are fractions must be added, we find that 0 + 1 / 10 + 2 / 10 + 3 / 10 = 6 / 10, which is just above 1/2. With four calls, our odds are better than even of getting an operator for a second time.
Notice that we cannot use simple squaring to estimate the number of operators in the call center. If we square the four calls that we will make, on average, before we are connected to one of the operators for the second time, we get 16 when there are only 10 operators.
This is an example of granularity. What we have to do is not count the first call because that just gives us our "starting point".
The method that I have come up with for estimating the number of operators in a call center, with equal random chance of answering, is to square the average number of calls that it takes to reach an operator that has already been talked to, not including the very first call. Remember that these are the odds only before the calls have begun.
The example described above would give us an estimate of nine operators, three calls squared, when there are actually ten. This is because of granularity. We are dealing with discrete units, each operator, rather than a homogeneous medium, such as empty space. But there would be less granularity if we had a higher number of operators in the call center and this method of squaring the average number of calls that it takes until we get an operator that we have previously talked to, minus the first call, would get more accurate.
The Inverse Square Law is a wonderful thing, but to make maximum use of it we have to take this principle of granularity into account because it will affect the results if we are dealing with a finite quantity or discrete units, rather than that which is either infinite or infinitesimal.
5) THE EVEN NUMBER BIAS
Numbers are manifested in the world all around us, that is why we find mathematics to be such a useful descriptive tool. I have concluded that, if we exclude the number 1, there must be more even numbers being manifested by the world and the universe around us than odd numbers.
At the lowest scale, that of the electric charges which comprise the universe, there is no such thing as odd numbers. There are two electric charges, negative and positive, which attract one another and must always be balanced. Two is thus the fundamental even number. Because there are two electric charges which comprise space and the universe, there must be two possible directions from a given point in a given dimension of space, and two is an even number.
At a higher scale, that of atoms, the even number bias still shows, but now odd numbers can be manifested as well. Of the ten most common elements in the universe, eight have even atomic numbers, which is the number of protons in the nucleus. The most stable atomic nuclei are those with an even number of protons and an even number of neutrons. The least stable atomic nuclei are those with an odd number of protons and an odd number of neutrons. This shows that the even number bias starts with the electric charges, but then shows up to a somewhat lesser extent in higher scales as well.
In a star exploding in a supernova, and scattering it's component matter across space, or the entire Big Bang for that matter, the least information state would be an equal thrust outward from opposite sides, and this can only mean an even number of fragments being thrown outward. This is in accordance with Newton's Law of Equal and Opposite Reactions.
Symmetry is of even numbers, asymmetry is of odd numbers. The universe is overall symmetric due to the action-reaction principle and due to the charge symmetry of even numbers. This also makes exponents of two more likely to be manifested than nearby multiples, but non-exponents of two. To have an odd number would require more information. Since we live in an even-numbered universe, founded on two electric charges, this makes odd numbers a higher-energy state.
The number system that we use is useful because it is an effective representation of the universe. Addition and multiplication operations show the bias toward even numbers. If we add two even numbers or two odd numbers, we get an even number. If we add an even and an odd number, then we get an odd number. This makes the operation of addition favor odd and even numbers equally.
But multiplication clearly favors even numbers. If we multiply two even numbers, or an even and an odd number, we get an even number. It is only when we multiply two odd numbers that we get an odd number.
This illustration of arithmetical operations shows why we should exclude 1 as an odd number, and say that odd numbers begin with 3. Addition favors neither odd nor even numbers, both are equal. But addition is one-dimensional in nature, while multiplication is two-dimensional. The two-dimensional multiplication clearly shows the bias toward even numbers, in a proportion of four to one.
Do you notice a relationship here? In addition, which is one-dimensional, even and odd numbers have an equal chance of being produced. But in multiplication, which is two-dimensional, even numbers have four times the chance of being produced as odd numbers. This is yet another application of the Inverse Square Law.
The reason that this even-numbered bias is not more apparent is the way we count and measure. This bias only shows up in "natural" numbers. Our number system is an artificial tool that we use to describe the reality around us. But this conceals the even-number bias.
To see the even-number bias more clearly, we would have to use only a "natural" number and measurement system. This means doing away with our artificial units of measure like meters, grams, pounds, minutes, degrees, etc. Also artificial are any kind of addressing system such as coordinates or altitude and longitude. Decimal is our artificial creation, in a natural number system, all portions of a number must be expressed in fractions. A "natural" measurement system is where everything is expressed in ratios and proportions, and not in artificial units.
Even numbers represent space, because it is composed of the two electric charges. Odd numbers are a higher information state, requiring an input of energy to bring about, and can be said to be represented by matter. There are no odd numbers in empty space along, as long as we exclude 1. It is only by joining like electric charges together to form matter that odd numbers can be manifested, and even then even numbers must still be predominant.
Do you see the connection, with regard to my cosmology theory, between this concept of even numbers representing space, while odd numbers represent matter, and the Inverse Square Law? We see the Inverse Square Law at work in how odd and even numbers are equally likely to be produced by addition, which is one-dimensional, while even numbers are four times as likely to be produced by multiplication, which is two-dimensional. My cosmology theory has a two-dimensional sheet of space disintegrating in one of it's two dimensions into the one-dimensional strings that compose matter as we know it.
This two-dimensional sheet was scattered by the dissolution, which we perceive as the explosion of the Big Bang, over four dimensions of background space. Four is twice as many as the two dimensions of the sheet. This is why the interaction of matter and space is governed by the Inverse square Law, in the same way that multiplication favors the production of even numbers (which we saw represent space) at a rate four times that of addition, which is one-dimensional..
Remember that, in my cosmology theory, matter began with a two-dimensional sheet of space that was within, but not contiguous to, the surrounding multi-dimensional background space. This was the first asymmetry of the universe. Matter originated from this sheet, and has been the basis for odd number manifestation ever since.
Odd numbers, as introduced into the universe, are a second-tier higher-information state. If we have twice (an even number) of one quantity than another, and the two quantities are combined together, the lower quantity will represent 1/3 of the total. In my cosmology theory, odd numbers come into play when it requires three times the energy to hold like charges together into matter than there is between the usual pattern of adjacent opposite charges. But matter is second-tier after space, just as odd numbers are second tier after even numbers.
6) PROGRESS AND THE INVERSE SQUARE LAW
In my complexity theory, the rule of how knowledge and technology affect society is that we can make life physically easier, but only at the expense of making it more complex. We can never, on a large scale, make life both physically easier and simpler.
The increased complexity that comes with technology most obviously appears as the requirement to learn more. We are exchanging some of the requirement of hard physical labor for the requirement to learn how to develop, build and, use technology. This also brings about the secondary requirements to learn how the technology might affect the natural environment and to learn about the increasing numbers of diverse people that it might bring us in contact with, and how it make us dependent on their natural resources.
But, according to my complexity theory, this increased complexity of an advanced society must be manifested in physical terms, and this obeys the Inverse Square Law.
I would like to show how life actually works like a lever, which is the most fundamental of the few basic simple machines upon which all technology is based.. We use a lever to pry something apart or move a heavy object. We are exchanging distance for force. We can get more force from a lever than we could have otherwise, but it comes at the expense of moving the handle of the lever further.
Life in an advanced society operates like a lever in that the complexity that we have accepted in order to make life physically easier has to be manifested, not only in the increased requirement of learning, but also in the distance of movement, and this is what is governed by the Inverse Square Law in the same way as energy or gravity across space.
Put simply, the more complex a society, due to adaptation of technology, the more movement there must necessarily be both of people and goods. This is the manifestation of the complexity. This movement brought about by complexity is actually relative movement because the wandering involved in a nomadic way of life would not indicate technical complexity. A nomadic group would wander together so that it would be their movement relative to one another, and not their total movement, which would be the manifestation of their low level of complexity.
