This Demonstration is based on a problem proposed by Plato more than 2000 years ago in Book VII of the celebrated work, The Republic, in a dialogue called "Allegory of the Cave". Imagine a film running inside a black box with a thin slit. Outside the box there is a source of parallel rays of light—the sun, for example. Between the source and the slit there is a prismatic object that can rotate on its axis, which is orthogonal to the slit and the direction of the light.
The opaque object leaves a shadow on the film that is moving below the slit in the black box. If we know the form of the object, its rotational velocity, and the speed of the film, we are able to determine the shadow's form.
The inverse problem is: knowing the shadow, is it possible to determine the prismatic section?
The direct problem is much simpler than this inverse problem. When the prism is regular, the solution of the inverse problem can be found in the work of Bevilacqua, Brandão, and Bassanezi . The direct problem was solved in the same work for a circular prism (cylinder) and those whose sections are ellipses and regular polygons. This Demonstration shows the corresponding shadows.