Hyperbolic Geometry in Art: Seven Stars After Dürer

How hyperbolic geometry changes a pattern

The hyperbolic plane has constant negative curvature. In its Poincaré disk model, the entire plane is represented inside a circular boundary. Angles are preserved, while distances and areas look increasingly compressed near the edge.

Its straightest paths, called geodesics, appear as circular arcs meeting the boundary at right angles, or as diameters through the center. The boundary itself lies infinitely far away in the hyperbolic metric. A print or carving can only show a finite portion at finite resolution.

This matters compositionally. The center can hold a recognizable motif while the surrounding repetitions become a dense fringe. I can work with that contrast without pretending a laser or a screen can draw infinitely small detail.

Seven Stars: knotwork translated into a disk

The source motif brings its own rhythm of interlaced strands and stars. My transformation changes the space those strands inhabit. The viewer can begin with the larger central forms and follow the knotwork outward until individual lines become difficult to separate.

The material adds another limit. A dense region that reads on a backlit screen can lose definition in wood. The finished work therefore needs to be assessed as an object with a surface and a practical detail limit, as well as a geometric image.

Reading tessellation notation

In standard Schläfli notation, {p,q} describes p-sided regular polygons with q meeting at each vertex. Thus {6,4} means four hexagons at a vertex, while {4,6} means six squares. Those are different tessellations. Douglas Dunham's Bridges paper gives the convention.

An earlier description of this work paired {6,4} with the incompatible phrase “six squares.” That wording was incorrect. I am leaving the specific tessellation assignment out of this account until it is reconciled with the original construction record; the image and exhibition history do not depend on that label.

Other transformation studies

I have also applied hyperbolic transformations to mandala imagery. These are contemporary image studies using historical source works. Their geometric structure does not validate a religious doctrine or prove that the source artists intended a hyperbolic construction.

For these studies, the useful question is what the transformation does to the viewer's reading of the image. A central figure becomes one part of a larger field, and the boundary changes the sense of scale. Historical claims require their own sources and cannot be inferred from the transformed image.

Where this sits in my work

Hyperbolic art concerns a different geometry of space. My 4D art explores an additional spatial dimension through projection. Both use mathematics as a structure for images and objects, but they should not be confused.

The CNC wall art case study follows geometric relief into wood. For another kind of pattern research, the seven-fold girih study examines how my own motifs meet across a changing field.