Notice in the posting "The Three Fundamental Costs", on the world and economics blog, that all three of the fundamental costs that compose the universal law of supply and demand revolve around distance. I define the three fundamental costs as: the price of land, the price of transportation and, the price of communication. I do not include the price of labor among the three fundamental costs because that is also composed of these three. Increased distance increases all three of the fundamental costs, but then technology works at bringing down the second and third ones.
In the economy of a simple village of centuries ago, where everything that is needed is made or grown locally and where there is not much variety, the distances that have to be traveled by people and products will be at a minimum. Such a simple economy will have few distinct job descriptions. As technology is adopted it will make life physically easier but will make more total knowledge necessary, which will mean more separate job descriptions.
Each job description in our simple economy will be either based at or performed at some node, or workplace. This simply means that cooking will be done in a kitchen, farming done on a farm, trades will be done in a specific workshop, and so on. But a more complex society will require more such nodes because there will have to be more job descriptions. Since there is usually a fixed amount of land, that will mean that the average distance that the average worker will have to travel to work will increase.
This is why the price of land is the most fundamental of the three fundamental costs, and the one of the three that usually cannot be reduced by technology. We can use less land for a given number of tasks by building multi-story buildings, but that increases the price of the land around the building because it will then have more potential for economic activity.
With the distance that each worker has to travel to work increasing, it will also mean that the average distance that the products that they make and food that they raise will also have further to travel to market. When the society advances to the point where specialized higher education is necessary, it will repeat the process of the average worker having further to travel with the average student having further to travel. Specialized education will also increase distance by prompting the move to another place for employment opportunity.
All of this is governed by how the Inverse Square Law applies to information or the complexity within some given units. If every unit has an inter-relationship with every other unit, and we double the number of units, it will mean that there are now four times the number of inter-relationships. Well, not exactly because the units are finite in number and so we must apply the principle of granularity. In terms of a progressing society, the increased number of inter-relationships translates to increased distance.
Increasing complexity means more information in how everything in our simple economy here relates to everything else. But this must always increase the total distance involved, because the increased complexity must be manifested in a physical way.
Remember the simple example of the lever, we can magnifying force to make work physically easier but only at the expense of moving the lever over a greater distance. This necessity of greater distance is brought about by the magnified complexity that we accepted, by adaptation of technology, in order to make life physically easier.
Distance traveled is what we accept in exchange for making work physically easier by way of technology. But this travel can itself be made easier by use of technology, this is what transportation technology does. But, like any technology, bringing this about necessitates more distance traveled by all of the workers in the transportation industry. This, again, is the Inverse Square Law at work. We develop technology to make life easier and more efficient, and then we develop the technology further. But the more technology, and the more information in that technology, the wider the field there is on which to make progress. This is akin to the widening sphere defined by the distance from a given point, according the the Inverse Square Law.
The process of increasing distance traveled with increasing complexity by use of technology does not, of course, work with perfect smoothness. Complexity means that more education is necessary, but the dispensing of education is somewhat repetitive and remember that repetition is not complexity. For one simple example, a set of thirty identical cars do not contain anywhere near thirty times as much information as one car because the design is repetitive.
For this reason, a society that is becoming more complex, and with more distance traveled, reaches a point due to the repetitive nature of education where more distance than is really necessary is being traveled and the opportunity arises for this to be reduced by further technology. Communication technologies, from mail and the telegraph and telephone to the internet, keeps the distances that have to be traveled in line with the true complexity of society. This allows us to exchange information without actually traveling the distance between us.
This is possible only because there is repetition in education and standardization in the way that things are done, which reduces complexity, and the communication technologies can then reduce the distances that have to be traveled, keeping these distances in line with the actual complexity, because distance traveled is a manifestation of the complexity of a technical society.
The repetition that comes with education and information in a technical society can further keep the distance that is necessary to travel in line with the true complexity of society by way of software. Where books transmit knowledge, software transmits skill. Being a form of what we could call multi-dimensional knowledge, software conveys skill by way of a structure of IF...THEN choices. This limiting of the distances traveled is, once again, to keep it in line with the complexity of a technical society that it is a reflection of because there is repetition in education and standardization in the way that things are done, and this limits complexity.
We could say that the most efficient society is one in which work is made as easy as possible by technology, harm to the natural environment and other side-effects of this technology are kept to a minimum, and the distance necessary to travel in this society is also at a minimum. But it all revolves around the Inverse Square Law just as much as light and gravity in space.
7) DO WE REALLY NEED CALCULUS?
I once took a class titled "Calculus-Based Physics". I was still learning calculus, and was more adept with spatial mathematics like geometry and trigonometry. I could not help noticing that just about anything that can be solved with calculus can also be solved without calculus. We live in a spatial universe, and the graphing used in calculus is just another way of solving spatial problems.
I found that an under-appreciated gem of basic physics is the Inverse Square Law. The Inverse Square Law states that an object that is twice as far away will appear as one-quarter the size or, if two radio antennae are broadcasting with equal strength, the signal from one twice as far away will have one-quarter the strength of the one that is closer.
I find it to be not surprising at all that both calculus and the Inverse Square Law are the products of the same mind, that of Sir Isaac Newton. Either one can be used to solve the same types of problems.
If we look at a building some distance away so that our distance from the building is equal to it's height, for example, the result is an isoceles triangle (one with two equal angles) with the observer at the point of the triangle and the height of the building forming the base of the triangle. This could also be expressed as a right triangle (one with a right angle) with the height of the building as the vertical axis of the triangle. The Inverse Square Law applies in that, if the building were twice as far away from the observer it would appear to the observer as having only a quarter it's former width, or height.
The reason for the Inverse Square Law is that the circumference of a circle is pi (3.1415927 is as many decimal places as I have it memorized) times the diameter of the circle, and the diameter is twice the radius. This means that if we double the radius, which represents the distance to the object, there are now four times the original radius in the diameter of the circle that the object lies on, with the observer at the center.
So, why can't we make use of the Inverse Square Law when dealing with anything that forms a triangle? It does not necessarily have to involve an actual spatial triangle, this law of physics can be applied to anything that forms a triangle in it's pattern of events. This opens up a whole world of possibilities.
Actually, anything which changes at a steady rate forms a triangle in it's pattern. Picture a right triangle, or a cone. If some entity begins at zero, and proceeds at a steady rate to some maximum, it can easily be expressed as a triangle. Let's replace the triangle formed by the observer looking at the building with the beginning at zero replacing the position at the point of the observer, and the maximum replacing the building.
Now, let's have a look at fractions. I find that fractions represent the way reality really operates. We count in tens, and so we prefer decimal expression. But that is an artificial numbering system and use of decimal tends to make patterns in numbers less apparent than if we used fractions.
Using a simple example of the Inverse Square Law, we can see that triangles have a very useful relationship with the squares of fractions.
Suppose that we have a right angle between two lines. The vertical line has a length of four units, and the horizontal line a length of six units. Let's draw a line from the end of the horizontal line to the top of the vertical line to form a right triangle. The triangle would have an area of twelve square units, since the triangle is half of what a square involving the two lines would be and such a square would have an area of 4 x 6 = 24 square units.
Next, let's consider the half of the horizontal line from the furthest point and moving toward the vertical line. This is the narrowest half of the triangle along the horizontal line. The vertical dimension of the triangle would be zero at the beginning point, and two units at the halfway point of the horizontal line of the triangle. This is because the vertical dimension reaches it's maximum of four units, and we have gone halfway there from the opposite point of the triangle.
The area of this narrow half of the triangle, along the horizontal axis line, would be half of six because 6 = 2 x 3. Thus, the area of the narrow horizontal half of the triangle would be three square units.
Do you see the Inverse Square Law at work? Starting at the narrow end of the triangle, we proceeded halfway toward the wide end of the triangle. In doing so, we passed one quarter of the area of the triangle because the total area of the triangle is twelve square units and the narrow half of the triangle, along the horizontal axis line, has a volume of three square units.
