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Goursat tetrahedron#Compact hyperbolic 3-space solutions

File:Coxeter-Dynkin 3-space groups.png

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, Euclidean 3-space, and hyperbolic 3-space. Coxeter named them after Édouard Goursat who first looked into these domains. It is an extension of the theory of Schwarz triangles for Wythoff constructions on the sphere.

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.

:100px

A 4-node Coxeter-Dynkin diagram represents this tetrahedral graph with order-2 edges hidden. If many edges are order 2, the Coxeter group can be represented by a bracket 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.

Extended symmetry

class=wikitable align=right width=600

|360px

|240px

colspan=2|The symmetry of a Goursat tetrahedron can be tetrahedral symmetry of any subgroup symmetry shown in this tree, with subgroups below with subgroup indices labeled in the colored edges.

An extended symmetry of the Goursat tetrahedron is a semidirect product of the Coxeter group symmetry and the fundamental domain symmetry (the Goursat tetrahedron in these cases). Coxeter notation supports this symmetry as double-brackets like [Y[X]] means full Coxeter group symmetry [X], with Y as a symmetry of the Goursat tetrahedron. If Y is a pure reflective symmetry, the group will represent another Coxeter group of mirrors. If there is only one simple doubling symmetry, Y can be implicit like X with either reflectional or rotational symmetry depending on the context.

The extended symmetry of each Goursat tetrahedron is also given below. The highest possible symmetry is that of the regular tetrahedron as [3,3], and this occurs in the prismatic point group [2,2,2] or [2[3,3]] and the paracompact hyperbolic group [3[3,3]].

See Tetrahedron#Isometries of irregular tetrahedra for 7 lower symmetry isometries of the tetrahedron.

Whole number solutions

The following sections show all of the whole number Goursat tetrahedral solutions on the 3-sphere, Euclidean 3-space, and Hyperbolic 3-space. The extended symmetry of each tetrahedron is also given.

The colored tetrahedal diagrams below are vertex figures for omnitruncated polytopes and honeycombs from each symmetry family. The edge labels represent polygonal face orders, which is double the Coxeter graph branch order. The dihedral angle of an edge labeled 2n is π/n. Yellow edges labeled 4 come from right angle (unconnected) mirror nodes in the Coxeter diagram.

=3-sphere (finite) solutions=

File:Coxeter diagram finite rank4 correspondence.png

The solutions for the 3-sphere with density 1 solutions are: (Uniform polychora)

class=wikitable

|+ Duoprisms and hyperprisms:

align=center

!Coxeter group
and diagram

|[2,2,2]
{{CDD|node|2|node|2|node|2|node}}

|[p,2,2]
{{CDD|node|p|node|2|node|2|node}}

|[p,2,q]
{{CDD|node|p|node|2|node|q|node}}

|[p,2,p]
{{CDD|node|p|node|2|node|p|node}}

|[3,3,2]
{{CDD|node|3|node|3|node|2|node}}

|[4,3,2]
{{CDD|node|4|node|3|node|2|node}}

|[5,3,2]
{{CDD|node|5|node|3|node|2|node}}

align=center

!Group symmetry order

|16

|8p

|4pq

|4p2

|48

|96

|240

align=center

!Tetrahedron
symmetry

|[3,3]
(order 24)
40px

|[2]
(order 4)
40px

|[2]
(order 4)
40px

|[2+,4]
(order 8)
40px

|[ ]
(order 2)
40px

|[ ]+
(order 1)
40px

|[ ]+
(order 1)
40px

align=center valign=top

!Extended symmetry

|[(3,3)[2,2,2]]
{{CDD|node_c1|2|node_c1|2|node_c1|2|node_c1}}
=[4,3,3]
{{CDD|node_c1|4|node|3|node|3|node}}

|[2[p,2,2]]
{{CDD|node_c1|p|node_c1|2|node_c2|2|node_c2}}
=[2p,2,4]
{{CDD|node|2x|p|node_c1|2|node|4|node_c2}}

|[2[p,2,q]]
{{CDD|node_c1|p|node_c1|2|node_c2|q|node_c2}}
=[2p,2,2q]
{{CDD|node|2x|p|node_c1|2|node|2x|q|node_c2}}

|[(2+,4)[p,2,p]]
{{CDD|node_c1|p|node_c1|2|node_c1|p|node_c1}}
=[2+[2p,2,2p]]
{{CDD|node|2x|p|node_c1|2|node_c1|2x|p|node}}

