Hi!....
I don't want you to think that I'm dribbling out my thought one at a time just to lead you on.
Symmetry is the tool being used to understand the universe and if there was an assumption underlying it (which could be wrong) then you can see that we would/could get wrong conclusions.
I have spent time doing that examination. Here is what I am concluding...
Packing and kissing numbers make symmetry. Symmetry arized from kissing numbers and packing.
I see this as the underlying and assumed principle. I have not found anything linking the two subjects,
I ask for your help in proving (right or wrong relevant or irrelevant). Maybe your seach will be better than mine.
The web is good but it does not have everything.
The mathematicians have probably got tons of papers on it.. (that my friend is a presumption)

Is this a good place to start an examination?
jal

The ancient Greeks (especially Euclid) kept numbers separated from magnitudes. Numbers to them were quantities, while magnitudes were geometric measures of lengths, areas, and volumes. Numbers were useful for counting magnitudes, and just as they could be used for counting books, lumber, or people, they could be used to count lengths, areas, and volumes, but they couldn’t be lengths, areas, or volumes.

The history of mathematics in physics is largely the attempt to generalize the number concept enough to merge it with the magnitude concept. The capability to algebraically manipulate geometric magnitudes as easily as numbers is intriguing.

However, while magnitudes and numbers are both quantities, magnitudes have other properties that numbers don't have. The properties of magnitudes are:

1) Quantity

2) Dimension

3) Polarity

Hence, adding dimension and polarity properties to numbers is the long-sought goal, or, to put it another way, if we can find numbers with these properties, then we can explore the magnitudes of geometry algebraically.

There is another property of geometric magnitudes that the Greeks were fond of and that modern man has rediscovered. This is the property of symmetry. The symmetry of nature can be seen everywhere and forms the core of what we consider beauty of form and harmony. Today, it is a guiding principle of mathematics and physics. Therefore, since symmetry is so powerful, let us start with numerical symmetry. This means finding the symmetrical relation of quantities initially, since that’s all we have to work with. The most obvious mathematical operation that will do this is an operation favored by the ancient Greeks, proportion; that is, equal proportions are the ultimate expression of symmetry: this can be expressed as n:n, which is different from the more familiar identity relation, where n=n. We can characterize this difference by noting that the identity relation equates the relative value of quantities, whereas the proportional relation evaluates the relative value of quantities. Thus, n=m is the same as m=n. However, n:m is the inverse of m:n, which is the simplest mathematical expression of the symmetry property obtainable.

Amazingly enough, though, there is one, and only one, case where m:n = n:m. This occurs only when m = n. Obviously, this is the simplest and oldest mathematical relation known to man. The ancients used it in the form of a scale, or a pan balance, to measure the relative proportions of trade goods. When the weight of goods on one side equals the weight of goods on the other side, the scale is balanced. If the weight of one side is more than the weight of the other, the difference is the same regardless of which side of the balance the goods are placed. Thus, we can see how the beautiful principle of symmetry relates the relative values of two quantities and, in effect, measures them.

Now, if the quantities we want to evaluate are constantly changing, then the principle of symmetry evaluates the rate of change, rather than the number or weight of things. On this basis, equal rates of change are balanced. The numerical expression of this is: n:m = m:n = 1/1 = 1, where n and m are the change rates of two, reciprocally, related quantities. In other words, in this case, the number 1, instead of representing a quantity of one, actually represents the equality of the change rates of two dynamic quantities in equilibrium. It is a mathematical expression of the balanced, or symmetrical, condition of a dynamic system.

Now, since we want numbers that can express the quantity, dimensions, and polarity of geometric magnitudes, we should be very impressed with a number capable of expressing a dynamic symmetry, because geometric magnitudes can only be measured dynamically; that is, we have to change something to measure length, area, or volume. For instance, one way to measure length, is to move a measuring device of known length until it is parallel and coincident to the length we want to measure.

So, what can we “move” in our symmetrical, dynamic, number, n:m = m:n = 1/1 = 1, to measure length magnitude? Well, obviously, the answer is either m or n, since these are the only two rates in our number. Ok, then, let's change m. Let's double it. We get:

n:m = 1/2.

