Monday, June 10, 2013

Cube corner pyramids

A manufactured surface decorated with cube corner pyramids.

Define a cube corner pyramid as a pyramid specified by taking one corner of a cube as the pyramid's apex, and, as the pyramid's base, the equilateral triangle formed by face diagonals of the three faces of the cube adjacent to the chosen corner.

This diagram (far right) clarifies the geometry:

A cube corner pyramid (far right.) Image quoted from Wolfram MathWorld.

Plugging in the formula for the volume of a pyramid, V = A* h/3, we can calculate from the numbers given in the Wolfram chart for a unit cube:

1/6 = h * sqrt(3) / (2 * 3),

so the height, h = 1/sqrt(3).

Since the sides of the equilateral triangle that is the base of the pyramid measure sqrt(2), the ratio of the height of the pyramid to one of the sides of its base is 1/sqrt(6) = 0.40825...

If a surface mesh is composed of equilateral triangles, then decorating the mesh with cube corner pyramids gives us a surface that we can weave with straight, constant-width weavers that always cross each other at right angles, over-one/under-one. This sort of weaving can be seen a tabby weave or kagome weave depending on how local your perspective is. I think it is easier to see it as corrugated kagome.

Whether to point a given pyramid inward or outward can be decided independently for each equilateral triangle in the mesh. Here are some simulations of corrugated kagome surfaces, mostly with points out. These surfaces are the same as those that can be assembled from Mitsunobu Sonobe's unit origami module.

An icosahedron decorated with cube corner pyramids.

A tetrahelix decorated with cube corner pyramids.

In the following two images the mesh triangles are not equilateral, so the weavers will not be straight or constant-width. One shows a mesh with points out, the other the same mesh with points in.

A non-equilateral decoration decorated with cube corner pyramids. (The resulting weavers are not straight or constant width.) Model courtesy INRIA.

The same model decorated with points-in cube corner pyramids.

A origami project using sonobe units by lonely-soldier.

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