|[1[3,3,2]]
{{CDD|node_c1|3|node_c2|3|node_c1|2|node_c3}}
=[4,3,2]
{{CDD|node|4|node_c1|3|node_c2|2|node_c3}}

|[4,3,2]
{{CDD|node_c1|4|node_c2|3|node_c3|2|node_c4}}

|[5,3,2]
{{CDD|node_c1|5|node_c2|3|node_c3|2|node_c4}}

align=center

!Extended symmetry order

|384

|32p

|16pq

|32p2

|96

|96

|240

class=wikitable
Graph type

!colspan=4|Linear

!Tridental

align=center

!Coxeter group
and diagram

|Pentachoric
[3,3,3]
{{CDD|node|3|node|3|node|3|node}}

|Hexadecachoric
[4,3,3]
{{CDD|node|4|node|3|node|3|node}}

|Icositetrachoric
[3,4,3]
{{CDD|node|3|node|4|node|3|node}}

|Hexacosichoric
[5,3,3]
{{CDD|node|5|node|3|node|3|node}}

|Demitesseractic
[31,1,1]
{{CDD|nodes|split2|node|3|node}}

colspan=6|Vertex figure of omnitruncated uniform polychora
align=center

!Tetrahedron

|80px

|80px

|80px

|80px

|80px

align=center

!Group symmetry order

|120

|384

|1152

|14400

|192

align=center

!Tetrahedron
symmetry

|[2]+
(order 2)
40px

|[ ]+
(order 1)
40px

|[2]+
(order 2)
40px

|[ ]+
(order 1)
40px

|[3]
(order 6)
40px

valign=top align=center

!Extended symmetry

|[2+[3,3,3]]
{{CDD|branch_c1|3ab|nodeab_c2}}

|[4,3,3]
{{CDD|node_c1|4|node_c2|3|node_c3|3|node_c4}}

|[2+[3,4,3]]
{{CDD|label4|branch_c1|3ab|nodeab_c2}}

|[5,3,3]
{{CDD|node_c1|5|node_c2|3|node_c3|3|node_c4}}

|[3[31,1,1]]
{{CDD|nodeab_c1|split2|node_c2|3|node_c1}}
=[3,4,3]
{{CDD|node_c2|3|node_c1|4|node|3|node}}

align=center

!Extended symmetry order

|240

|384

|2304

|14400

|1152

=Euclidean (affine) 3-space solutions=

File:Coxeter diagram affine rank4 correspondence.png

Density 1 solutions: Convex uniform honeycombs:

class=wikitable
Graph type

!Linear
Orthoscheme

!Tri-dental
Plagioscheme

!Loop
Cycloscheme

!colspan=3|Prismatic

!Degenerate

align=center

!Coxeter group
Coxeter diagram

|[4,3,4]
{{CDD|node|4|node|3|node|4|node}}

|[4,31,1]
{{CDD|nodes|split2|node|4|node}}

|[3[4]]
{{CDD|branch|3ab|branch}}

|[4,4,2]
{{CDD|node|4|node|4|node|2|node}}

|[6,3,2]
{{CDD|node|6|node|3|node|2|node}}

|[3[3],2]
{{CDD|branch|split2|node|2|node}}

|[∞,2,∞]
{{CDD|node|infin|node|2|node|infin|node}}

colspan=8|Vertex figure of omnitruncated honeycombs
align=center

!Tetrahedron

|80px

|80px

|80px

|

|

|

|

align=center

!Tetrahedron
Symmetry

|[2]+
(order 2)
40px

|[ ]
(order 2)
40px

|[2+,4]
(order 8)
40px

|[ ]
(order 2)
40px

|[ ]+
(order 1)
40px

|[3]
(order 6)
40px

|[2+,4]
(order 8)
40px

valign=top align=center

!Extended symmetry

|[(2+)[4,3,4]]
{{CDD|branch_c2|4-4|nodeab_c1}}

|[1[4,31,1]]
{{CDD|nodeab_c1|split2|node_c2|4|node_c3}}
=[4,3,4]
{{CDD|node|4|node_c1|3|node_c2|4|node_c3}}

|[(2+,4)[3[4]]]
{{CDD|branch_c1|3ab|branch_c1}}
=[2+[4,3,4]]
{{CDD|branch_c1|4-4|nodes}}