If we change n instead, we get:

n:m = 2/1,

but what does this have to do with length magnitude? Answer, everything. Think of n:m = m:n = 1/1 = 1, as a point, a balanced point. Now, the two unbalanced points n:m = 1/2 and n:m = 2/1 are unbalanced in two, opposite, “directions” from n:m = m:n = 1/1 = 1, the balance point. If we plot them on a line, we get

1/2 1/1 2/1,

where the imbalance between 1/2 and 1/1 is one unit on the line, and the imbalance between 1/1 and 2/1 is one unit on the line as well, but on the other side of 1/1. Therefore, what we have here is a numerical expression of a length magnitude; that is, this number has three properties:

1) quantity
2) dimension
3) polarity

The value of the number's quantity property is three. The value of its dimension property is one, and the value of its polarity property is two; that is, the three units of quantity (1/2, 1/1,and 2/1) are two opposed quantities measured from 1/1, like the two opposite ends of a unit length (maybe a stick or a rod), measured from its center. We can express this as a combination of integers as follows:

(1/2 + 1/1 + 2/1) = (-1 + 0 + 1),

but where it is regarded as one composite number, not a total of three separate numbers. In other words, we can think of it, as we think of complex numbers, which were invented using the "imaginary" number i and have the form:

(a + ib),

which is one composite number consisting of two different types of numbers that don’t sum to a total quantity of one type, but express the result of combining two related types of numbers. Thus, we can think of our reciprocal number as a new complex number with the form:

(aL + bM + cR),

where L, M, and R, indicate left, middle, and right respectively. Recall that these complex reciprocal numbers are numbers representing a symmetrical condition. Hence, they are numbers with three properties, only one of which is quantity. There are not just three quantities here. There are two, opposing, quantities the sum of which balance. In this type of number, the symmetrical condition can be either balanced, or unbalanced. If it is unbalanced, it can be unbalanced toward one end or the other, but not both. In the case of (1/2 + 1/1 + 2/1), the number is balanced. Therefore, the imbalance is zero, but not the number itself! We say that the zero sum of its two polarized quantities means that it is in numerical equilibrium, not that it doesn’t exist. Thus, while the integer value of

(-1 +0 + 1),

is 0, the rational sum of

(1/2 + 1/1 + 2/1),

is 4/4 = 1/1 = 1. In other words, this composite number is a one-dimensional, balanced, number, with two opposite polarities with respect to a monopole. Since ancient balances have been replaced by more modern methods of measuring proportions, we haven’t used these types of numbers much in modern times, but in building a TOE, we are looking for numerical symmetry as a starting point and this numerical symmetry has amazing powers.

For instance, we can see how its property of symmetry just keeps on giving in the binomial/trinomial expansion, where the dimension property of a reciprocal number determines the value of the other two properties, its quantity and polarity properties.

For example, recall that the dimension of our reciprocal number above is 1 and the values of its corresponding quantity and polarity properties are 3 and 2, respectively; that is, it has three quantity terms, two of which are polarized with respect to a third, non-polarized, quantity. Thus, it is a complex number composed of two types of numbers. In other words, just as the familiar complex number is a composite of two types of numbers, a real type and an imaginary type, the complex reciprocal number is also composed of two types of numbers, a unipolar type and a bipolar type. Therefore, we can say, in general, that the value of a complex, reciprocal, number consists of the values of its two properties, quantity and polarity, which are determined by the value of its dimension property. In the 1D case, the value of one of these properties, polarity, is 2^1 = 2, and the value of the other, quantity, is 3^1 = 3.

In the case of the ordinary, quantity (scalar), or non-reciprocal, numbers that we are all familiar with, increasing the dimensions of these numbers from 0 to 3, is interpreted as a general change in the type of the number; that is, the type of number goes from real to complex, from complex to quaternion, and from quaternion to octonion, etc. Each type of number has a different set of properties and algebraic rules, called normed division algebra.

However, because the reciprocal number is a numerical expression of symmetry, the result is different. As the dimensions increase, the type of number doesn't change per se, but its two properties, quantity and polarity, change value.

