A Woven Klein Quartic

Abstract

We describe a new method of weaving a model of the Klein quartic, a highly symmetric, but abstract genus-3 surface akin to a platonic polyhedron, with negatively-curved geometry. The Klein quartic cannot be realized in its fully symmetric form in three-dimensional space, but this model exhibits the most rigid symmetry that is possible. With remarkably little time and material you can have a Klein quartic of your own!

Figures (4)

• Figure 1: At left a portion of the hyperbolic plane with underlying *732 symmetry. At right is a model tiled by heptagons and triangles bounded by geodesics, assembled at the Gathering For Gardner, February 24, 2024. The Klein quartic is formed by identifying like triangles on the boundary of the region at left (two pairs marked), or by identifying opposite blue rings on the model at right. In both, there are 21 colored geodesics, in seven colors of three each. Seven triples of colors do not appear, in a Fano plane of the colors, each pair missing from exactly one triple.
• Figure 2: At left a model of the Klein quartic with tetrahedral symmetry, formed from twenty-four heptagons meeting in threes, in eight colors. At right, controlling the surface curvature with the weaving pattern, positive curvature and pentagonal holes at lower right; the negative curvature with heptagonal holes at top; the kogame lattice at right..
• Figure 3: Templates, at top for the "rings", middle for the "short strips", and bottom for the "long strips".
• Figure 4: Tips for constructing the model are described in the text.