With the right set of curvatures, it’s possible to make a clear object project an image that’s not visible until light shines through it. Science educator Steve Mould explains the optical and mathematical properties of these uniquely engineered lenses. It turns out the problem has a lot to do with moving the minimal amount of dirt to build a structure and was studied extensively by mathematicians who called it optimal transport. These transport theory problems have a number of solutions and applications.

A similar effect can be created with mirrors and reflected light. Rayform specializes in the technique for a wide variety of luxury and architectural items.