|[1[4,4,2]]
{{CDD|node_c1|4|node_c2|4|node_c1|2|node_c3}}
=[4,4,2]
{{CDD|node|4|node_c1|4|node_c2|2|node_c3}}

|[6,3,2]
{{CDD|node_c1|6|node_c2|3|node_c3|2|node_c4}}

|[3[3[3],2]]
{{CDD|branch_c1|split2|node_c1|2|node_c2}}
=[3,6,2]
{{CDD|node|3|node|6|node_c1|2|node_c2}}

|[(2+,4)[∞,2,∞]]
{{CDD|node_c1|infin|node_c1|2|node_c1|infin|node_c1}}
=[1[4,4]]
{{CDD|node|4|node_c1|4|node}}

=Compact hyperbolic 3-space solutions=

Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter diagram#Compact (Lannér simplex groups))

class=wikitable

|+ Rank 4 Lannér simplex groups

Graph type

!colspan=3|Linear

!Tri-dental

align=center

!Coxeter group
Coxeter diagram

|[3,5,3]
{{CDD|node|3|node|5|node|3|node}}

|[5,3,4]
{{CDD|node|5|node|3|node|4|node}}

|[5,3,5]
{{CDD|node|5|node|3|node|5|node}}

|[5,31,1]
{{CDD|node|5|node|split1|nodes}}

|

colspan=6| Vertex figures of omnitruncated honeycombs
align=center

!Tetrahedron

|80px

|80px

|80px

|80px

|

align=center

!Tetrahedron
Symmetry

|[2]+
(order 2)
40px

|[ ]+
(order 1)
40px

|[2]+
(order 2)
40px

|[ ]
(order 2)
40px

|

valign=top align=center

!Extended symmetry

|[2+[3,5,3]]
{{CDD|label5|branch_c1|3ab|nodeab_c2}}

|[5,3,4]
{{CDD|node_c1|5|node_c2|3|node_c3|4|node_c4}}

|[2+[5,3,5]]
{{CDD|branch_c1|5a5b|nodeab_c2}}

|[1[5,31,1]]
{{CDD|node_c1|5|node_c2|split1|nodeab_c3}}
=[5,3,4]
{{CDD|node_c1|5|node_c2|3|node_c3|4|node}}

|

Graph type

!colspan=7|Loop

align=center

!Coxeter group
Coxeter diagram

|[(4,3,3,3)]
{{CDD|label4|branch|3ab|branch}}

|[(4,3)2]
{{CDD|label4|branch|3ab|branch|label4}}

|[(5,3,3,3)]
{{CDD|label5|branch|3ab|branch}}

|[(5,3,4,3)]
{{CDD|label5|branch|3ab|branch|label4}}

|[(5,3)2]
{{CDD|label5|branch|3ab|branch|label5}}

colspan=8| Vertex figures of omnitruncated honeycombs
align=center

!Tetrahedron

|80px

|80px

|80px

|80px

|80px

align=center

!Tetrahedron
Symmetry

|[2]+
(order 2)
40px

|[2,2]+
(order 4)
40px

|[2]+
(order 2)
40px

|[2]+
(order 2)
40px

|[2,2]+
(order 4)
40px

valign=top align=center

!Extended symmetry

|[2+[(4,3,3,3)]]
{{CDD|label4|branch_c1|3ab|branch_c2}}

|[(2,2)+[(4,3)2]]
{{CDD|label4|branch_c1|3ab|branch_c1|label4}}

|[2+[(5,3,3,3)]]
{{CDD|label5|branch_c1|3ab|branch_c2}}

|[2+[(5,3,4,3)]]
{{CDD|label5|branch_c1|3ab|branch_c2|label4}}

|[(2,2)+[(5,3)2]]
{{CDD|label5|branch_c1|3ab|branch_c1|label5}}

=Paracompact hyperbolic 3-space solutions=

File:Hyperbolic subgroup tree 36.png

thumb

Density 1 solutions: (See Coxeter diagram#Paracompact (Koszul simplex groups))

