Thursday, December 11, 2014

Weaving tetrakis vox surfaces

Another way to deal with the lack of surface corrugation on some vox solids is to corrugate each exposed voxel . When the corrugation takes the form of a square pyramid added to each exposed voxel face, the adjective tetrakis, usually applied to polyhedra, can be applied to the vox surface.

A parameter is the height of the pyramid, or, what is easier to measure, the altitude/base ratio, a/b, of its isosceles triangle faces.

When a/b = 0.5, the pyramid has zero height: we have a cube that has been decorated with crossing diagonals. The universal weave of a cube naturally produces this shape with its four straight weavers crossing on every face.

When a/b = √5 / 4 = 0.5590, all the dihedral angles are equal and we have a tetrakis hexahedron. The weave is the same as the universal weave of the cube, but the weavers are a little crooked. This is an appropriate a/b for folding.

When a/b = √2 / 2 = 0.7071, the original cube edges momentarily disappear and we have a rhomic dodecahedron. This is the tallest pyramid we can use without encountering mechanical interference at sharp internal corners. Again, the weave is the same as the universal weave of the cube except that the weavers are crooked and, in this case, some folds are missing, leaving rhombic rather than triangular facets.

When a/b = √3 / 2 = 0.8660, the facets are all equilateral triangles. The vox solid cannot have any internal corners—assuming it does not, the resulting  tetrakis surface is a deltahedron.

All the weaves described here are coarse weaves: the universal weaves of these shapes would require a fabric opening in the center of each triangular facet.

All of these tetrakis vox surfaces are in theory completely foldable (the rhombic dodecahedron would need to permit folding along its short diagonals.)

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