This means that we can do all manner of measurements of anything forming a triangle using the squares of fractions. If the narrow half of a triangle (or cone) contains one quarter of it's area or volume, it must mean that the widest half of the triangle contains 3/4 of it's area or volume. Likewise, the narrowest 1/3 of a triangle contains 1/9 of it's total area or volume. The widest 1/9 of the triangle contains 1/3 of it's total area or volume, and so on.
Now, let's move on. No one says that this very useful Inverse Square Law has to be limited to actual spatial applications. It must also apply to anything that forms a triangle in pattern, even if it does not involve an actual triangle in space. When you think about it, anything that proceeds between zero, or a minimum, and a maximum forms a triangle pattern if it were displayed on a graph.
An object in motion with a steady acceleration or deceleration forms a triangle, with the minimum at the point of the triangle and the maximum at it's widest part. Of course, if the minimum is other than zero, all we have to do is to add a rectangle beneath the triangle so that the width of the rectangle represents the value of the minimum. The most common use of calculus is to measure change, and change proceeds between a maximum and a minimum.
A falling object forms a definite triangle. The acceleration of falling due to gravity is the well-known 32 feet per second squared ( I won't convert this to metric because it is easier to express in feet). This means that if an object is dropped, it will go into the first second of fall with a velocity of zero feet per second and end the first second with a velocity of 32 feet per second, with the increase coming at a steady rate. This means that the average velocity of the object, in it's first second of fall, will be 16 feet per second. So, it will fall 16 feet in the first second.
The object enters it's second second of fall with a velocity of 32 feet per second, and ends the second with a velocity of 64 feet per second. This means that it's average velocity throughout the second second of fall was 48 feet per second. So, it fell 48 feet in the second second of fall.
In two seconds, the object has fallen 64 feet. The 16 feet that it fell in the first of the two seconds is 1/4 of 64. Can you see the triangle that is formed in this pattern, and the applicability of the Inverse Square Law?
(By the way, this 16 feet would be a very useful unit of vertical measurement because of how it relates to the velocity of falling objects. I named this unit a "grav", for gravity, and described it's use in the posting on this blog, "The Way Things Work", and in the book "The Patterns Of New Ideas").
A rising ballistic object forms a triangle in reverse to that of a falling object. Throw a ball into the air and it will form one triangle on the way up, by starting at a maximum vertical velocity and proceeding to zero as a result of the action of gravity, and then another triangle on the way down as it's velocity starts from zero, at the maximum altitude, and proceeds to a maximum.
Something like a ball rolling across the ground, with a steady deceleration, also forms a neat triangle that can readily be measured with this method.
What about a dam holding back a body of water? The pressure of the water against the dam also forms a triangle. The water pressure starts at zero at the surface of the water, and proceeds steadily to a maximum at the bottom of the water.
Anything spreading steadily along a circular front, such as an oil spill, forms the base of a cone in pattern that we can easily measure using this method. When half of the time from the beginning of the spill, if it was from an area of zero, to now had elapsed, the area covered by the spill was 1/4 of what it is now. We can also measure withdrawal at a steady rate in the same way.
Possibly the most useful application of the Inverse Square Law and the squares of fractions involves the total earnings of money which earns interest, with the interest rolled back in. This also forms a triangle, increasing at a steady rate between from minimum to maximum.
To find the sum total of any calculation, how much distance has been covered in the case of velocity, or how much money has been earned in the case of interest, just form a triangle and find the area under the triangle.
So far, we have seen how calculations can be done on anything involving change at a steady rate by using the Inverse Square Law of fundamental physics and basic fractions, with no need whatsoever to use calculus. But now, let's get a little bit more complicated.
It is easy enough to do measurements involving constant change, such as acceleration. But what if the rate of change is itself changing? For example, a graph of velocity will appear as a straight horizontal line for constant, unchanging velocity and a slanted line for constant change in velocity (acceleration or deceleration). But if the rate of acceleration was also changing, a graph of the velocity would show as a curve. The area under the curve would represent the total distance travelled. The trouble is that we do need calculus to find the area under a curve.
But a simple curve is the synthesis of two straight lines, with one of the lines changing in length, which forms the two axes of a triangle. There may be constant acceleration, or change of some kind, which would be expressed as a straight slanted line on a graph, which could be the hypotenuse of a right triangle. But there may be a change in the rate of acceleration, or a change in the change in the rate of acceleration. There may even be a change in the change of the change in the rate of change or acceleration.
To dispense with calculus, all we need to do is to arrive at triangles on our graph so that we can easily find the total distance travelled (or money earned, etc.) using ordinary geometry. No matter how complex the curve, we can find this by simply using multiple triangles and then adding their values to get a total. Of course, we would subtract the value that we get from the area under a triangle if it represented a negative value, such as deceleration, instead of positive acceleration.
Suppose that we wanted to find the total distance travelled by a moving object over a given period of time. But, the velocity of the object was contantly changing.
We would start with one triangle representing the acceleration of the object at the beginning. It would be graphed as a rectangle if it were a contant velocity, without acceleration.
If the object began to accelerate at a given point in time, we would start another triangle beginning at an axis representing the point in time at which the acceleration began.
If that acceleration rate was changing, rather than acting at a contant rate, we would set up another triangle representing that change, and continuing between the appropriate points in time represented by the common vertical axes of the triangles.
If there was a change in that rate, we would set up yet another triangle to represent it. If there was deceleration, we could set that up with the common time axis at the top, instead of the bottom of the graph, and subtract that from our final total rather than adding it.
Isn't this easier, and more enjoyable, than using calculus?
Hopefully now you have a renewed appreciation of that simple wonder of science and mathematics, known as the Inverse Square Law.
8) THE FIRST ATOMS AND THE INVERSE SQUARE LAW
There is a posting, "The Special Numbers" that is within "The Lowest Information Point". The Special Numbers referred to here are those described in that posting.
The first atoms are believed to have formed in space within minutes of the Big Bang. We can presume that the first atom to form was hydrogen, which is the simplest atom with just one proton and one electron. But there was enough heat energy in space to fuse many of those hydrogen atoms into three other stable atoms.
The other three atoms that formed in what is known as Big Bang Nucleo-Synthesis are:
1) An isotope of hydrogen known as deuterium. Like ordinary hydrogen, deuterium has one proton and one electron. The difference is that it also has one neutron in the nucleus. A neutron is made by crunching an electron into a proton, in the process referred to as K-capture.
2) Helium. A helium atom has two protons, two electrons and two neutrons. Neutrons are necessary in atoms with more than one proton because the neutrally-charged neutrons hold the positively-charged protons together so that they do not mutually repel due to their like electric charge. Four hydrogen atoms can be crunched together to create one helium atom and that is what the sun is doing now, turning hydrogen into helium. There is energy leftover when four hydrogen atoms are turned into one helium atom. The energy is released as radiation and that is why the sun shines.
3) Lithium is the element with three protons and four neutrons. Small amounts of it were produced following the Big Bang, but it still counts as a point of information.
It is true that actually five atoms form, because there is the stable isotope of helium with two protons and only one neutron, helium-3. But nature seems to regard that as an incomplete version of the far more common helium-4, with two neutrons, and the universe operates, as we will see, as if there were four original atoms.
The Inverse Square Law describes how if a light is twice as far away, it will have only one-quarter of the brightness. It is called inverse square because we must square the proportional increase in distance and then invert it. Twice as far so that 2 x 2 = 4, which inverted becomes 1 / 4.
But the Inverse Square Law governs the relationship between radiation and distance in space, it also works with gravity. Since the first atoms that were formed from hydrogen were put together in space by radiant energy from the Big Bang, shouldn't we also expect to find that the Inverse Square law applies to the formation of the first atoms in the universe?
We see that the maximum number of electrons that can be held in an atomic orbital, moving outward from the nucleus and beginning with 2, is governed by the inverse of the Inverse Square Law, so I suppose that we can just call it the Square Law. We begin with 2 because electrons usually exist in pairs, with opposite spins that balance out.
