Taking noncommutative paths of length $d$ in $n$ variables to their endpoint monomials results in a planar complex of measured simplices. The standard simplices form one column, as shown in the table. If we replace $X$ by a zero and $Y$ by a $1$, the row of measured intervals condenses the $d$ qubit objects. The secondary polytope in this row (see the green book) is the cube. For example, the interval of length $3$ maps to the square, because the triangulations of this simplex are given by the partitions of $3$, namely the partitions $(1,1,1)$, $(2,1)$, $(1,2)$ and $(3)$. Moreover, this square secondary polytope also comes from the noncommutative paths for the interval of length two, since we can draw Young diagrams around the nodes of the planar lattice of noncommutative paths. In higher rows, these standard partition maps may still be applied to the edges of a higher dimensional simplex.

The first instance of a simplex with central vertices is the tetractys. This simplex came from the $27$ length $3$ paths on a cubic lattice. The central vertex stands for the six paths in three distinct variables, namely the six permutations in $S_3$. This is the first vertex weight that is the product of two prime numbers. Volumes of pieces in a triangulation (of a simplex) are also naturally expressed in terms of prime factors, once a minimal volume is normalised to $1$. Such volumes determine coefficients for generalised entropy measures.

7 years ago

The tetractys, the Pythagorean theory of massivation: http://en.wikipedia.org/wiki/Classical_element

ReplyDeleteThis is also Lie algebra.

But the elements could not be the first cause, to that would be needed 'the endless' infinity. Chaos is the original dark void (aether, dark energy today). In ancient Greece it first meant the initial state of the universe, and, by extension, space, darkness. Metamorphoses-book describes the initial state of the Universe as a disorganised mixture of the four elements:

Rather a rude and indigested mass:

A lifeless lump, unfashion'd, and unfram'd,

Of jarring seeds; and justly Chaos nam'd.

No sun was lighted up, the world to view;

No moon did yet her blunted horns renew:

Nor yet was Earth suspended in the sky,

Nor pois'd, did on her own foundations lye:

Nor seas about the shores their arms had thrown;

But earth, and air, and water, were in one.

Thus air was void of light, and earth unstable,

And water's dark abyss unnavigable.

Plato: As I said at first, when all things were in disorder God created in each thing in relation to itself, and in all things in relation to each other, all the measures and harmonies which they could possibly receive. For in those days nothing had any proportion except by accident; nor did any of the things which now have names deserve to be named at all — as, for example, fire, water, and the rest of the elements. All these the creator first set in order, and out of them he constructed the universe.

http://en.wikipedia.org/wiki/Arche

This arche is called "eternal and ageless". How I admire the ancient philosophers. They did not know quantum physics but still talks of it :) The same with the meridians. How could they see so clearly?

This leads directly to the Platonic solids and the elemental (stoicheion) symbol is a triangular one.

There may also be alink to QCD?

http://en.wikipedia.org/wiki/Aether_theories

I just found your blog via others and I can't wait to start absorbing your perspective into my already quite cluttered mind...so much to learn/ so little time.

ReplyDeleteHi Sphere Coupler. Ah yes, my mind feels quite cluttered too. I guess one cannot expect otherwise at over 40. But then age does also have its benefits. As for time, I would not worry so much about it.

ReplyDeleteHi Ulla. It is unfortunate that so much of that age was lost, but no doubt we will slowly recover more of it. And now it seems that the tetractys is indeed a natural symbol of a deep, but nonetheless outdated, 'commutative' mindset ... and we have the freedom to enlarge this universe.

ReplyDeleteYes, your right, time will take care of itself, as I too am over 40, as for the benefits I would agree that seniority has it's perks and that comes with age, yet to be 20 with the expanse of knowledge that I now possess...anyway I will be reading more than commenting and mostly beginning with your archive until I feel that I can add something of worth and (OT)someday I hope to visit your fair Country.

ReplyDeleteI see you have been blogging since 2006, could you link me to lesson M Theory Lesson #1 and I will begin my assimilation.

Always best to start from the beginning.

I wish I could more math, but this looks so exciting.

ReplyDeleteCommutative is exactly what quantum gravity is about :) Classical gravity is also very commucative, and I have thought it can be explained as quantum mechanics too, because of that. Maybe it is the general relativity that is a bit wrong after all. What would it be without gravity? Directly special relativity?

Wait for how this unfolds :)

To be honest, SC, I don't remember when I started labeling posts M Theory, but the discussion goes back before the oldest posts at AF. Best just to browse from the beginning.

ReplyDeleteCould these 'commutative' bits also be p-adic? or other primes? The expanding follows a pattern?

ReplyDelete