The Larson Research Center

The eighth post in the BAUT forum follows.  These posts are a continuation of the RST thread in the “Against the Mainstream” forum of BAUT.

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(continued from previous post)

What I have done, in answering antoniseb’s question, seeking to understand the context in which Larson’s RST is applicable to physics, is to show that a new, operationally interpreted, view of the numbers in this hypercomplex structure, reveals that there is an astonishing simplicity existing in it, when it is expressed in terms of “non-rotatable” (i.e. non-vector), but nevertheless n-dimensional, scalar numbers, called “Reciprocal Numbers” (RNs).

The power of the RNs should be clearly manifest in this, because, on the one hand, Baez asserts that, referring to the perplexing structure of hypercomplex numbers where n < 3, “You can classify all these things,” but when n is equal to, or greater than, 3, then the classification breaks down, because “3-dimensional manifolds are a lot more complicated: nobody knows how to classify them; 4-dimensional manifolds are a lot more complicated: you can prove that it’s impossible to classify them - that’s called Markov’s Theorem.”

Yet on the other hand, we can show that they are easily classified, in the new system of reciprocal numbers. The first four dimensions (counting the zero dimension) are classified by the first four RNs:

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

where

1. = RN⁰
2. = RN¹
3. = RN²
4. = RN³

The next four dimensions are then:

1. 2⁴ (1/1) = 256/256
2. 2⁵ (256/512 + 256/256 + 512/256) = 1024/1024
3. 2⁶ (1024/2048 + 1024/1024 + 2048/1024) = 4096/4096
4. 2⁷ (4096/8192 + 4096/4096 + 8192/4096) = 16384/16384

where

1. = RN⁴
2. = RN⁵
3. = RN⁶
4. = RN⁷

and so on, up the ladder of dimensions, ad infinitum. We can see the structure more clearly when the ascending powers of the RNs are expanded in terms of the powers of the first RN:

1. = (1/2+1/1+2/1)⁰ = 1/1
2. = (1/2+1/1+2/1)¹ = 4/4
3. = (1/2+1/1+2/1)² = 16/16
4. = (1/2+1/1+2/1)³ = 64/64
5. = (1/2+1/1+2/1)⁴ = 256/256
6. = (1/2+1/1+2/1)⁵ = 1024/1024
7. = (1/2+1/1+2/1)⁶ = 4048/4048
8. = (1/2+1/1+2/1)⁷ = 16384/16384

The reason for the mysterious connection of Bott periodicity and the rotation groups that mathematicians work with in topology, is that, as Raul Bott proved, there are no new phenomena beyond three geometric dimensions in nature. However, as we can now see, that applies to the vectorial aspect of the structure only, it does not apply to the scalar aspect.

From the scalar point of view of n-dimensional numbers, the Bott periodicity theorem’s assertion that there is a limit at three dimensions, after which things repeat, is clearly seen when we factor out the value of 2⁴ = 256 from the RNs in each group. This shows us that each 3D group is based on powers of 256:

256⁰

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

256¹

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

256²

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

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.
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256ⁿ

1. 2⁰ (1/1) = 1/1
2. 2¹ (1/2+1/1+2/1) = 4/4
3. 2² (4/8+4/4+8/4) = 16/16
4. 2³ (16/32+16/16+32/16) = 64/64

which is why Bott’s period 8 periodicity theorem holds.

It is the lack of recognition, on the part of the physicists, of the nature of space, as the reciprocal aspect of time, in scalar motion, together with the lack of recognition, on the part of the mathematicians, of the operational interpretation of rational numbers, as integers, that is the root cause of the present confusion, both physical and mathematical.

What we have discovered is that the vectorial science is related to the scalar science; that is, vectorial science emerges from scalar science, just as vectors emerge from scalars. The key is to understand and recognize that the duality of numbers is built in, it doesn’t have to be an add on through an ad hoc invention such as imaginary numbers.

The contribution that Hestenes has made, in bringing to light the advantage of using Clifford algebras in dealing with vectors, extends beyond the utilitarian aspect, regarding the use of different approaches to vector science, and ultimately lies in the unification of vectorial and scalar mathematical concepts.

However, this is not clear until the reciprocal ideas of Larson are applied to numbers, as well as space and time, and this is accomplished through following Hestenes’ example in exploiting the idea of the operational interpretation of number. For this reason, the value of Hestenes’ work is not fully appreciated yet, because the focus has been on GA defined as a better language for vectorial physics, not as a generalization of the number concept to accommodate the magnitude concept.

This is why Baez observes:

What we have found is that advocating GA as THE approach to teaching vectorial physics probably has great merit, because it vastly clarifies the relation of duality and dimension, when these aspects of numbers are properly understood. However, the only way to see what is really going on in three dimensions, is to see that the octonions contain all the properties of 3D numbers, including the “directions” of the scalars and pseudoscalars, not just the 1D and 2D directions of vectors.

In other words, the fact that the 3D dimensions of the octonions makes for a great way to do 2D physics, is greatly eclipsed by the fact that they make it possible to do 3D, or scalar, physics!

What we want to show next is how, even though restricted to integers, nature provides us with a continuum, and how that union actually exists in an integer number system like the RSM, reflecting Kronecker’s astute observation:

Stay tuned.

Excal