Universality of Polyhedral Linkages

TL;DR

This paper extends Kempe's Universality Theorem from planar linkages to polyhedral linkages in three or more dimensions, demonstrating that complex semi-algebraic sets can be realized with higher-dimensional rigid structures.

Contribution

The paper generalizes Kempe's Universality Theorem to polyhedral linkages in higher dimensions, expanding the theoretical understanding of linkage realizations.

Findings

Polyhedral linkages can realize complex semi-algebraic sets in higher dimensions.

Kempe's Universality Theorem is successfully extended beyond planar cases.

The results provide a foundation for designing higher-dimensional mechanical linkages.

Abstract

Planar linkages are a rich area of study motivated by practical applications in engineering mechanisms. A central result is Kempe's Universality Theorem, which states that semi-algebraic sets can be realized by planar linkages. Polyhedral linkages are generalizations of planar linkages to higher dimensions, where the faces are required to be rigid. In this paper, we generalize Kempe's Universality Theorem to polyhedral linkages with an embedded construction in dimension three and above.