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 or to represent imaginary
landscapes only computers can handle.
Things become yet more interesting, when you can transform your
two-dimensional dream objects in real three dimensional sculptures.
You can then handle your creations, and look at them from an infinity
of view angles.
Hypocycloids are basically two-dimensionnal curves. We can add the third dimension by moving the pen up and down while drawing it. One can then imagine two points moving on this curve, and draw a line between these two points at regular intervals. Playing with the curve and the speed of the moving points makes one explore an infinite variety of shapes.
Ring cyclides are images of tori under sphere inversion. If certain conditions are fullfilled, a torus can contain a set of tangents spheres. Since the tangency property is preserved by inversion, this set of tangent spheres find its place inside the cyclide.