Mobile Icosapods

TL;DR

This paper characterizes the complete set of mobile 20-pods, a class of mechanical devices with a maximum of 20 legs, and demonstrates the possibility of constructing such pods with all real coordinates.

Contribution

It proves that Borel's 1904 construction encompasses all mobile 20-pods and shows they can be realized with real coordinates.

Findings

Borel's construction characterizes all mobile 20-pods.

All mobile 20-pods can be realized with real coordinates.

The maximum number of legs for a mobile pod is 20.

Abstract

Pods are mechanical devices constituted of two rigid bodies, the base and the platform, connected by a number of other rigid bodies, called legs, that are anchored via spherical joints. It is possible to prove that the maximal number of legs of a mobile pod, when finite, is 20. In 1904, Borel designed a technique to construct examples of such 20-pods, but could not constrain the legs to have base and platform points with real coordinates. We show that Borel's construction yields all mobile 20-pods, and that it is possible to construct examples with all real coordinates.