On Some Non-Rigid Unit Distance Patterns

TL;DR

This paper explores geometric patterns with fixed distances on spheres and in 3D space, proving bounds on their maximum numbers.

Contribution

The paper provides sharp bounds for unit distance paths and cycles on specific spheres and examines 3-regular unit distance graphs in 3D.

Findings

Sharp bounds are proven for unit distance paths and cycles on spheres of radius 1/√2.

A related problem about 3-regular unit distance graphs in 3D space is analyzed.

Abstract

A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger, and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in 3-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1/{\sqrt{2}}$$\end{document}1/2. We also consider a similar problem about 3-regular unit distance graphs in \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {R}^3$$\end{document}R3.

Figures