The Larson Research Center

Scientific American recently published an article by John Baez and John Huerta on the use of octonions in string theory. Their motivation was that octonions “may explain why the universe has the number of dimensions it does,” if string theory is right.

The number of dimensions of the universe they refer to is either ten, which includes the eight dimensions associated with the LST tetraktys, one real and one, three or seven imaginary dimensions, plus two more swept out through space as a 1D string propagates, or else eleven dimensions, which includes one more, when the 2D membranes of M theory propagate through space over time.

These ten (string) or eleven (membrane) dimensions are to be understood in terms of the mathematical operations used in describing a unified picture of a physical universe that has two sectors, one sector consisting of the observed matter particles (spin 1/2), the other a mirror image of the first, but consisting of force particles (spin 1), an idea called supersymmetry in string theory.

If it weren’t for the extra dimensions that strings or membranes sweep out over time, say the authors, the interactions of force and matter particles can be described with simple multiplication within the tetraktys (thus providing a unified description of nature), but “[The evolution over time] changes the dimensions in which supersymmetry arises, by adding two—one for the string and one for time. Instead of supersymmetry in dimension one, two, four or eight [of the tetraktys], we get supersymmetry in dimension three, four, six or ten [for strings, or four, five, seven, or eleven for membranes.]”

In other words, they need to keep the mathematics confined to the dimensions of the tetraktys (of course, they don’t use the word tetraktys, but the shortcut is useful in referring to “the standard collection of one, two, four and eight dimensions.”) This is understandable, because the Bott periodicity theorem proves that there are no new phenomena beyond the dimensions of the tetraktys. Yet, instead of accepting this, they spend billions of dollars and decades of time looking for the evidence that the universe can escape the tetraktys!

But it is the eight dimensional octonions of the tetraktys that works out for strings. Using the four-dimensional quaternions, or the two-dimensional complexes, or the one-dimensional reals introduces anamolies, in which string theory breaks down. String theory and M theory (presumably) are only self-consistent and anamoly free, when the system is described using the eight dimensional octonions.

“So,” they conclude, “if string theory is right, the octonions are not a useless curiosity: on the contrary, they provide the deep reason why the universe must have 10 dimensions: in 10 dimensions, matter and force particles are embodied in the same type of numbers—the octonions.”

Regular readers of the LRC’s three blogs will probably be wondering why in the world do these people insist on complicating the algebraic picture, by counting the real and imaginary numbers as mathematical dimensions that correspond with physical dimensions? I cannot for the life of me answer that question. It is a complete mystery to me why they can’t see that the three physical dimensions of space and the one of time are embodied in the tetraktys.

It’s clear that there are two inherent “directions” of dimensions; that, instead of the numbers one, two, four and eight of the tetraktys representing physical dimensions, these numbers represent the physical “directions” of space and time, the 20 = 1, 21 = 2, 22 = 4, and 23 = 8, “directions” of the 4D tetraktys, corresponding to the point, line, area and volume of geometry.

When we construct the right lines and circles of Larson’s Cube, with its two balls (eeew, that’s hard to write!), we get a wonderful picture of the discrete and continuous structure of the physical tetraktys, which corresponds perfectly to the observed space and time of our universe. 

The only thing that remains is to set it in motion; that is, describe how it changes over time. However, it’s not the vectorial motion of the LST we should envision, which is so misleading, but rather the scalar motion of the RST, which does not add an extra two, or three, dimensions to the 4D tetraktys, thus eliminating the vexation of extra dimensions that is so perplexing to the LST community.

The idea of supersymmetry, that there are material and cosmic twins, one the inverse of the other, in all but magnitude, then falls out within the four space/time dimensions of the tetraktys, revealing an inverse tetraktys with four time/space dimensions, if you will, in which the dimensions of space and time are swapped, where time has three dimensions and space has one dimension.

It is just so simple, but don’t look for it to appear in a Scientific American article any time soon. 

Update: I should point out that the expansion/collapse of the 3D oscillation adds two “directions” to the eight “directions” of the tetraktys. If we call the eight diagonals in Larson’s Cube dimensions, which is what the LST would do, then the inward and outward “directions” of these over time would constitute two additional dimensions in that sense, I suppose.

I can see how this thinking evolved from the correlation of 1D vector motion with numbers on the number line, but when it is realized that scalar changes in space and time are legitimate instances of motion, as well, it clarifies the whole picture.