class=wikitable

|+ Rank 4 Koszul simplex groups

Graph type

!colspan=7|Linear graphs

align=center

!Coxeter group
and diagram

|[6,3,3]
{{CDD|node|6|node|3|node|3|node}}

|[3,6,3]
{{CDD|node|3|node|6|node|3|node}}

|[6,3,4]
{{CDD|node|6|node|3|node|4|node}}

|[6,3,5]
{{CDD|node|6|node|3|node|5|node}}

|[6,3,6]
{{CDD|node|6|node|3|node|6|node}}

|[4,4,3]
{{CDD|node|4|node|4|node|3|node}}

|[4,4,4]
{{CDD|node|4|node|4|node|4|node}}

align=center

!Tetrahedron
symmetry

|[ ]+
(order 1)
40px

|[2]+
(order 2)
40px

|[ ]+
(order 1)
40px

|[ ]+
(order 1)
40px

|[2]+
(order 2)
40px

|[ ]+
(order 1)
40px

|[2]+
(order 2)
40px

align=center valign=top

!Extended symmetry

|[6,3,3]
{{CDD|node_c1|6|node_c2|3|node_c3|3|node_c4}}

|[2+[3,6,3]]
{{CDD|label6|branch_c1|3ab|nodeab_c2}}

|[6,3,4]
{{CDD|node_c1|6|node_c2|3|node_c3|4|node_c4}}

|[6,3,5]
{{CDD|node_c1|6|node_c2|3|node_c3|5|node_c4}}

|[2+[6,3,6]]
{{CDD|branch_c1|6a6b|nodeab_c2}}

|[4,4,3]
{{CDD|node_c1|4|node_c2|4|node_c3|3|node_c4}}

|[2+[4,4,4]]
{{CDD|label4|branch_c1|4-4|nodeab_c2}}

Graph type

!colspan=8|Loop graphs

align=center

!Coxeter group
and diagram

|[3[ ]×[ ]]
{{CDD|node|split1|branch|split2|node}}

|[(4,4,3,3)]
{{CDD|node|split1-44|nodes|split2|node}}

|[(43,3)]
{{CDD|label4|branch|4-4|branch}}

|[4[4]]
{{CDD|label4|branch|4-4|branch|label4}}

|[(6,33)]
{{CDD|label6|branch|3ab|branch|2}}

|[(6,3,4,3)]
{{CDD|label6|branch|3ab|branch|label4}}

|[(6,3,5,3)]
{{CDD|label6|branch|3ab|branch|label5}}

|[(6,3)[2]]
{{CDD|label6|branch|3ab|branch|label6}}

align=center

!Tetrahedron
symmetry

|[2]
(order 4)
40px

|[ ]
(order 2)
40px

|[2]+
(order 2)
40px

|[2+,4]
(order 8)
40px

|[2]+
(order 2)
40px

|[2]+
(order 2)
40px

|[2]+
(order 2)
40px

|[2,2]+
(order 4)
40px

align=center valign=top

!Extended symmetry

|[2[3[ ]×[ ]]]
{{CDD|node_c2|split1|branch_c1|split2|node_c2}}
=[6,3,4]
{{CDD|node|6|node_c1|3|node_c2|4|node}}

|[1[(4,4,3,3)]]
{{CDD|node_c1|split1-44|nodeab_c3|split2|node_c2}}
=[3,41,1]
{{CDD|node|4|node_c3|split1-43|nodeab_c1-2}}

|[2+[(43,3)]]
{{CDD|label4|branch_c1|4-4|branch_c2}}

|[(2+,4)[4[4]]]
{{CDD|label4|branch_c1|4-4|branch_c1|label4}}
=[2+[4,4,4]]
{{CDD|label4|branch_c1|4-4|nodes}}

