Points at rational distances from the vertices of certain geometric objects

TL;DR

This paper investigates the existence and properties of points with rational distances from vertices of specific geometric objects like squares, rectangles, cubes, and tetrahedra within rational coordinate spaces.

Contribution

It explores the problem of locating points with rational distances from vertices of various geometric shapes in rational coordinate spaces, extending previous work to new shapes and dimensions.

Findings

Identifies conditions for rational points at specific distances

Provides constructions or proofs for certain geometric configurations

Highlights open problems in rational distance geometry

Abstract

We consider various problems related to finding points in $\Q^{2}$ and in $\Q^{3}$ which lie at rational distance from the vertices of some specified geometric object, for example, a square or rectangle in $\Q^{2}$, and a cube or tetrahedron in $\Q^{3}$.