Expansive periodic mechanisms

TL;DR

This paper explores expansive behaviors in periodic frameworks across dimensions, extending the understanding from two-dimensional pseudo-triangulations to higher dimensions where new challenges arise.

Contribution

It introduces a theory for expansive periodic frameworks in dimensions three and above, highlighting differences from the two-dimensional case.

Findings

Expansive frameworks exist in higher dimensions with distinct properties.

The theory extends the concept of pseudo-triangulations to higher dimensions.

Challenges in higher dimensions are identified and contrasted with 2D cases.

Abstract

A one-parameter deformation of a periodic bar-and-joint framework is expansive when all distances between joints increase or stay the same. In dimension two, expansive behavior can be fully explained through our theory of periodic pseudo-triangulations. However, higher dimensions present new challenges. In this paper we study a number of periodic frameworks with expansive capabilities in dimension $d\geq 3$ and register both similarities and contrasts with the two-dimensional case.