The first electron shell, denoted as N = 1, can hold a maximum of 2 electrons.
The second electron shell, in any atom, can hold a maximum of 8 electrons.
The third electron shell can hold a maximum of 18 electrons.
The fourth electron shell can hold a maximum of 32 electrons.
From there, the maximum remains at 32 for all electron shells. The maximum number of electrons in the outermost shell of any atom is always 8.
Can you see how the "Square Law" applies?
First Shell- 2 x (1 x 1) = 2
Second Shell- 2 x (2 x 2) = 8
Third Shell- 2 x (3 x 3) = 18
Fourth Shell- 2 x (4 x 4) = 32
Notice something interesting here. The ordinary fusion process in stars goes only as far as iron, which has 26 each protons and electrons. The first three electron shells add up to 26 but since the outermost shell can never have more than 8 electrons, which is why there are 8 columns in the Periodic Table of the Elements, iron requires four electron shells.
Elements heavier than iron, up to uranium with 92 each protons and electrons, are created only during the brief time that a large star is actually exploding in a supernova. After the fourth electron shell, the maximum number of electrons in a shell remains at 32. Continuing the same progression of the Square Law, the fifth shell should have a maximum of 50 electrons. The reason that the maximum is 32 is that there is not enough quantum addresses for more and every electron in an atom must have a unique quantum address.
Could it be that the universe didn't "plan on" there being the additional elements that are created by a supernova so that the electron shells after the fourth can still have a maximum of only 32 electrons?
There are my "Special Numbers" that govern so many of the limits of the numbers of protons and neutrons in the different stages of production of new atoms. There are also the Magic Numbers that are already well-known.
A Magic Number applies to either the number of protons or the number of neutrons and confers a special stability to the atom, thus making it a preferred state. If an atom happens to have a Magic Number of both, that is even better for stability and is referred to a "Double Magic" nucleus.
The well-known Magic Numbers are 2, 8, 20, 28, 50, 82 and, 126.
Lead is one of the heaviest of elements, only ten below uranium which is the heaviest. Lead, like other elements heavier than iron, is formed only during the brief time that a supernova is actually exploding because a tremendous amount of energy is being released during that time. But that means that these heavier elements are usually exponentially less common then iron and the ones lighter than it. But lead happens to be a Double Magic element because the number of protons and the number of neutrons in the nucleus of it's common isotope are both magic, 82 protons and 126 neutrons. This is why lead is much more common than it's atomic number would indicate that it should be.
The Special Numbers that I have pointed out have two separate roots, the Nucleon Root and the Proton Root. The Nucleon Root is 4 and 14, and multiples of these numbers tend to govern the total number of nucleons in different stages of the fusion process. 4 x 14, for example, = 56. Iron, which is as far as the ordinary fusion process goes, has a total of 56 nucleons, which are protons and neutrons, in it's nucleus. The 4 and 14 come from the 4 original atoms which had a total of 14 nucleons.
The Proton Root of the Special Numbers is 2 or 3 and 11. This is because the four original atoms were all of different mass. The heavier two, lithium and helium, were much more likely than the two hydrogen isotopes to be pulled into fusion into heavier elements due to their greater mass. These heaviest two atoms of the original four had a total of 11 nucleons. The 2 comes from these two elements and the 3 for adding the heavier isotope of hydrogen, the deuterium, that was more likely than the lighter one to be pulled into fusion.
Notice that the first two magic numbers, 2 and 8, are the same as the maximum number of electrons in the first two electron shells, which operate by the Square Law. In the first four atoms, as described above, only the first two electron shells were necessary as lithium, the heaviest of these first four, has only three electrons.
My Special Numbers did not become important until stars formed and fusion of these first atoms into heavier atoms began to take place. The Special Numbers, in determining the limits of the numbers of protons and total nucleons in different stages of the fusion process, operate more by multiplication, but the Magic Numbers by addition. The reason is the way atoms were being formed. The first hydrogen was crunched together into the heavier three by collisions in space. Such a collision, with momentum from only one direction represents addition.
The difference between addition and multiplication is that addition is one-dimensional while multiplication is at least two-dimensional. When stars formed, and atoms were crunched together into larger ones, the gravitational pressure was from all around, not from a one-dimensional vector of momentum. This pressure from all around represents multiplication and is why the Special Numbers, which do not take effect until fusion to heavier atoms in stars begins, operate mainly by multiplication rather than addition.
We can see how the Magic Numbers accumulate by addition, except that we are allowed to multiply by 2 because of the information of the two particles that make up the nucleus, protons and neutrons. The Magic Numbers are 2, 8, 20, 28, 50, 82 and, 126.
(2 + 8) x 2 = 20.
20 + 8 = 28.
28 + 20 + 2 = 50
50 + 8 + 8 + 8 + 8 = 82 or 50 + 28 + (2 x 2) = 82
50 + 28 + 28 + 20 = 126
But in the maximum numbers of electrons in successive electron shells, which operate by the Square Law as we have seen above, start with the same as the Magic Numbers, 2 and 8, but then switch to additives of my Special Numbers, which begin with 4 and 14. This is because the Special Numbers did not become important until fusion of new and heavier elements in stars began. Remember that 4 and 14 are so important because there were 4 different original atoms with a total of 14 nucleons.
The maximum number of electrons in successive shells are 2, 8, 18 and 32. 2 and 8 are the same as the Magic Numbers but 18 = 14 + 4 and, adding 14 again, gives us 32. If there were enough quantum addresses available in the universe, the next electron shell would hold a maximum of 50, which we would get by adding another 14 and 4 to the 32.
Also, 22 new elements are added to the original 4 to make the 26 elements, the heaviest being iron, which concludes how far the ordinary fusion process in stars goes. Notice that 22 and 26 are additives of 4 and 14, and do not follow the course of the Magic Numbers.
In fact, all of the maximums of electrons in orbitals as well as all of the Magic Numbers are additives of the 4 and 14 that are the root of my Special Numbers.
Maximum electrons in an orbital shell, 8 = 4 + 4. 18 = 14 + 4. 32 = 14 + 14 + 4
Magic Numbers, 8 = 4 + 4. 20 = 4 x 5. 28 = 14 + 4. 50 = (14 x 3) + (4 x 2) 82 = (14 x 5) + (4 x 3) 126 = 9 x 14
What all of this tells us is that the reason 4 original atoms formed following the Big Bang, which had 14 total nucleons, is that so the atoms of matter in the universe would operate by the Square Law, which is the inverse of the Inverse Square Law, with a starting point of 2, because there would be 2 particles composing the nucleus, protons and neutrons, and electrons exist in pairs with opposite spin. The numbers produced by the successive squares, beginning with 2, are always composed of the numbers 4 and 14.
The method that I have come up with for estimating the number of operators in a call center, with equal random chance of answering, is to square the average number of calls that it takes to reach an operator that has already been talked to, not including the very first call. Remember that these are the odds only before the calls have begun.
The example described above would give us an estimate of nine operators, three calls squared, when there are actually ten. This is because of granularity. We are dealing with discrete units, each operator, rather than a homogeneous medium, such as empty space. But there would be less granularity if we had a higher number of operators in the call center and this method of squaring the average number of calls that it takes until we get an operator that we have previously talked to, minus the first call, would get more accurate.
The Inverse Square Law is a wonderful thing, but to make maximum use of it we have to take this principle of granularity into account because it will affect the results if we are dealing with a finite quantity or discrete units, rather than that which is either infinite or infinitesimal.
5) THE EVEN NUMBER BIAS
Numbers are manifested in the world all around us, that is why we find mathematics to be such a useful descriptive tool. I have concluded that, if we exclude the number 1, there must be more even numbers being manifested by the world and the universe around us than odd numbers.