With the non-reciprocal numbers, the invention of the imaginary number compensates for the natural symmetry of the reciprocal number, but in the reciprocal number the three quantities, arising out of the symmetry, is a result of natural reflection. We can place signs on the two opposite quantities and call the unbalanced term on the left negative, and the unbalanced term on the right, positive, and the balanced term in the center neutral, or one bipolar term and one unipolar term. However, scalar numbers, being quantity only, don't have this capability, so the polarity property had to be invented for them.

Thus, the way you get to the opposite quantity with scalars is you just change the sign and say you did it by multiplying it by the square of an imaginary number, i. In this way, you can make two types of numbers (positive and negative) out of one type of number (positive). It seems kind of hokey now, but it has worked for two centuries and today it is regarded as arguably the greatest leap of imagination in the history of mankind. Go figure!

Anyway, once this was done, why stop there? If you think of i^2 as a 180 degree rotation from the positive side of zero to the negative side of zero, then a rotation of i is a 90 degree rotation. So, what happens when you increase the dimension of these numbers? You increase the number of imaginary numbers! In other words, increasing the dimensions of these numbers increases the different types of numbers.

For example, increasing the dimensions of the non-reciprocal number from 0 to 1 increases the quantity of imaginary numbers from 0 to 1, creating a new type of non-reciprocal number with opposite polarity from the one with 0 polarity, but now conveniently considered as possessing positive polarity. These are the familiar complex numbers. They are a composite number with the positive, or real, type of numbers, and the negative, or imaginary, type of number. Incrementing the number of dimensions from 1 to 2 adds two more imaginary types of numbers, to the real type and the first imaginary type. In this manner, one can form complex numbers with three positive (real) and three negative (imaginary) terms.

These numbers are called quaternions. Again, the quaternions have three types of numbers, the real number type, the complex number type, with one imaginary number, and the quaternion type, with two imaginary numbers.

Finally, incrementing from 2 to 3 dimensions brings us to the octonions, but these are regarded, as a combination of two sets of quaternions, since the quaternions have all the imaginary numbers required in a three-dimensional system. This might seem complicated to explain, but we can put it all together in the first four levels of the binomial expansion known as Pascal's triangle:

0 2^0 = 1 = 1 type (1 2^0 (real))
1 2^1 = 11 = 2 types (1 2^0 (real) and 1 2^1 (complex))
2 2^2 = 121 = 3 types (1 2^0 (real), 2 2^1 (complex), 1 2^2 (quaternion))
3 2^3 = 1331 = 4 types (1 2^0 (real), 3 2^1 (complex), 3 2^2 (quaternion), 1 2^3 (octonion))

Now, clearly there is geometric information in these numbers. If you start with the 2^0 positive scalars (reals), you can regard them as geometric points that have no polarity, then comes the 2^1 complexes. Think of these as 1D lines (a line between two points). Next the quaternions are 2D planes (four lines between four points), and then the octonions are cubes (eight lines between eight points) formed from two intersecting planes (quaternions), forming the three, orthogonal, axes of a 3D volume. It's all kind of messy and unsatisfying and mysterious, but perhaps you can see why: the principle of symmetry is missing from this interpretation of numbers. The ad hoc invention of imaginary numbers enabled mathematicians to compensate for the lack of symmetry in their numbers, but, as a result, the union of number and geometric magnitude is incomplete and confused.

Ok, so let me show you the same thing now, but this time in terms of the reciprocal numbers, the numerical expression of the equilibrium stemming from the symmetry of proportions. Remember, these numbers also have two properties, quantity and polarity, the values of which are determined by the dimensional property of the number, a characteristic that emerges from the intrinsic symmetry of the reciprocal number. As the dimensions increase from 0 to 3, the value of the quantity property increases exponentially with base 3, and the value of the polarity property increases exponentially with base 2. (notice that there is 1 quantity associated with every pole of a multipole, including the 1 quantity associated with the monopole, 1/1, term).

