Carlos Castro Perelman, Fang Fang, Klee Irwin (2013)Inspired by the Sum of the Squares law obtained in the paper titled “The Sum of Squares Law“ by J. Kovacs, F. Fang, G. Sadler and K. Irwin, we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.
Law of Sums of the Squares of Areas,
Volumes and Hyper Volumes of Regular
Polytopes from Clifford Algebras
Carlos Castro Perelman∗, Fang Fang, Klee Irwin
Quantum Gravity Research Group, Topanga, California 90290, USA
November 2012
Abstract
Inspired by the recent sums of the squares law obtained by [1] we derive
the law of the sums of the squares of the areas, volumes and hyper-volumes
associated with the faces, cells and hyper-cells of regular polytopes in
diverse dimensions after using Clifford algebraic methods [3].
1
Introduction
The sums of squares law derived by [1] states that the ratio of the sums of
the squares of the edge-lengths of a regular polytope, before and after their
orthogonal projection from D to lower D′-dimensions, is given by the ratio of
dimensions D′/D. Such sums of squares law was a direct consequence of the
Schur orthogonality formula and the edge-transitive property of the point group
G that allows to break the order of the group |G| into an integer multiple of the
number edges |G| = kE, and to recast any inner product of edges < eb, ea >
in the form < Rb(ea), ea >, where Rb(ea) is the linear map (represented as a
D ×D matrix) which sends edge ea into edge eb.
The Schur orthogonality relations [2] (see Appendix) among the matrix com-
ponents R
(g)
nm and associated with the real-valued D-dimensional irreducible rep-
resentation of the point group G are given by
|G|∑
g
R(g)
nm R
(g)
n′m′ = δnn′ δmm′
|G|
dim V
= δnn′ δmm′
|G|
D
(1a)
where one has dim V = D. In particular
|G|∑
g
R
(g)
n1 R
(g)
n′1 = δnn′ δ11
|G|
D
(1b)
∗perelmanc@hotmail.com
1
where one sums over the number of group elements g denoted by |G|. If one
takes the trace over the n, n′ indices but now with the restriction that n, n′
range from 1 to D′ ( since we are projecting down to the subspace S of D′-dim
), and if instead of summing over all the group elements from 1 to |G|, one sums
only over the E group elements associated with one representative of each one
of the respective E cosets gH corresponding to the (normal) subg