Mobile Icosapods
This solves a theoretical geometry problem for mechanical design, but it is incremental as it builds on Borel's 1904 work.
The paper tackled the problem of constructing mobile 20-pods with real coordinates, proving that Borel's construction yields all such pods and enabling examples with real coordinates.
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.