# Dugan Hammock

My research involves geometric constructions that involve slices and projections of shapes from higher-dimensional spaces. Great amounts of time and care are taken in implementing the prescribed algorithms computationally using tools such as Mathematica or Matlab.

While writing the code I have no idea what the output is going to look like, so it's always a wild surprise once I see the results.

The constructions always involve a certain variability in the parameters that are used. When I am exploring the possibilities I will often introduce some amount of randomness while preserving some amount of desired symmetries, and a good balance makes are an aesthetically pleasing design.

Twixt chaos and order, beauty lies.

This quasicrystal tiling covers the entire plane without translational symmetry, it uses a finite number of tile shapes, and possesses a single point of 30-fold rotational symmetry at its center. It is derived as a projection of the E8 lattice onto an appropriate Coxeter plane which exhibits this symmetry. Around each E8 lattice point is an 8-dimensional Voronoi domain which the dual to the Gosset polytope. Only those lattice points whose Voronoi domain intersect the Coxeter plane are selected to be included in the projection. After all the appropriate points are mapped to the plane, a two-dimensional Voronoi tessellation is calculated to generate the tiles as shown. The tiles are colored as a function of area.

This rhombic tiling is contained within a regular 98-sided
polygon.

There are 4,704 rhombi in total which come in 24 different sizes
and they are colored by area.

The outer layers are identical to the fully symmetric standard
rosette construction, but the tiles toward the center are permuted
so as to preserve 7-fold rotational symmetry.