# Tetrahedra

This project was a collaboration with the California based artist Ann Preston in 2009. In the last several years she has worked with geometrical forms based on the Golden Mean, especially tetrahedra. She approached me with the challenge of constructing irregular tetrahedra out of wood, later to be gessoed and painted by the artist, as an exploration for more complex sculpture. The problem was how to construct these tetrahedra in a wooden form.

A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or “equilateral”. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as triangular pyramid.

The tetrahedra for this project were more complex because of the fact of being irregular tetrahedra: most of the triangles comprising the sides are very irregular, having sides and angles of varying dimensions, but needed to be constructed to very precise specifications. What method of construction would yield very precise irregular tetrahedra?

Ann sent me a package with precise dimensions and angles for four tetrahedra that would be the basis for many of the tetrahedra in the sculpture. Another feature of tetrahedra is that they can be folded out of a single sheet of paper, and I also received scale models of the four prototypes to fold and construct for better visualization. Ann also included puzzle pieces from a Zome tool, a kind of toy kit with precisely made parts that can be used to construct models of crystals and, not surprisingly, tetrahedra. Most of the tetrahedra were approximately a meter on their longest side.

I began working with plywood (initially 3/4 inch stock but quickly switching to ½ Baltic birch plywood) to see how it could be cut into triangles and mitered at difficult angles, and to experiment with different ways of holding the miters together. My initial thought was to cut the forms of the triangles with a circular saw and then to cut the angles that would form the joints or miters between adjacent sides and on a table saw or band saw. Accuracy in the mitered angles would be very important, and it soon became apparent that no saw would be able to cut the required acute angles.

Cutting out the triangles that form the sides proved not too difficult, once a suitable and reliable protractor was found for the precise measuring of the angles. Using my plunge cut Festool circular saw with guide, I was not only able to cut the forms of the triangles, but I was also able to rough cut the miters, and even do finish cuts up to 45 degrees. Beyond that, the solution for the precise cutting of the more acute angles turned out to be hand planning.

Once the triangular side were cut and mitered, I developed a system of splines to join the sides. The grooves for the splines were cut on the router table with a jig for supporting the sides at the correct angle while pushing the side through a 1/8^{th} upcut spiral bit. It sounds easy, but to align the splines/grooves accurately for the multiple sides in each tetrahedron was quite challenging. And the glue up was equally challenging.

The first four forms were named Alpha, Beta, Delta and Gamma and represented the modules upon which all future tetrahedra would be based. Upon completing these four forms, Ann specified a more complex form with multiple joined tetrahedra, a sort of complex crystal/diamond shape to be finished with book matched veneers. This final form was built and subsequently painted/gessoed, and exhibited in her solo exhibition at the Rosamund Felsen Gallery in Santa Monica in February 2010.