Icosahedral Symmetry: A Review

Abstract

Abstract.This Review covers over 40 years of research on using the algebras of Quarternions E 6 , E 8 to model Elementary Particle physics. In particular the Binary Icosahedral group is isomorphic to the Exceptional Lie algebra E 8 by the MacKay correspondence. And the toric graph of E 8 in Fig.2

What Are Quarks?

“…Three namely {E 23 ,E 14 ,E 05 }, { E 13 ,E 24 ,E 05 } and{ E 12 , E 12 ,E 05 },(where Ɣ ν = E 0ν , ν =1,··· ,5 ) describe nucleons with spin E 23 ,parity E 14 and charge E 05 [4], and each member is the product of the other two. While another set{ E 14, E 25, E 05 }, { E 24 ,E 35 , E 01 } ,{ E 34 ,E 15 ,E 02 } have spin rotations on the imaginary axes in 4-, 5-, and 6-space ,but have no charge and could represent Dark Matter.…”

“…It is also a Hessian polyhedron with 3 complex coordinates i,j,k (Coxeter [2],Section 12.3, Fig.12.3 A) originally found by the same Author in 1940 [1] so we propose that the quarks and anti-quarks belong to this complex plane which accounts for their tiny masses that are only estimated.When E6 is blown down to 3 real axes x,y,z we encounter the massive baryons ,protons and neutrons described in [4]. Leptons are not composed of quarks and so all appear on the inner ring of Fig.1 Also the Big Bang took place in (CP)³ where quarks existed in imaginary space which was blown down to P³with cosmic inflation leading to nucleons.…”

Particle Knots in Toric Modular Space

“…Charge conjugation belongs to a second set of 3 synthemes with the same rotations in a 3-space but a parity 5-space in another charge space [7]. according to the relationship…”

An Exact Derivation of The Quark Coupling Constant Without QCD

“…is the lattice of E6 with quarks assigned to the vertices, based on a modelby Slansky [9],which has been shown to agree with the Standard Model in severalpapers summarised in [6]. The vertices are also labeled by 0, according to anotation adopted by Coxeter [4],Section 12.3,where 0; can assume the values0,2,3 indicating rotations through 120 and 240 degrees.…”

“…Here 27 is the number of quarks and leptons of the Standard Model(alsothe number of vertices in Fig.1 together with the 3 leptons in the center) and216 is the order of the subalgebra (su 3 ) rotation + (su 3 ) isospin + (su 3 ) color ofE6(cf. [6]). …”