Goursat tetrahedron

In geometry, a Goursat tetrahedron is a tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, the Euclidean 3-space, and hyperbolic 3-space. Coxeter named after Edouard Goursat who first looked into these domains. It is an extension of the theory of Schwarz triangles for Wythoff constructions on the sphere.

Contents 1 Graphical representation 2 Solutions 2.1 3-sphere (finite) solutions 2.2 Euclidean (affine) 3-space solutions 2.3 Hyperbolic 3-space solutions 3 References

Graphical representation

A Goursat tetrahedron can be represented graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face (mirror) of the Goursat tetrahedron. Each edge is labeled by a rational value corresponding to the reflection order, being π/dihedral angle.

A 4-node Coxeter-Dynkin diagram represents this tetrahedral graphs with order-2 edges hidden.

If some edges are order 2, the Coxeter group can be used for a simpler notation.

Existence requires each of the 3-node subgraphs of this graph, (p q r), (p u s), (q t u), and (r s t), must correspond to a Schwarz triangle.

Solutions

There are many hundreds of solutions with finite densities.

3-sphere (finite) solutions

The solutions for the 3-sphere with density 1 solutions are:

1. Duoprisms: [p]x[q],
2. Hyperprisms:
   • [3,3]x[],
   • [3,4]x[],
   • [3,5]x[],
3. Linear graphs: (Uniform polychora)
   • [3,3,3],
   • [3,3,4],
   • [3,4,3],
   • [3,3,5],
4. Tri-dental graphs:
   • [3^1,1,1],

There are hundreds of rational solutions for the 3-sphere, including these 6 linear graphs which generate the Schläfli-Hess polychora, and 11 nonlinear ones from Coxeter:

• Linear graphs
  1. Density 4: [3,5,5/2]
  2. Density 6: [5,5/2,5]
  3. Density 20: [5,3,5/2]
  4. Density 66: [5/2,5,5/2]
  5. Density 76: [5,5/2,3]
  6. Density 191: [3,3,5/2]
• Loop-n-tail graphs:
  1. Density 2:
  2. Density 3:
  3. Density 5:
  4. Density 8:
  5. Density 9:
  6. Density 14:
  7. Density 26:
  8. Density 30:
  9. Density 39:
  10. Density 46:
  11. Density 115:

Euclidean (affine) 3-space solutions

Density 1 solutions:

1. Convex uniform honeycomb:
   • Linear graph: [4,3,4],
   • Tri-dental graph: [4,3^1,1],
   • Loop graph: (3 3 3 3),

Hyperbolic 3-space solutions

Density 1 solutions:

1. Compact hyperbolic groups: (Convex uniform honeycombs in hyperbolic space)
   • Linear graphs:
     • [3,5,3],
     • [5,3,4],
     • [5,3,5],
   • Tri-dental graph:
     • [5,3^1,1],
   • Loop graphs:
     • (4 3 3 3),
     • (4 3 4 3),
     • (5 3 3 3),
     • (5 3 4 3),
     • (5 3 5 3),
2. Noncompact hyperbolic groups (with affine subgraphs)
   • Linear graphs: , , , , , ,
   • Tri-dental graphs: , ,
   • Loop graphs: , , , , , ,
   • Loop-n-tail graphs: , , ,
   • Two-loop graph:
   • Simplex graph:

References

• Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (page 280, Goursat's tetrahedra) [1]
• Goursat, Edouard, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'École Normale Supérieure, Sér. 3, 6 (1889), (pp. 9–102, pp. 80–81 tetrahedra)
• Dynkin Diagrams
• Richard Klitzing, Dynkin diagrams, Goursat tetrahedra