I took as inspiration for this piece the following image by Laidlaw and Kocak (for Thomas Banchoff).

Foliations of the Three-Sphere overlaid atop ``Inside the Hypersphere,'' David H. Laidlaw, Huseyin Kocak; The Mathematical Tourist, Ivars Peterson, W. H. Freeman and Co., New York, 1988, p. 96.

Thomas Banchoff's conception of the HyperSphere has the structure of a simple torus, infinitely nested within itself.

I have done work animating the metamorphosis of the Costa's Three-Ended minimal surface from a simple torus.

Costa's Three-Ended minimal surface is described as a Torus with three points removed (from 3-space).

The three points which are 'removed' are flung to infinity, dragging and deforming the surface with it.

This piece proposes that the same operation can, in principle, be done to the Hypersphere, resulting in what I am calling a HyperCosta surface.

I have geometrically constructed a three-dimensional representation of a hypersphere, by nesting tori of varying diameters within themselves.

The folowing is the Hypersphere rendeered in laser micro-cavitated crystal.

It is also possible to construct a topologically-equivalent representation of a hyper-manifold in a number of discrete sheets, based upon Costa's Genus One Three-Ended minimal surface.

The following is two copies of Costa's Genus One Three-Ended minimal surface, generated in Mathematica, displaced at every point normal from the original surface, in opposite directions. The original Costa's Genus One Three-Ended minimal surface lies in the space half-way between the two sheets.

The result is a pair of Costa's minimal surfaces symmetrically intertwined with each other. It is theoretically possible to repeat this operation any number of times.

Here is the Genus 1 HyperCosta surface of two sheets replicated in Z Corp's 3D Printing by the Visualization, Media and Imaging Laboratory, Imaging Technology Group, Beckman Institute, University of Illinois at Urbana-Champaign :

Here is the same operation performed on the Costa-Hoffman-Meeks Genus 2, 3-Ended Minimal Surface (geometry by James Hoffman).