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Andrius Kulikauskas

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See: Math, Divisions

Bott periodicity

Bott periodicity has to do with topological K-theory, infinite unitary group, classifying spaces.

Complex case: 2-periodicity - divisions having 4 (nežinojimas) or 2 (žinojimas) representations. Real case: 8-peridocity.

Clifford algebra periodicity

  • C0 R
  • C1 C
  • C2 H
  • C3 H + H
  • C4 H(2)
  • C5 C(4)
  • C6 R(8)
  • C7 R(8) + R(8)
  • C8 R(16)

''C_{n+8} consists of 16 x 16 matrices with entries in Cn ! For a proof you might try

2) H. Blaine Lawson, Jr. and Marie-Louise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", Springer-Verlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of n-dimensional manifolds is very similar to the topology of (n+8)-dimensional manifolds in some subtle but important ways!'' Physics of fermions.

Introduction to rotation groups Triality of octonions. More generally, it turns out that the representation theory of Spin(n) depends strongly on whether n is even or odd. When n is even (and bigger than 2), it turns out that Spin(n) has left-handed and right-handed spinor representations, each of dimension 2^{n/2 - 1}. When n is odd there is just one spinor representation. Of course, there is always the representation of Spin(n) coming from the vector representation of SO(n), which is n-dimensional. This leads to something very curious. If you are an ordinary 4-dimensional physicist you undoubtedly tend to think of spinors as "smaller" than vectors, since the spinor representations are 2-dimensional, while the vector representation is 3-dimensional. However, in general, when the dimension n of space (or spacetime) is even, the dimension of the spinor representations is 2^(n/2 - 1), while that of the vector representation is n, so after a while the spinor representation catches up with the vector representation and becomes bigger! This is a little bit curious, or at least it may seem so at first, but what's really curious is what happens exactly when the spinor representation catches up with the vector representation. That's when 2^(n/2 - 1) = n, or n = 8. The group Spin(8) has three 8-dimensional irreducible representations: the vector, left-handed spinor, and right-handed spinor representation. While they are not equivalent to each other, they are darn close; they are related by a symmetry of Spin(8) called "triality". And, to top it off, the octonions can be seen as a kind of spin-off of this triality symmetry... as one might have guessed, from all this 8-dimensional stuff. One can, in fact, describe the product of octonions in these terms. So now let's dig in a bit deeper and describe how this triality business works. For this, unfortunately, I will need to assume some vague familiarity with exterior algebras, Clifford algebras, and their relation to the spin group. But we will have a fair amount of fun getting our hands on a 24-dimensional beast called the Chevalley algebra, which contains the vector and spinor representations of Spin(8)!

Kirby,

You encouraged me to look at Clifford algebras at some point. Today I took a quick look at the Bott periodicity theorem: https://en.wikipedia.org/wiki/Bott_periodicity_theorem Which, of course, I don't understand 95% of. Bott periodicity came up in one of the videos I listend to as something fundamental.

Well, what struck me is that there is an 8-cycle called the Bott periodicity clock, also known as the Clifford algebra clock.

In Andrius Philosphy World there is an 8-cycle of the divisions of everything:

  • into 0 perspectives (God)
  • into 1 perspective (Everything)
  • into 2 perspectives (like free will and fate, for Being)
  • into 3 perspectives (taking a stand, falling through, reflecting, for Participating)
  • into 4 perspectives (whether, what, how, why) for Knowledge
  • into 5 perspectives (for perspectives in time (alternatively, space))
  • into 6 perspectives (for Morality)
  • into 7 perspectives (for a closed Logical system with some Slack inside)

An 8th perspective would be "all is good and all is bad" which means that the system is empty, we have a contradiction, and we go back to 0 perspectives.

Now on this 8-cycle there are 3 shifts. They are for reflections that add +1 perspective or +2 perspectives or +3 perspectives. The latter +3 is consciousness. For example, 2+3 = 5 means that consciousness (+3) of issues of Being (2) is given by issues of Time/Space (5). Basically, it says that your mind is like a trolley that moves from one trolley stop to the next. We use trolley stop 5 to describe your consciousness (+3) of trolley stop 2. I've spent my whole life mapping this out but this is just a quick prompt for your imagination.

