### Spatial Graph Conformations

The model shown in the image is a 3D spatial representation of the complete bipartite graph \(K_{4,4}\). When drawn in the plane, this graph necessarily intersects itself, and it has quite a few overlapping edges, for example in this configuration:

When working in three dimensions, we can arrange the vertices so that the edges do not intersect. Determining the “best” way to do this is an aesthetic and artistic challenge.

We already have working OpenSCAD code for creating spatial graph models. Next steps include cleaning up this code to have a friendlier interface and working on finding pleasing spatial conformations for other non-planar graphs. We will prototype designs in the Lab and then send final models to Shapeways for printing on their industrial SLS machines.

Current students: Hannah Critchfield (\(K_{4,4}\) and \(K_6\))

Past students: Ben Flint (generalized Petersen graphs)