My art plays two roles in my mathematics: investigation and communication. I love mathematics where small examples are valuable starting points for building broader understanding. When these examples are 3D, I make them to get to know them on a practical and computational level. Once these objects exist, it’s easier to ask and investigate certain questions. As they live on, they help communicate this single visual perspective, letting me physically hand over one way I’m thinking.
Slip casting creates ceramic copies of an object. As I learned this process, I chose to slip cast 3D permutahedra since I encountered the permutahedron in many contexts, from group theory to combinatorics to algebraic geometry.
Artworks
Polytopes amidst a Permutahedral Tiling
20.0 x 40.0 x 22.5 cm
Ceramic
2024
These compound forms were inspired by the fact that permutahedra tile space in any dimension. To emphasize connections from permutahedra to tetrahedra, cubes, and octahedra, I chose subsets of the 3D tiling where the permutahedra's centers are the vertices of another polytope.
In any dimension, a permutahedron can be constructed by truncating the faces of a simplex in order of increasing dimension. In 3D or any dimension where the vertices of a hypercube contain a simplex, the tiling contains a subset of permutahedra whose centers form the vertices of a cube and octahedron. Each piece highlights different connections through glazing and structure. I see one cube and one octahedron. How many tetrahedrons can you find?
Permutahedron as a Cayley Graph
7.5 x 30.0 x 22.5 cm
Ceramic
2024
A permutahedron is one way to visualize the structure of the symmetric group as a Cayley graph using adjacent transpositions as generators. Each vertex is a way to rearrange 4 colors: brown, purple, blue, green. Each edge corresponds to the action of an adjacent transposition. While on paper left action and right action are separated by a typo, the making process demands attention to differences.
Making a permutahedron with vertices color-coded by conjugacy classes let me envision it for the first time. Its structure reflects the relationship between these generators and conjugation. A visual doesn’t simplify all ideas; it highlights one perspective. What a joy of written mathematics to switch perspectives when it offers the most insight.
