Lot’s of exciting things have happened in the last few months, which unfortunately have led to a lot less blogging. Over the last few weeks I completely overhauled my website (davidbachmandesign.com), including pictures of all of my latest work, and a new online “store.” In November I was busy with my first (non-local) solo exhibition in Los Angeles, (pictures from the show are here) and in the months before that I was creating a host of new pieces for it. In this post I’ll describe how I made four of these new pieces, shown here.

The theme that ties these four objects together is *Symmetry*. Each piece has symmetries from two platonic solids, one of which is visually dominant. Here I will go through, in detail, the story behind the first piece pictured here, *Octoplex*. This has symmetries from both an octahedron and a cube, with the octahedron being dominant.

The story begins with the artist Bathsheba Grossman. Bathsheba is a true pioneer of using 3D Printing to combine Mathematics and Art. Many of the pieces featured at Shapeways are hers. Years ago I went to a 3D Printing industry convention, and half of the floor samples were her designs. She’s got a series of pieces that have always intrigued me: *Ora*, *Metatron*, *Metatrino*, *Quintron*, and *Quintrino*. Here are three different pictures of *Metatrino*, shown here with her permission.

It took me an embarrassingly long time to come to some understanding of this form. I’ll do my best here to explain it. First, start with an octahedron: a 3-dimensional shape with eight congruent triangular sides. The centers of each side lie at the corners of a cube (called it’s *dual*). Both the octahedron and its dual cube are shown here.

If you cut off the corners of the octahedron, you create six little squares. Similarly, cutting off the corners of the cube creates eight little triangles. These squares and triangles are shown in yellow here.

Note that each yellow square has four edges, and since there are six such squares, there are a total of 24 square-edges. Each yellow triangle has three edges, and since there are eight of them, there are also 24 triangle-edges. The fact that there are the same number of each is the key to making the construction work.

The next step is to pick one edge of one yellow square, and one edge of one yellow triangle, and connect them with some ribbon-shape, as pictured here.

Now we get to use symmetry. There are lots of ways to rotate the octahedron so that it lands in a similar position, but the ribbon surface gets moved somewhere else. Applying all such rotations to the ribbon surface gives the following shape.

This is somewhat of a tangled mess, but it’s essentially the whole idea. The shape of the final form is completely determined by the shape of the initial ribbon between one square-edge and one triangle-edge. Creating an appealing final form is a matter of sculpting this ribbon surface appropriately. Bathsheba’s *Metatrino* comes from one very beautiful band shape. To create something truly different, I had to come up with another. For my piece, I created the ladder-like surface pictured here.

Getting the shape of this right was extremely difficult. It is made so that none of the copies of the ladder will intersect each other in the final form. Adjacent copies also weave together at each end in a way that I thought would be novel. Here is the original ladder with all of the copies coming from the symmetries (rotations) of the octahedron. You can still just barely make out the green lines showing the original octahedron and cube.

The final form was 3D printed by Shapeways in their “Sandstone” material, which is a gypsum powder fused with some binder (essentially glue). During the printing process they also add colored ink, as specified in the design. The resulting piece is about 7.5″ in diameter, and has a nice, almost ceramic, feel. Here are three images, showing the same symmetries as the images of Bathsheba’s piece shown above.

To create *Hexaplex*, shown at the beginning of this post, I could have started with a cube on the outside, and constructed it’s dual (another octahedron) on the inside. However, I decided to use this opportunity to demonstrate another kind of symmetry. Mathematicians would call this “reflection in a sphere.” For non-mathematicians, I would describe this as a reflection in a plane in four-dimensions. Similarly, the other two pieces in the series, *Dodecaplex* and *Icosaplex*, both come from an icosahedron and a dodecahedron, and they are related to each other by reflection in a sphere.