At the lowest scale, that of the electric charges which comprise the universe, there is no such thing as odd numbers. There are two electric charges, negative and positive, which attract one another and must always be balanced. Two is thus the fundamental even number. Because there are two electric charges which comprise space and the universe, there must be two possible directions from a given point in a given dimension of space, and two is an even number.
At a higher scale, that of atoms, the even number bias still shows, but now odd numbers can be manifested as well. Of the ten most common elements in the universe, eight have even atomic numbers, which is the number of protons in the nucleus. The most stable atomic nuclei are those with an even number of protons and an even number of neutrons. The least stable atomic nuclei are those with an odd number of protons and an odd number of neutrons. This shows that the even number bias starts with the electric charges, but then shows up to a somewhat lesser extent in higher scales as well.
In a star exploding in a supernova, and scattering it's component matter across space, or the entire Big Bang for that matter, the least information state would be an equal thrust outward from opposite sides, and this can only mean an even number of fragments being thrown outward. This is in accordance with Newton's Law of Equal and Opposite Reactions.
Symmetry is of even numbers, asymmetry is of odd numbers. The universe is overall symmetric due to the action-reaction principle and due to the charge symmetry of even numbers. This also makes exponents of two more likely to be manifested than nearby multiples, but non-exponents of two. To have an odd number would require more information. Since we live in an even-numbered universe, founded on two electric charges, this makes odd numbers a higher-energy state.
The number system that we use is useful because it is an effective representation of the universe. Addition and multiplication operations show the bias toward even numbers. If we add two even numbers or two odd numbers, we get an even number. If we add an even and an odd number, then we get an odd number. This makes the operation of addition favor odd and even numbers equally.
But multiplication clearly favors even numbers. If we multiply two even numbers, or an even and an odd number, we get an even number. It is only when we multiply two odd numbers that we get an odd number.
This illustration of arithmetical operations shows why we should exclude 1 as an odd number, and say that odd numbers begin with 3. Addition favors neither odd nor even numbers, both are equal. But addition is one-dimensional in nature, while multiplication is two-dimensional. The two-dimensional multiplication clearly shows the bias toward even numbers, in a proportion of four to one.
Do you notice a relationship here? In addition, which is one-dimensional, even and odd numbers have an equal chance of being produced. But in multiplication, which is two-dimensional, even numbers have four times the chance of being produced as odd numbers. This is yet another application of the Inverse Square Law.
The reason that this even-numbered bias is not more apparent is the way we count and measure. This bias only shows up in "natural" numbers. Our number system is an artificial tool that we use to describe the reality around us. But this conceals the even-number bias.
To see the even-number bias more clearly, we would have to use only a "natural" number and measurement system. This means doing away with our artificial units of measure like meters, grams, pounds, minutes, degrees, etc. Also artificial are any kind of addressing system such as coordinates or altitude and longitude. Decimal is our artificial creation, in a natural number system, all portions of a number must be expressed in fractions. A "natural" measurement system is where everything is expressed in ratios and proportions, and not in artificial units.
Even numbers represent space, because it is composed of the two electric charges. Odd numbers are a higher information state, requiring an input of energy to bring about, and can be said to be represented by matter. There are no odd numbers in empty space along, as long as we exclude 1. It is only by joining like electric charges together to form matter that odd numbers can be manifested, and even then even numbers must still be predominant.
Do you see the connection, with regard to my cosmology theory, between this concept of even numbers representing space, while odd numbers represent matter, and the Inverse Square Law? We see the Inverse Square Law at work in how odd and even numbers are equally likely to be produced by addition, which is one-dimensional, while even numbers are four times as likely to be produced by multiplication, which is two-dimensional. My cosmology theory has a two-dimensional sheet of space disintegrating in one of it's two dimensions into the one-dimensional strings that compose matter as we know it.
This two-dimensional sheet was scattered by the dissolution, which we perceive as the explosion of the Big Bang, over four dimensions of background space. Four is twice as many as the two dimensions of the sheet. This is why the interaction of matter and space is governed by the Inverse square Law, in the same way that multiplication favors the production of even numbers (which we saw represent space) at a rate four times that of addition, which is one-dimensional..
Remember that, in my cosmology theory, matter began with a two-dimensional sheet of space that was within, but not contiguous to, the surrounding multi-dimensional background space. This was the first asymmetry of the universe. Matter originated from this sheet, and has been the basis for odd number manifestation ever since.
Odd numbers, as introduced into the universe, are a second-tier higher-information state. If we have twice (an even number) of one quantity than another, and the two quantities are combined together, the lower quantity will represent 1/3 of the total. In my cosmology theory, odd numbers come into play when it requires three times the energy to hold like charges together into matter than there is between the usual pattern of adjacent opposite charges. But matter is second-tier after space, just as odd numbers are second tier after even numbers.
6) PROGRESS AND THE INVERSE SQUARE LAW
In my complexity theory, the rule of how knowledge and technology affect society is that we can make life physically easier, but only at the expense of making it more complex. We can never, on a large scale, make life both physically easier and simpler.
The increased complexity that comes with technology most obviously appears as the requirement to learn more. We are exchanging some of the requirement of hard physical labor for the requirement to learn how to develop, build and, use technology. This also brings about the secondary requirements to learn how the technology might affect the natural environment and to learn about the increasing numbers of diverse people that it might bring us in contact with, and how it make us dependent on their natural resources.
But, according to my complexity theory, this increased complexity of an advanced society must be manifested in physical terms, and this obeys the Inverse Square Law.
I would like to show how life actually works like a lever, which is the most fundamental of the few basic simple machines upon which all technology is based.. We use a lever to pry something apart or move a heavy object. We are exchanging distance for force. We can get more force from a lever than we could have otherwise, but it comes at the expense of moving the handle of the lever further.
Life in an advanced society operates like a lever in that the complexity that we have accepted in order to make life physically easier has to be manifested, not only in the increased requirement of learning, but also in the distance of movement, and this is what is governed by the Inverse Square Law in the same way as energy or gravity across space.
Put simply, the more complex a society, due to adaptation of technology, the more movement there must necessarily be both of people and goods. This is the manifestation of the complexity. This movement brought about by complexity is actually relative movement because the wandering involved in a nomadic way of life would not indicate technical complexity. A nomadic group would wander together so that it would be their movement relative to one another, and not their total movement, which would be the manifestation of their low level of complexity.
Notice in the posting "The Three Fundamental Costs", on the world and economics blog, that all three of the fundamental costs that compose the universal law of supply and demand revolve around distance. I define the three fundamental costs as: the price of land, the price of transportation and, the price of communication. I do not include the price of labor among the three fundamental costs because that is also composed of these three. Increased distance increases all three of the fundamental costs, but then technology works at bringing down the second and third ones.
In the economy of a simple village of centuries ago, where everything that is needed is made or grown locally and where there is not much variety, the distances that have to be traveled by people and products will be at a minimum. Such a simple economy will have few distinct job descriptions. As technology is adopted it will make life physically easier but will make more total knowledge necessary, which will mean more separate job descriptions.
Each job description in our simple economy will be either based at or performed at some node, or workplace. This simply means that cooking will be done in a kitchen, farming done on a farm, trades will be done in a specific workshop, and so on. But a more complex society will require more such nodes because there will have to be more job descriptions. Since there is usually a fixed amount of land, that will mean that the average distance that the average worker will have to travel to work will increase.
This is why the price of land is the most fundamental of the three fundamental costs, and the one of the three that usually cannot be reduced by technology. We can use less land for a given number of tasks by building multi-story buildings, but that increases the price of the land around the building because it will then have more potential for economic activity.
With the distance that each worker has to travel to work increasing, it will also mean that the average distance that the products that they make and food that they raise will also have further to travel to market. When the society advances to the point where specialized higher education is necessary, it will repeat the process of the average worker having further to travel with the average student having further to travel. Specialized education will also increase distance by prompting the move to another place for employment opportunity.