0 2^0 = 1 = 1 polarity (balanced polarity), 3^0 = 1 quantity
1 2^1 = 11 = 2 polarities, 3^1 = 3 quantities
2 2^2 = 121 = 4 polarities, 3^2 = 9 quantities
3 2^3 = 1331 = 8 polarities, 3^3 = 27 quantities

Here, we have a binomial/trinomial expansion, as the two properties, quantity, and polarity, expand exponentially. Now, behold the magic of symmetry:

1) Line 0 is a 0D reciprocal number corresponding to a geometric point magnitude, a balanced number equivalent to the magnitude of one point, with no dimensions, and 1, one-quantity, monopole:

RN^0 = (1/1) => 2^0 = 1 = 1 => 3^0 = 1 quantity

2) Line 1 is a 1D reciprocal number corresponding to a geometric line magnitude, a balanced number equivalent to the magnitude of unit length, with one dimension and 1, one-quantity, monopole and 1, two-quantity, dipole:

RN^1 = (1/2 + 1/1 + 2/1) => 2^1 = 11 = 2 => 3^1 = 3 quantities

3) Line 2 is a 2D reciprocal number corresponding to a geometric plane magnitude, a balanced number equivalent to the magnitude of unit area, with two dimensions and 1, one-quantity, monopole, 2, two-quantity, dipoles, and 1, four-quantity, quadrapole:

RN^2 = (1/2 + 1/1 + 2/1)^2 => 2^2 = 121 = 4 => 3^2 = 9 quantities

4) Line 3 is a 3D reciprocal number corresponding to a geometric volume magnitude, a balanced number equivalent to the magnitude of unit volume, with three dimensions and 1, one-quantity, monopole, 3, two-quantity dipoles, 3, four-quantity, quadrapoles, and 1, eight-quantity, octopole:

RN^3 = (1/2 + 1/1 + 2/1)^3 => 2^3 = 1331 = 8 => 3^3 = 27 quantities

Thus, the long, elusive, goal of mathematical physics, to unify number and magnitude, is reached at last through the principle of symmetry. To fully appreciate this will take some time, but let me help you get started:

Recall that the three quantities and two polarities of the 1D reciprocal number completely define a unit line as two opposite numbers, 1/2 and 2/1, equi-distant from the center, 1/1. Now, the 2D reciprocal number must do the same for the unit plane, and the 3D reciprocal number must do it for the unit volume. If they do this, the numbers and geometric magnitudes are equivalent.

1) The RN^2 reciprocal number, corresponding to the plane unit magnitude, has four polarities (2^2 = 4), and nine associated quantities (3^2 = 9), which can be represented as a 3x3 matrix, or combination of nine quantities, each with its corresponding polarity:

|+-|+|++|
| - |0| + |
|--| - |-+|

where ‘+’ is the positive polarity of a dipole, ‘-‘ is the negative polarity of a dipole, ‘++,’ ‘--,‘ ‘+-,’ and ‘-+’ are the four polarities of a quadrapole, and 0 is the balanced, or non-polarity, of a monopole.

2) ) The RN^3 reciprocal number, corresponding to the volume unit magnitude, has eight polarities (2^3 = 8), and 27 associated quantities (3^3 = 27), which can be represented as a 3x3x3 matrix, or combination of 27 quantities, each with its corresponding polarity:

|+-|+|++|
| - |0| + |
|--| - |-+|

|+-|+|++|
| - |0| + |
|--| - |-+|

|+-|+|++|
| - |0| + |
|--| - |-+|

You have to use your imagination here a little, because I’ve separated out the three, orthogonal, dimensions of the 3x3x3 matrix for simplicity, but you should be able to see that the four, quadrapole, poles in any given plane will combine with two, orthogonal, dipole poles to form the eight poles of the octopole:

1) +++
2) ---
3) ++-
4) --+
5) -+-
6) +-+
7) +--
8) -++

The interesting and unusual feature of all these RNs is that they each contain the monopole at the center, which, of course, is the source of their symmetry, and, as such, are indispensable.

There is so much more to say about these numbers, but this is more than enough for now. The thing is, they give our TOE an enormous advantage, something undreamed of in current theories.

Excal

Excal....
It seems that you have expanded from your earlier presentation of

Your post is one of the reasons that I wanted to have this discussion on "symmetry 101". I could not find your concept when I investigated symmetry.
Maybe, there should be a new definition of symmetry?
I hope that you'll stay around to make everyone understand your points.
Have you developed your thinking to encompass what I was saying,

QUOTE

Posted: Oct 23 2005, 09:27 AM