Julio Kovacs, Fang Fang, Garrett Sadler, Klee Irwin (2012)
This paper shows that when projecting an edge-transitive N-dimensional polytope onto an M-dimensional subspace of RN, the sums of the squares of the original and projected edges are in the ratio N/M.
arXiv:1210.1446v1 [math.MG] 2 Oct 2012The Sum of Squares Law
Julio Kovacs∗, Fang Fang, Garrett Sadler, and Klee Irwin†
Quantum Gravity Research, Topanga, CA, U.S.
27 September 2012
We show that when projecting an edge-transitiveN -dimensional polytope onto anM -dimensional
subspace of RN , the sums of the squares of the original and projected edges are in the ratio N/M .
Let X ⊂ RN a set of points that determines an N -dimensional polytope. Let E denote the number of
its edges, and σ the sum of the squares of the edge lengths. Let S be an M -dimensional subspace of
N , and σ′ the sum of the squares of the lengths of the projections, onto S, of the edges of X.
Let G be the group of proper symmetries of the polytope X (that is, no reflections). If G acts
transitively on the set of edges of X, then:
σ′ = σ ·
The orthogonality relations
The basic result used in our proof is the so-called orthogonality relations in the context of representations
of groups. The form of these relations that we need is the following:
Theorem 1. Let Γ : G → V ×V be an irreducible unitary representation of a finite group G. Denoting
by Γ(R)nm the matrix elements of the linear map Γ(R) with respect to an orthonormal basis of V , we
Γ(R)∗nmΓ(R)n′m′ = δnn′δmm′
where the ∗ denotes complex conjugation.
A proof of these relation can be found in standard books on representation theory, for instance [1, p.
79] or [2, p. 14]. See also theWikipedia article http://en.wikipedia.org/wiki/Schur_orthogonality_relations.
∗Corresponding author. Email: email@example.com
†Group leader. Email: firstname.lastname@example.org
Proof of the sum of squares law
The idea is apply the orthogonality relations (1) to the group G of proper symmetries of the polytope
X, considering its standard representation on the space RN (i.e., R · x = R(x)). This representation
is clearly unitary, since the elements of the group are rotations and hence orthogonal transformations.
Also, the representation i