All of this is governed by how the Inverse Square Law applies to information or the complexity within some given units. If every unit has an inter-relationship with every other unit, and we double the number of units, it will mean that there are now four times the number of inter-relationships. Well, not exactly because the units are finite in number and so we must apply the principle of granularity. In terms of a progressing society, the increased number of inter-relationships translates to increased distance.
Increasing complexity means more information in how everything in our simple economy here relates to everything else. But this must always increase the total distance involved, because the increased complexity must be manifested in a physical way.
Remember the simple example of the lever, we can magnifying force to make work physically easier but only at the expense of moving the lever over a greater distance. This necessity of greater distance is brought about by the magnified complexity that we accepted, by adaptation of technology, in order to make life physically easier.
Distance traveled is what we accept in exchange for making work physically easier by way of technology. But this travel can itself be made easier by use of technology, this is what transportation technology does. But, like any technology, bringing this about necessitates more distance traveled by all of the workers in the transportation industry. This, again, is the Inverse Square Law at work. We develop technology to make life easier and more efficient, and then we develop the technology further. But the more technology, and the more information in that technology, the wider the field there is on which to make progress. This is akin to the widening sphere defined by the distance from a given point, according the the Inverse Square Law.
The process of increasing distance traveled with increasing complexity by use of technology does not, of course, work with perfect smoothness. Complexity means that more education is necessary, but the dispensing of education is somewhat repetitive and remember that repetition is not complexity. For one simple example, a set of thirty identical cars do not contain anywhere near thirty times as much information as one car because the design is repetitive.
For this reason, a society that is becoming more complex, and with more distance traveled, reaches a point due to the repetitive nature of education where more distance than is really necessary is being traveled and the opportunity arises for this to be reduced by further technology. Communication technologies, from mail and the telegraph and telephone to the internet, keeps the distances that have to be traveled in line with the true complexity of society. This allows us to exchange information without actually traveling the distance between us.
This is possible only because there is repetition in education and standardization in the way that things are done, which reduces complexity, and the communication technologies can then reduce the distances that have to be traveled, keeping these distances in line with the actual complexity, because distance traveled is a manifestation of the complexity of a technical society.
The repetition that comes with education and information in a technical society can further keep the distance that is necessary to travel in line with the true complexity of society by way of software. Where books transmit knowledge, software transmits skill. Being a form of what we could call multi-dimensional knowledge, software conveys skill by way of a structure of IF...THEN choices. This limiting of the distances traveled is, once again, to keep it in line with the complexity of a technical society that it is a reflection of because there is repetition in education and standardization in the way that things are done, and this limits complexity.
We could say that the most efficient society is one in which work is made as easy as possible by technology, harm to the natural environment and other side-effects of this technology are kept to a minimum, and the distance necessary to travel in this society is also at a minimum. But it all revolves around the Inverse Square Law just as much as light and gravity in space.
7) DO WE REALLY NEED CALCULUS?
I once took a class titled "Calculus-Based Physics". I was still learning calculus, and was more adept with spatial mathematics like geometry and trigonometry. I could not help noticing that just about anything that can be solved with calculus can also be solved without calculus. We live in a spatial universe, and the graphing used in calculus is just another way of solving spatial problems.
I found that an under-appreciated gem of basic physics is the Inverse Square Law. The Inverse Square Law states that an object that is twice as far away will appear as one-quarter the size or, if two radio antennae are broadcasting with equal strength, the signal from one twice as far away will have one-quarter the strength of the one that is closer.
I find it to be not surprising at all that both calculus and the Inverse Square Law are the products of the same mind, that of Sir Isaac Newton. Either one can be used to solve the same types of problems.
If we look at a building some distance away so that our distance from the building is equal to it's height, for example, the result is an isoceles triangle (one with two equal angles) with the observer at the point of the triangle and the height of the building forming the base of the triangle. This could also be expressed as a right triangle (one with a right angle) with the height of the building as the vertical axis of the triangle. The Inverse Square Law applies in that, if the building were twice as far away from the observer it would appear to the observer as having only a quarter it's former width, or height.
The reason for the Inverse Square Law is that the circumference of a circle is pi (3.1415927 is as many decimal places as I have it memorized) times the diameter of the circle, and the diameter is twice the radius. This means that if we double the radius, which represents the distance to the object, there are now four times the original radius in the diameter of the circle that the object lies on, with the observer at the center.
So, why can't we make use of the Inverse Square Law when dealing with anything that forms a triangle? It does not necessarily have to involve an actual spatial triangle, this law of physics can be applied to anything that forms a triangle in it's pattern of events. This opens up a whole world of possibilities.
Actually, anything which changes at a steady rate forms a triangle in it's pattern. Picture a right triangle, or a cone. If some entity begins at zero, and proceeds at a steady rate to some maximum, it can easily be expressed as a triangle. Let's replace the triangle formed by the observer looking at the building with the beginning at zero replacing the position at the point of the observer, and the maximum replacing the building.
Now, let's have a look at fractions. I find that fractions represent the way reality really operates. We count in tens, and so we prefer decimal expression. But that is an artificial numbering system and use of decimal tends to make patterns in numbers less apparent than if we used fractions.
Using a simple example of the Inverse Square Law, we can see that triangles have a very useful relationship with the squares of fractions.
Suppose that we have a right angle between two lines. The vertical line has a length of four units, and the horizontal line a length of six units. Let's draw a line from the end of the horizontal line to the top of the vertical line to form a right triangle. The triangle would have an area of twelve square units, since the triangle is half of what a square involving the two lines would be and such a square would have an area of 4 x 6 = 24 square units.
Next, let's consider the half of the horizontal line from the furthest point and moving toward the vertical line. This is the narrowest half of the triangle along the horizontal line. The vertical dimension of the triangle would be zero at the beginning point, and two units at the halfway point of the horizontal line of the triangle. This is because the vertical dimension reaches it's maximum of four units, and we have gone halfway there from the opposite point of the triangle.
The area of this narrow half of the triangle, along the horizontal axis line, would be half of six because 6 = 2 x 3. Thus, the area of the narrow horizontal half of the triangle would be three square units.
Do you see the Inverse Square Law at work? Starting at the narrow end of the triangle, we proceeded halfway toward the wide end of the triangle. In doing so, we passed one quarter of the area of the triangle because the total area of the triangle is twelve square units and the narrow half of the triangle, along the horizontal axis line, has a volume of three square units.
This means that we can do all manner of measurements of anything forming a triangle using the squares of fractions. If the narrow half of a triangle (or cone) contains one quarter of it's area or volume, it must mean that the widest half of the triangle contains 3/4 of it's area or volume. Likewise, the narrowest 1/3 of a triangle contains 1/9 of it's total area or volume. The widest 1/9 of the triangle contains 1/3 of it's total area or volume, and so on.
Now, let's move on. No one says that this very useful Inverse Square Law has to be limited to actual spatial applications. It must also apply to anything that forms a triangle in pattern, even if it does not involve an actual triangle in space. When you think about it, anything that proceeds between zero, or a minimum, and a maximum forms a triangle pattern if it were displayed on a graph.
An object in motion with a steady acceleration or deceleration forms a triangle, with the minimum at the point of the triangle and the maximum at it's widest part. Of course, if the minimum is other than zero, all we have to do is to add a rectangle beneath the triangle so that the width of the rectangle represents the value of the minimum. The most common use of calculus is to measure change, and change proceeds between a maximum and a minimum.
A falling object forms a definite triangle. The acceleration of falling due to gravity is the well-known 32 feet per second squared ( I won't convert this to metric because it is easier to express in feet). This means that if an object is dropped, it will go into the first second of fall with a velocity of zero feet per second and end the first second with a velocity of 32 feet per second, with the increase coming at a steady rate. This means that the average velocity of the object, in it's first second of fall, will be 16 feet per second. So, it will fall 16 feet in the first second.
