Integral Planes and Unit-Norm Polytopes

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.