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How to Make a 3-D Tessellation

Three-dimensional (3-D) tessellations, tile, or pack, a volume with geometric shapes evolved from polyhedra. Unlike two-dimensional (2-D) tessellations, which have 17 ways to tile a plane, 3-D tessellations have hundreds of ways to pack a volume, including using cubes, a combination of tetrahedra and octahedra, a combination of octahedra and cubes, and convex polyhedra. However, of these many ways, only cubes and the combination of tetrahedra and octahedra use regular polyhedra. To create 3-D tessellations, the shape of the polyhedra is modified reciprocally, where the deformation of one face or edge must be accommodated in the deformation of the abutting faces or edges.

Things You'll Need

Foam blocks
Battery-powered, hand-held foam cutter, available at most hobby stores
Foam glue
Color markers

Instructions

- 1
  Select the polyhedra to pack a volume. Cubes are the easiest to stack and construct, but regular tetrahedra and octahedra stack easily, creating a triangulated pattern. If you prefer a greater challenge, there are many convex and irregular polygons that fill space.
- 2
  Cut the polyhedra, approximately 3 inches to 4 inches high or wide, from foam blocks, using a hand-held foam cutter. Cut straight, flat surfaces before attempting to deform the faces of the polyhedra.
- 3
  Select one foam polyhedron from those cut. With the foam cutter, remove a part of a face or edge of the polyhedron and glue the piece onto the filling pattern's neighboring polyhedron, so the augmented, neighboring polyhedron nests precisely with the modified, initial polyhedron. Continue removing and gluing pieces from one neighboring polyhedron to another to deform the polyhedra and create a unique 3-D tessellation. Ensure all the polyhedra in the packing pattern are cut and augmented the precise way as the first, modified polyhedra.
- 4
  Create surface patterns on the newly modified forms to accentuate the 3-D tessellation structure or cut away the interior volumes from the modified forms with the foam cutter. Possible surface patterns include single color faces, geometric patterns that conform with the faces of the 3-D tessellation module, or modified form, as well as naturalistic depictions, such as animals, people, plants and trees. Repeat the same surface patterns on each tessellation module to draw attention to the repetitive, space-filling pattern.

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