The object enters it's second second of fall with a velocity of 32 feet per second, and ends the second with a velocity of 64 feet per second. This means that it's average velocity throughout the second second of fall was 48 feet per second. So, it fell 48 feet in the second second of fall.
In two seconds, the object has fallen 64 feet. The 16 feet that it fell in the first of the two seconds is 1/4 of 64. Can you see the triangle that is formed in this pattern, and the applicability of the Inverse Square Law?
(By the way, this 16 feet would be a very useful unit of vertical measurement because of how it relates to the velocity of falling objects. I named this unit a "grav", for gravity, and described it's use in the posting on this blog, "The Way Things Work", and in the book "The Patterns Of New Ideas").
A rising ballistic object forms a triangle in reverse to that of a falling object. Throw a ball into the air and it will form one triangle on the way up, by starting at a maximum vertical velocity and proceeding to zero as a result of the action of gravity, and then another triangle on the way down as it's velocity starts from zero, at the maximum altitude, and proceeds to a maximum.
Something like a ball rolling across the ground, with a steady deceleration, also forms a neat triangle that can readily be measured with this method.
What about a dam holding back a body of water? The pressure of the water against the dam also forms a triangle. The water pressure starts at zero at the surface of the water, and proceeds steadily to a maximum at the bottom of the water.
Anything spreading steadily along a circular front, such as an oil spill, forms the base of a cone in pattern that we can easily measure using this method. When half of the time from the beginning of the spill, if it was from an area of zero, to now had elapsed, the area covered by the spill was 1/4 of what it is now. We can also measure withdrawal at a steady rate in the same way.
Possibly the most useful application of the Inverse Square Law and the squares of fractions involves the total earnings of money which earns interest, with the interest rolled back in. This also forms a triangle, increasing at a steady rate between from minimum to maximum.
To find the sum total of any calculation, how much distance has been covered in the case of velocity, or how much money has been earned in the case of interest, just form a triangle and find the area under the triangle.
So far, we have seen how calculations can be done on anything involving change at a steady rate by using the Inverse Square Law of fundamental physics and basic fractions, with no need whatsoever to use calculus. But now, let's get a little bit more complicated.
It is easy enough to do measurements involving constant change, such as acceleration. But what if the rate of change is itself changing? For example, a graph of velocity will appear as a straight horizontal line for constant, unchanging velocity and a slanted line for constant change in velocity (acceleration or deceleration). But if the rate of acceleration was also changing, a graph of the velocity would show as a curve. The area under the curve would represent the total distance travelled. The trouble is that we do need calculus to find the area under a curve.
But a simple curve is the synthesis of two straight lines, with one of the lines changing in length, which forms the two axes of a triangle. There may be constant acceleration, or change of some kind, which would be expressed as a straight slanted line on a graph, which could be the hypotenuse of a right triangle. But there may be a change in the rate of acceleration, or a change in the change in the rate of acceleration. There may even be a change in the change of the change in the rate of change or acceleration.
To dispense with calculus, all we need to do is to arrive at triangles on our graph so that we can easily find the total distance travelled (or money earned, etc.) using ordinary geometry. No matter how complex the curve, we can find this by simply using multiple triangles and then adding their values to get a total. Of course, we would subtract the value that we get from the area under a triangle if it represented a negative value, such as deceleration, instead of positive acceleration.
Suppose that we wanted to find the total distance travelled by a moving object over a given period of time. But, the velocity of the object was contantly changing.
We would start with one triangle representing the acceleration of the object at the beginning. It would be graphed as a rectangle if it were a contant velocity, without acceleration.
If the object began to accelerate at a given point in time, we would start another triangle beginning at an axis representing the point in time at which the acceleration began.
If that acceleration rate was changing, rather than acting at a contant rate, we would set up another triangle representing that change, and continuing between the appropriate points in time represented by the common vertical axes of the triangles.
If there was a change in that rate, we would set up yet another triangle to represent it. If there was deceleration, we could set that up with the common time axis at the top, instead of the bottom of the graph, and subtract that from our final total rather than adding it.
Isn't this easier, and more enjoyable, than using calculus?
Hopefully now you have a renewed appreciation of that simple wonder of science and mathematics, known as the Inverse Square Law.
8) THE FIRST ATOMS AND THE INVERSE SQUARE LAW
There is a posting, "The Special Numbers" that is within "The Lowest Information Point". The Special Numbers referred to here are those described in that posting.
The first atoms are believed to have formed in space within minutes of the Big Bang. We can presume that the first atom to form was hydrogen, which is the simplest atom with just one proton and one electron. But there was enough heat energy in space to fuse many of those hydrogen atoms into three other stable atoms.
The other three atoms that formed in what is known as Big Bang Nucleo-Synthesis are:
1) An isotope of hydrogen known as deuterium. Like ordinary hydrogen, deuterium has one proton and one electron. The difference is that it also has one neutron in the nucleus. A neutron is made by crunching an electron into a proton, in the process referred to as K-capture.
2) Helium. A helium atom has two protons, two electrons and two neutrons. Neutrons are necessary in atoms with more than one proton because the neutrally-charged neutrons hold the positively-charged protons together so that they do not mutually repel due to their like electric charge. Four hydrogen atoms can be crunched together to create one helium atom and that is what the sun is doing now, turning hydrogen into helium. There is energy leftover when four hydrogen atoms are turned into one helium atom. The energy is released as radiation and that is why the sun shines.
3) Lithium is the element with three protons and four neutrons. Small amounts of it were produced following the Big Bang, but it still counts as a point of information.
It is true that actually five atoms form, because there is the stable isotope of helium with two protons and only one neutron, helium-3. But nature seems to regard that as an incomplete version of the far more common helium-4, with two neutrons, and the universe operates, as we will see, as if there were four original atoms.
The Inverse Square Law describes how if a light is twice as far away, it will have only one-quarter of the brightness. It is called inverse square because we must square the proportional increase in distance and then invert it. Twice as far so that 2 x 2 = 4, which inverted becomes 1 / 4.
But the Inverse Square Law governs the relationship between radiation and distance in space, it also works with gravity. Since the first atoms that were formed from hydrogen were put together in space by radiant energy from the Big Bang, shouldn't we also expect to find that the Inverse Square law applies to the formation of the first atoms in the universe?
We see that the maximum number of electrons that can be held in an atomic orbital, moving outward from the nucleus and beginning with 2, is governed by the inverse of the Inverse Square Law, so I suppose that we can just call it the Square Law. We begin with 2 because electrons usually exist in pairs, with opposite spins that balance out.
The first electron shell, denoted as N = 1, can hold a maximum of 2 electrons.
The second electron shell, in any atom, can hold a maximum of 8 electrons.
The third electron shell can hold a maximum of 18 electrons.
The fourth electron shell can hold a maximum of 32 electrons.
From there, the maximum remains at 32 for all electron shells. The maximum number of electrons in the outermost shell of any atom is always 8.
Can you see how the "Square Law" applies?
First Shell- 2 x (1 x 1) = 2
Second Shell- 2 x (2 x 2) = 8
Third Shell- 2 x (3 x 3) = 18
Fourth Shell- 2 x (4 x 4) = 32
Notice something interesting here. The ordinary fusion process in stars goes only as far as iron, which has 26 each protons and electrons. The first three electron shells add up to 26 but since the outermost shell can never have more than 8 electrons, which is why there are 8 columns in the Periodic Table of the Elements, iron requires four electron shells.
Elements heavier than iron, up to uranium with 92 each protons and electrons, are created only during the brief time that a large star is actually exploding in a supernova. After the fourth electron shell, the maximum number of electrons in a shell remains at 32. Continuing the same progression of the Square Law, the fifth shell should have a maximum of 50 electrons. The reason that the maximum is 32 is that there is not enough quantum addresses for more and every electron in an atom must have a unique quantum address.
Could it be that the universe didn't "plan on" there being the additional elements that are created by a supernova so that the electron shells after the fourth can still have a maximum of only 32 electrons?
