# Spectrality of polytopes and equidecomposability by translations

## TL;DR

This paper extends the understanding of spectral polytopes by proving face-area symmetry across all dimensions and shows that such polytopes can be dissected and rearranged into a cube using translations.

## Contribution

It generalizes previous face-area symmetry results to all face dimensions and establishes that spectral polytopes are equidecomposable to cubes via translations.

## Key findings

- Face-area symmetry extends to all face dimensions
- Spectral polytopes can be dissected into smaller parts
- Rearrangement of parts forms a cube

## Abstract

Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis (2002) asserts that if $A$ is a spectral polytope, then the total area of the $(d-1)$-dimensional faces of $A$ on which the outward normal is pointing at a given direction, must coincide with the total area of those $(d-1)$-dimensional faces on which the outward normal is pointing at the opposite direction. In this paper, we prove an extension of this result to faces of all dimensions between $1$ and $d-1$. As a consequence we obtain that any spectral polytope $A$ can be dissected into a finite number of smaller polytopes, which can be rearranged using translations to form a cube.