Integral Planes and Unit-Norm Polytopes

TL;DR

This paper introduces integral planes in real composition algebras, linking them to classical and exceptional polytopes, and proves a non-existence theorem for a specific octonion order.

Contribution

It unifies the construction of root systems and polytopes via integral planes and establishes a new non-existence result for a rank-eight golden octonion order.

Findings

Recovered classical root systems and polytopes within a uniform framework.

Identified the algebraic Hopf map as a finite principal fibration.

Proved that no indecomposable rank-eight golden octonion order exists.

Abstract

We introduce and study integral planes associated with crystallographic and non-crystallographic integral systems in real composition algebras. For an integral order $\Order$ in such an algebra we define the plane $\Order^{2}$ with quadratic form $Q(x,y)=\NN(x)+\NN(y)$, the axis shell, the balanced shell, and the corresponding unit-normalised spherical polytopes. For ten crystallographic orders we recover, in one uniform construction, the orthogonal-direct-sum root systems $2A_{1}$, $A_{2}\oplus A_{2}$, $4A_{1}$, $D_{4}\oplus D_{4}$, $16A_{1}$, and $E_{8}\oplus E_{8}$ (with classical-polytope realisations including the square, the 16-cell, the 24-cell, and the Gosset polytope $4_{21}$); for two non-crystallographic orders we obtain $H_{2}\oplus H_{2}$ (decagonal tegum) and $H_{4}\oplus H_{4}$ (600-cell tegum) over $\Z[\golden]$. We prove a rank-obstruction theorem that closes, unconditionally and by a purely Coxeter-theoretic argument, the existence question for an indecomposable rank-eight golden octonion order: no such order can exist. On the balanced shell side, we identify the genuine algebraic Hopf map $\Hopfmap_{A}(a,b)=(2a\bar b,\NN(a)-\NN(b))$ and prove that its restriction to the balanced shell is a finite principal fibration of the unit loop, valid both for the associative case and for the alternative Moufang case.