Starfish Triply Periodic Minimal Surfaces

Cubelets

This family of triply periodic minimal surfaces has a kaleidoscopic cell of a tetragonal disphenoid with two C2 axes. This page shows "cubelets" outlined by C2 axes; cubelets fit together in a face-centered-cubic lattice to make the complete surface. This is a two-parameter family, members of which are also shown in tables of kaleidoscopic cells, cubic unit cells, and rhombic dodecahedra, labeled by their genus. The family may be parameterized by (p,q), where p is the number of holes along the cubelet edge, and q along the cubelet diagonal. Putative members of the family may not actually exist; attempts to get all the edges of the fundamental region to match up properly (called "period killing" by the cognoscenti) may leave a gap. Starfish 4-2 fails to period kill by only 0.005 (so far).

The degrees of freedom I use in the datafiles are the lengths of the boundary edges of the fundamental region; starfish p-q has p+q-1 independent degrees of freedom, and there are p+q-1 gaps to eliminate. Unfortunately, the mapping from parameter space to gap space is sometimes highly nonlinear in the neighborhood of a solution, making finding the solution (if it even exists) difficult. And it means that there may be multiple solutions, although I have not found any yet.

For Surface Evolver datafiles, get starfish.tar and see Readme.txt therein.


Starfish 2-1 genus 31 Buy	Starfish 2-2 genus 47	Starfish 2-3 genus 63	Starfish 2-4 genus 79

Starfish 3-1 genus 43	Starfish 3-2 genus 59	Starfish 3-3 genus 75	Starfish 3-4 genus 91

Starfish 4-1 genus 55	Starfish 4-2 genus 71 (fake)	Starfish 4-3 genus 87	Starfish 4-4 genus 103

Starfish 5-1 genus 67	Starfish 5-2 genus 83 (not quite)	Starfish 5-3 genus 99 (not quite)	Starfish 5-4 genus 115

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