Squared edge lengths of regular simplices with rational vertices

TL;DR

This paper characterizes the set of rational numbers that can be squared edge lengths of regular simplices with rational vertices, revealing a stabilization phenomenon in higher dimensions.

Contribution

It provides a complete classification of rational squared edge lengths for regular simplices, linking geometric realizability to quadratic form theory.

Findings

For n - d ≥ 3, all positive rational numbers occur as squared edge lengths.

Explicit algebraic conditions govern realizability in low codimensions.

The problem reduces to the Hasse--Minkowski classification of rational quadratic forms.

Abstract

We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational number occurs, while codimensions $0$, $1$, and $2$ are governed by explicit square-class, norm-group, and Hilbert-symbol conditions. The proof reduces simplex realizability to the Hasse--Minkowski classification of rational quadratic forms.