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Octonions are alternative and thus ŽDŽDŽ is associative but ABC... is not.

Geometry based on projective modules.See Daniel Chan on vector bundles.

How do classifying spaces relate to my video on Sheffer polynomials as space builders?

Building up Bott periodicity is a lot like building up a basis. The first vector can be anything. But with each new basis vector you have to check that its not dependent on the other ones. Similarly with the linear complex structures, or with the generators of a Clifford algebra, or in the iterated loop space, they come in sequence.

A circle is really a line with a point of infinity. It should be understand as a map relating to a point. And that point which is involved in that map, and the circle, they have a relationship. And together that seems what a perspective is all about.

A projective line is nondirected but it can be broken down into a line going forward and a line going backward. It is the same topologically as {$S^1$} but that is the difference being captured.

Gauge theories became even more attractive when it was realized that non-abelian gauge theories reproduced a feature called asymptotic freedom. (Comparw with threesome or fivesome)

Are symmetric spaces (quotients of Lie groups) related to Lie algebras and how?

THE HOPF INVARIANT ONE PROBLEM RAFAEL M. SAAVEDRA parallizability of spheres

A fundamental theorem in K- theory which, in its simplest form, states that for any (compact) space X there exists an isomorphism between the rings K(X)⊗K(S2) and K(X×S2). More generally, if L is a complex vector bundle over X and P(L⊕1) is the projectivization of L⊕1, then the ring K(P(L⊕1)) is a K(X)- algebra with one generator [H] and a unique relation ([H]−[1])([L][H]−[1])=0, where [E] is the image of a vector bundle E in K(X) and H−1 is the Hopf fibration over P(L⊕1). This fact is equivalent to the existence of a Thom isomorphism in K- theory for complex vector bundles. Encyclopedia of Mathematics

In quantum mechanics, the Riemann sphere is known as the Bloch sphere, and the Hopf fibration describes the topological structure of a quantum mechanical two-level system or qubit. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration {$ S^{3}\hookrightarrow S^{7}\to S^{4}$} (4+3=7 sevensome relates knowledge (foursome) and consciousness (three minds))

(Mosseri & Dandoloff 2001). Moreover, the Hopf fibration is equivalent to the fiber bundle structure of the Dirac monopole.[7]

Rudolf Steiner. Human and Cosmic Thought. Lecture II.

How to parametrize the circles of Apollonius? And what does that say about the Hopf fibration?

Daniel Friedman's relationship with truth relates the first mind's truth and the second mind's truth.

Clean Up