Abstract

Geometry is about lengths and angles, and how, when spatially varying, these connections give twist, curvature, ... to a space. Bodies that energetically resist changes in lengths and angles, that is solids, have an associated mechanics that must clearly be related somehow to the foundations of geometry.

Studying the relation is easy and I now wish I had started mechanics with this connection made from the outset. I also wish I had studied at an earlier age the machinery for describing the relevant geometry, that is differential geometry.

I will outline the connection between mechanics and geometry, with physical and mathematical examples, and show how it then leads to new frontiers in more sophisticated solids: For instance they can reversibly develop intrinsic curvature in response to light. I will have to use simple geometric ideas since I have come to the field so late!

These solids can apparently solve the map-maker’s problem of resolving planes with curved space. I will say how this unique property might be exploited.