Francesco De Comité
Manipulation of digital images, and use of ray-tracing software can help you to concretize mathematical concepts. Either for giving you an idea of how a real object will look like, or to represent imaginary landscapes only computers can handle. 3D printing has opened new opportunities to make mathematical objects real; I am just beginning to explore this new world.
Cyclid Islands fishermen were famous in Ancient Greece for their
science of weaving fishtraps and baskets. Living under the
protection of King Atreus and Queen Aerope of Mycenae, they put
all their knowledge in building a very special rocking horse for
their first son, the future king Agamemnon.
Recent mathematical studies reveal that this toy, as all other examples of Cyclid islands weavers artworks, are in fact Dupin cyclids, defined with their four families of circles.
This particular model is a part of an infinite cyclid, limited by a bounding sphere.
Dupin cyclids are images of tori under sphere inversion. Since inversion preserves circles, cyclids contain four families of circles, like tori : parallels, meridians and images of Villarceau circles. This model is build using only both families of Villarceau circles.
It is not evident, at first sight, that this object, another Dupin cyclid, is made only from circles. The inside part, when seen from a short range is quite fascinating. That is what makes 3D printing interesting : the ability of manipulating objects, and find unexpected perspectives.