Alan Schoen’s I-WP surface from 1970 is a triply periodic minimal surface of genus 4. One can think of it as a sphere extending handles towards the vertices of a cube.

The Letters WP stand for wrapped package, the appearance of the skeletal graph of the surface near a vertex, where four sticks are bundled together. After identifying opposite faces of a cubical cell the surface has genus 7, but half of such a cell already constitutes a translational fundamental domain, bringing the genus down to 4.

The Gauss map has degree 3 and realizes the compact quotient surface as a 3-fold cover cyclically branched over the vertices of a regular octahedron. The order 4 rotations of this octahedron lift to the genus 4 surface as order 12 rotations, representing the surface also as a 12-fold cyclically branched cover over a thrice punctured sphere, as one can read off from the hyperbolic fundamental domain. This means that I-WP is conformally considerably more symmetric than it appears, and that it can be tiled by copies of (conformally regular) 12-gons, as indicated by the image on the right. It took me a while to convince myself that this is again I-WP.

There is a simple 1-parameter deformation that keeps the symmetries of a box over a square. This deformation limits is a Traizet limit of horizontal planes joined by catenoidal necks, and a singly periodic Karcher-Scherk surfaces with 8 ends.


Mathematica Notebooks cube and 12-gon

Wolfram CDF

PoVRay Sources cube and 12-gon

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