Well, lo and behold, there are "clock-shift" operators in the Generalized Clifford Algebra: https://en.wikipedia.org/wiki/Generalized_Clifford_algebra Some matrices describe the 8 cycle clock (the trolley stops) and other matrices describe the 3 shifts (the trolley cars of different increments +1, +2, +3). These are generalized Pauli matrices: https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices

Weird.

Just to add more of what I'm looking for. There are 4 representations of the first four divisions (trolley stops) and 2 representations of the next four divisions. For example, on questions of Knowledge, we have to choose between thinking in terms of questions (Whether? What? How? Why?) like an idealist, with Whether? being the uninteresting ground floor, or in terms of answers (Whether! What! How! Why!) like a materialist, with Why being uninteresting. Thus there are 4+2 = 6 ways of looking at the whole of the division. Also, there are 12 "topologies", ways of snatching out a single perspective.

So these 8 divisions, 6 representations, 12 topologies are the fundamental "tables of perspectives" that describe the mind statically. Well, by coincidence, a cube has 8 corners, 6 faces and 12 edges. In fact, 24 / 2 = 3 x 4 = 12 24 / 3 = 4 x 2 = 8 24 / 4 = 2 x 3 = 6 There also seem to be 3 dynamic languages:

  • argumentation taking us from 12 topologies to 6 representations
  • verbalization taking us from 8 divisions to 12 topologies
  • narration taking us from 6 representations to 8 divisions

These 3 static tables and 3 dynamic languages are "Third-Person (He/She)" structures. There are also 4 "Second-Person (You)" structures of 8 perspectives each. It seems that pairs of the latter 4 generate the former 6. Together they are like the 10 commandments: 4 for loving God, 6 for not hurting your neighbor.

Each of the 4 "You" structures links a null perspective (God) followed by a backbone of six perspectives and ending with a seventh perspective (-1 = slack = good). If we say it is the same God and the same good then we have 6x4 = 24 + 1 + 1 = 26. So we have the 6x4 = 24 + 2 dimensions which seems to come up in String theory and the monster group.

The 8 fold You structure and the 10=4+6 He/She structures and a 3-cycle and three states 1+1+1 together combine to give the 24 ways of figuring things out that I had said I think are at work in each "world".

What I have been working most intensely on is the key to all of this structure. I call it "God's dance". I start by imagining how God gets going. God asks, is God necessary? Would there be God if there was no God? So God removes himself, but being God, has to reappear. But is the first God (who understands) the same God (who comes to understand)? Yes because they understand the same God. So these 3 angles on God are the same God. But that's how it looks like to God #1 when God is "I" and God loves himself. For God #2, God is "You". This is the case that we find ourselves, the most unfavorable circumstances for us to be God. God #2 and God #1 are one through their perspectives on each other, and they love each other. This takes place through and 8 fold You structure. Then God #3 sees this same but now from the side, objectively, so that God is third-person "He/She". Then God #1 and #2 are the same through their non-existence, that is, their circumstances, which are one. And they love all, either by loving God (4 ways) or loving neighbor (6 ways). Finally these three unities (of God, of Person, of People) are united by a human three-cycle of taking a stand, following through, reflecting. So all together in God's dance we have 3 + 8 + 10 + 3 = 24 expressions of God in my imagination.

Mathematically, it is a state of contradiction (God) relating with itself so as to produce a state of non-contradiction, a system of truth (goodness).

So you can imagine that the same kinds of small numbers that are so highly restrictive of advanced mathematics 2, 3, 4 are similarly the same kinds of small numbers that are highly restrictive of my imagination as well, and I think, of all imaginations. You can see why I'm optimistic that at the bottom they are rooted in the same. But I think I have the more powerful way to approach it because I'm working on a model of human life. My work is defined implicitly (with reference to the imagination) whereas math is nowadays supposed to be completely explicit (written down - ignoring the role of imagination in interpreting it).

But the Clifford algebra clock is a coincidence of higher order. Also, it points to the importance of very particular structures, some of them the Lie groups (unitary, orthogonal, symplectic) and others like the quaternions.

So it will be exciting to choose this as a central point in Math and show how it is a hub for whatever. And it's that much more motivation to learn it and try to link it to my own model of human life.

BottPeriodicity


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Puslapis paskutinį kartą pakeistas 2017 sausio 24 d., 00:53
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