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 cubelet starfish 2-2 cubelet starfish 2-3 cubelet starfish 2-4 cubelet
Starfish 2-1 genus 31 Buy Starfish 2-2 genus 47 Starfish 2-3 genus 63 Starfish 2-4 genus 79
starfish 3-1 cubelet starfish 3-2 cubelet starfish 3-3 cubelet starfish 3-4 cubelet
Starfish 3-1 genus 43 Starfish 3-2 genus 59 Starfish 3-3 genus 75 Starfish 3-4 genus 91
starfish 4-1 cubelet starfish 4-2 cubelet starfish 4-3 cubelet starfish 4-4 cubelet
Starfish 4-1 genus 55 Starfish 4-2 genus 71 (fake) Starfish 4-3 genus 87 Starfish 4-4 genus 103
starfish 5-1 cubelet starfish 5-2 cubelet starfish 5-3 cubelet starfish 5-4 cubelet
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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