Abstract Dimensions

Abstract Dimensions was a national juried exhibition at the Epperson Gallery in Crockett, California, open to work across the full range of abstraction, from the barely representational to the wholly non-objective. The artist Diane Williams juried it. The show ran October 4 to November 3, 2024, and the opening and awards were held on the evening of October 5.

Kleinian Creatures from the 4th Dimension was among the selected works. I submitted it because it takes the show’s title fairly literally. A Kleinian group is a discrete group of symmetries of hyperbolic space, and its limit set is the fractal on which its dynamics pile up. For groups acting on hyperbolic 3-space the limit set lives on an ordinary sphere, which is where the familiar gaskets and circle packings come from. This one comes from a group acting on hyperbolic 4-space, so its limit set lives on the three-sphere instead and has to be pushed down into ordinary space before it can be looked at. Its dimension is fractional wherever it sits — that is what makes it a fractal — and the projection into ordinary space leaves that unchanged.

The rendering treats the object as translucent, letting light scatter through it with some probability at each step of a random walk rather than reflecting cleanly off a surface. That is what gives it the fleshy, undersea look the title refers to.