There are my "Special Numbers" that govern so many of the limits of the numbers of protons and neutrons in the different stages of production of new atoms. There are also the Magic Numbers that are already well-known.
A Magic Number applies to either the number of protons or the number of neutrons and confers a special stability to the atom, thus making it a preferred state. If an atom happens to have a Magic Number of both, that is even better for stability and is referred to a "Double Magic" nucleus.
The well-known Magic Numbers are 2, 8, 20, 28, 50, 82 and, 126.
Lead is one of the heaviest of elements, only ten below uranium which is the heaviest. Lead, like other elements heavier than iron, is formed only during the brief time that a supernova is actually exploding because a tremendous amount of energy is being released during that time. But that means that these heavier elements are usually exponentially less common then iron and the ones lighter than it. But lead happens to be a Double Magic element because the number of protons and the number of neutrons in the nucleus of it's common isotope are both magic, 82 protons and 126 neutrons. This is why lead is much more common than it's atomic number would indicate that it should be.
The Special Numbers that I have pointed out have two separate roots, the Nucleon Root and the Proton Root. The Nucleon Root is 4 and 14, and multiples of these numbers tend to govern the total number of nucleons in different stages of the fusion process. 4 x 14, for example, = 56. Iron, which is as far as the ordinary fusion process goes, has a total of 56 nucleons, which are protons and neutrons, in it's nucleus. The 4 and 14 come from the 4 original atoms which had a total of 14 nucleons.
The Proton Root of the Special Numbers is 2 or 3 and 11. This is because the four original atoms were all of different mass. The heavier two, lithium and helium, were much more likely than the two hydrogen isotopes to be pulled into fusion into heavier elements due to their greater mass. These heaviest two atoms of the original four had a total of 11 nucleons. The 2 comes from these two elements and the 3 for adding the heavier isotope of hydrogen, the deuterium, that was more likely than the lighter one to be pulled into fusion.
Notice that the first two magic numbers, 2 and 8, are the same as the maximum number of electrons in the first two electron shells, which operate by the Square Law. In the first four atoms, as described above, only the first two electron shells were necessary as lithium, the heaviest of these first four, has only three electrons.
My Special Numbers did not become important until stars formed and fusion of these first atoms into heavier atoms began to take place. The Special Numbers, in determining the limits of the numbers of protons and total nucleons in different stages of the fusion process, operate more by multiplication, but the Magic Numbers by addition. The reason is the way atoms were being formed. The first hydrogen was crunched together into the heavier three by collisions in space. Such a collision, with momentum from only one direction represents addition.
The difference between addition and multiplication is that addition is one-dimensional while multiplication is at least two-dimensional. When stars formed, and atoms were crunched together into larger ones, the gravitational pressure was from all around, not from a one-dimensional vector of momentum. This pressure from all around represents multiplication and is why the Special Numbers, which do not take effect until fusion to heavier atoms in stars begins, operate mainly by multiplication rather than addition.
We can see how the Magic Numbers accumulate by addition, except that we are allowed to multiply by 2 because of the information of the two particles that make up the nucleus, protons and neutrons. The Magic Numbers are 2, 8, 20, 28, 50, 82 and, 126.
(2 + 8) x 2 = 20.
20 + 8 = 28.
28 + 20 + 2 = 50
50 + 8 + 8 + 8 + 8 = 82 or 50 + 28 + (2 x 2) = 82
50 + 28 + 28 + 20 = 126
But in the maximum numbers of electrons in successive electron shells, which operate by the Square Law as we have seen above, start with the same as the Magic Numbers, 2 and 8, but then switch to additives of my Special Numbers, which begin with 4 and 14. This is because the Special Numbers did not become important until fusion of new and heavier elements in stars began. Remember that 4 and 14 are so important because there were 4 different original atoms with a total of 14 nucleons.
The maximum number of electrons in successive shells are 2, 8, 18 and 32. 2 and 8 are the same as the Magic Numbers but 18 = 14 + 4 and, adding 14 again, gives us 32. If there were enough quantum addresses available in the universe, the next electron shell would hold a maximum of 50, which we would get by adding another 14 and 4 to the 32.
Also, 22 new elements are added to the original 4 to make the 26 elements, the heaviest being iron, which concludes how far the ordinary fusion process in stars goes. Notice that 22 and 26 are additives of 4 and 14, and do not follow the course of the Magic Numbers.
In fact, all of the maximums of electrons in orbitals as well as all of the Magic Numbers are additives of the 4 and 14 that are the root of my Special Numbers.
Maximum electrons in an orbital shell, 8 = 4 + 4. 18 = 14 + 4. 32 = 14 + 14 + 4
Magic Numbers, 8 = 4 + 4. 20 = 4 x 5. 28 = 14 + 4. 50 = (14 x 3) + (4 x 2) 82 = (14 x 5) + (4 x 3) 126 = 9 x 14
What all of this tells us is that the reason 4 original atoms formed following the Big Bang, which had 14 total nucleons, is that so the atoms of matter in the universe would operate by the Square Law, which is the inverse of the Inverse Square Law, with a starting point of 2, because there would be 2 particles composing the nucleus, protons and neutrons, and electrons exist in pairs with opposite spin. The numbers produced by the successive squares, beginning with 2, are always composed of the numbers 4 and 14.
9) THE AVERAGE DISTANCE OF A PLANET FROM THE SUN
Here is something relatively simple that I cannot see has ever been pointed out.
We know that planets revolve around the sun in ellipses, rather than circles. While a circle has one focal point, it's center, an ellipse has two, with the sun at one of the focal points. This means that the distance between the planet and the sun varies. The point in the orbit at which the planet is closest to the sun is called perihelion, and the furthest point is aphelion.
The planet moves in it's orbit fastest when it is closer to the sun, and more slowly when it is further away. One of Kepler's Laws of Planetary Motion describes it mathematically as "A line between the planet and the sun will sweep over equal areas of space in equal periods of time".
In school we learn the distances of the planets from the sun. At perihelion the earth is 91 million miles from the sun and at aphelion 95 million miles. So the distance that we learn of the earth from the sun is the average distance of 93 million miles.
Except that this is not really correct.
Sorry but when I was 15 years old neighboring Canada converted to the Metric System, and at that point I learned the Metric System as well. But the things I learned before that, including the distances of the planets from the sun, I remember in miles. You do not need a conversion of units to understand this.
It is true that the average distance of the earth's orbit from the sun is 93 million miles. But that cannot really be considered as a satisfactory answer. The earth moves fastest through it's orbit when it is closest to the sun and slowest when it is furthest from the sun. This means that the earth spends more time further from the sun and less time closer.
We can say that the average distance of the earth's orbit from the sun is 93 million miles but that is not true if we measured the distance from the earth to the sun at regular intervals over the course of the year and took an average, simply because the earth spends more time further than the average than closer.
If we measured the distance from the earth to the sun every day over the course of the year, and took an average, we would get a figure higher than the 93 million miles. This is because, while the earth rotates at a constant rate it doesn't revolve around the sun at a constant rate. It spends more time further from the sun because it is moving through it's orbit more slowly than when it is closer to the sun.
Suppose that we wanted to measure the average depth of an area of water. If we simply measured the greatest depth and the shallowest depth, and averaged the two, it would in no way give a satisfactory answer. But that is what we do when we say that the average distance from the sun is 93 million miles. To get a satisfactory answer it would be necessary to take depth measurements at regular intervals and then average them together. A greater number of measurements would give greater accuracy.
My reasoning is that, to find the true day-by-day average distance between the earth and the sun we have to use squares. This shouldn't come as a surprise because it is the Inverse Square Law that describes space so well.
We take the distance at perihelion and the distance at aphelion and square both of them. Then we average them and our answer is the square root of the average.
That gives us an answer of just over 94 million miles as the average distance between the earth and the sun if distance measurements are taken at regular intervals throughout the year, such as every day.