### M Theory Lesson 167

At PF, Lawrence B. Crowell taught us about the remarkable invention of non-commutative geometry by the great Hamilton, the inventor of the quaternions. But I do not refer to the quaternions themselves. Rather, as Janet Heine Barnett explains in a beautiful article on the icosian game, in Hamilton's own words:

a. reverse the edge (eg. $ST \mapsto TS$)

b. rotate (say left) around the endpoint (eg. $HG \mapsto BG$)

c. move one edge (to the right) along a pentagon (eg. $BZ \mapsto ZQ$) At least one crazy retired physicist has incorporated this calculus into a spacetime model for the leptons and quarks, in which the $E8$ lattice magically appears out of paired quaternion like (ie. octonion) operations. A triality involving three $E8$s is briefly discussed.

Actually, it was supposedly Hamilton who first considered the complex numbers algebraically as an ordered pair of reals, in a paper entitled, Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time. Hamilton's next publication was entitled, On the Propagation of Light in Vacuo. (I almost wish I was 15 again so that I had time to read more.)

I have lately been led to the conception of a new system, or rather family of systems, of non-commutative roots of unity, which are entirely distinct from the i j k of quaternions, though having some general analogy thereto.The basic icosian calculus describes moves through the vertices of a dodecahedron and is generated by three kinds of move, let us say $a$, $b$ and $c$, such that $a^{2} = 1$, $b^{3} = 1$, $c^{5} = 1$ and $c = ab$. Observe the appearance of the rules for the modular group. All these moves apply to the oriented graph and are given by

a. reverse the edge (eg. $ST \mapsto TS$)

b. rotate (say left) around the endpoint (eg. $HG \mapsto BG$)

c. move one edge (to the right) along a pentagon (eg. $BZ \mapsto ZQ$) At least one crazy retired physicist has incorporated this calculus into a spacetime model for the leptons and quarks, in which the $E8$ lattice magically appears out of paired quaternion like (ie. octonion) operations. A triality involving three $E8$s is briefly discussed.

Actually, it was supposedly Hamilton who first considered the complex numbers algebraically as an ordered pair of reals, in a paper entitled, Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time. Hamilton's next publication was entitled, On the Propagation of Light in Vacuo. (I almost wish I was 15 again so that I had time to read more.)

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