# 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.

## Key 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}
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\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}
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\begin{document}$$\mathbb {R}^3$$\end{document}R3.