|[2+[(6,33)]]
{{CDD|label6|branch_c1|3ab|branch_c2|2}}

|[2+[(6,3,4,3)]]
{{CDD|label6|branch_c1|3ab|branch_c2|label4}}

|[2+[(6,3,5,3)]]
{{CDD|label6|branch_c1|3ab|branch_c2|label5}}

|[(2,2)+[(6,3)[2]]]
{{CDD|label6|branch_c1|3ab|branch_c1|label6}}

Graph type

!colspan=3|Tri-dental

!colspan=4|Loop-n-tail

!Simplex

align=center

!Coxeter group
and diagram

|[6,31,1]
{{CDD|node|6|node|split1|nodes}}

|[3,41,1]
{{CDD|node|3|node|split1-44|nodes}}

|[41,1,1]
{{CDD|node|4|node|split1-44|nodes}}

|[3,3[3]]
{{CDD|node|3|node|split1|branch}}

|[4,3[3]]
{{CDD|node|4|node|split1|branch}}

|[5,3[3]]
{{CDD|node|5|node|split1|branch}}

|[6,3[3]]
{{CDD|node|6|node|split1|branch}}

|[3[3,3]]
{{CDD|branch|splitcross|branch}}

align=center

!Tetrahedron
symmetry

|[ ]
(order 2)
40px

|[ ]
(order 2)
40px

|[3]
(order 6)
40px

|[ ]
(order 2)
40px

|[ ]
(order 2)
40px

|[ ]
(order 2)
40px

|[ ]
(order 2)
40px

|[3,3]
(order 24)
40px

align=center valign=top

!Extended symmetry

|[1[6,31,1]]
{{CDD|node_c1|6|node_c2|split1|nodeab_c3}}
=[6,3,4]
{{CDD|node_c1|6|node_c2|3|node_c3|4|node}}

|[1[3,41,1]]
{{CDD|node_c1|3|node_c2|split1-44|nodeab_c3}}
=[3,4,4]
{{CDD|node_c1|3|node_c2|4|node_c3|4|node}}

|[3[41,1,1]]
{{CDD|node_c1|4|node_c2|split1-44|nodeab_c1}}
=[4,4,3]
{{CDD|node_c2|4|node_c1|4|node|3|node}}

|[1[3,3[3]]]
{{CDD|node_c1|3|node_c2|split1|branch_c3}}
=[3,3,6]
{{CDD|node_c1|3|node_c2|3|node_c3|6|node}}

|[1[4,3[3]]]
{{CDD|node_c1|4|node_c2|split1|branch_c3}}
=[4,3,6]
{{CDD|node_c1|4|node_c2|3|node_c3|6|node}}

|[1[5,3[3]]]
{{CDD|node_c1|5|node_c2|split1|branch_c3}}
=[5,3,6]
{{CDD|node_c1|5|node_c2|3|node_c3|6|node}}

|[1[6,3[3]]]
{{CDD|node_c1|6|node_c2|split1|branch_c3}}
=[6,3,6]
{{CDD|node_c1|6|node_c2|3|node_c3|6|node}}

|[(3,3)[3[3,3]]]
{{CDD|branch_c1|splitcross|branch_c1}}
=[6,3,3]
{{CDD|node_c1|6|node|3|node|3|node}}

Rational solutions

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:

class=wikitable
valign=top

|Linear graphs

  1. Density 4: [3,5,5/2] {{CDD|node|3|node|5|node|5|rat|d2|node}}
  2. Density 6: [5,5/2,5] {{CDD|node|5|node|5|rat|d2|node|5|node}}
  3. Density 20: [5,3,5/2] {{CDD|node|5|node|3|node|5|rat|d2|node}}
  4. Density 66: [5/2,5,5/2] {{CDD|node|5|rat|d2|node|5|node|5|rat|d2|node}}
  5. Density 76: [5,5/2,3] {{CDD|node|5|node|5|rat|d2|node|3|node}}
  6. Density 191: [3,3,5/2] {{CDD|node|3|node|3|node|5|rat|d2|node}}

|Loop-n-tail graphs:

  1. Density 2: {{CDD|label3-2|branch|split2|node|5|node}}
  2. Density 3: {{CDD|label5|branch|split2-5t|node|3|node}}
  3. Density 5: {{CDD|label5-3|branch|split2-53|node|3|node}}
  4. Density 8: {{CDD|label5-4|branch|split2-55|node|3|node}}
  5. Density 9: {{CDD|branch|split2-p3|node|3|node}}
  6. Density 14: {{CDD|label5|branch|split2-p3|node|5|node}}
  7. Density 26: {{CDD|label5-3|branch|split2-p3|node|5|node}}
  8. Density 30: {{CDD|branch|split2-5p|node|3|node}}
  9. Density 39: {{CDD|label3-2|branch|split2-53|node|3|node}}
  10. Density 46: {{CDD|label5|branch|split2-5t|node|5|rat|d2|node}}
  11. Density 115: {{CDD|label5|branch|split2-p3|node|3|node}}

In all, there are 59 sporadic tetrahedra with rational angles, and 2 infinite families.https://arxiv.org/abs/2011.14232 Space vectors forming rational angles, Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, Michael Rubinstein, 2020

See also

  • Point group for n-simplex solutions on (n−1)-sphere.

References

{{reflist}}

  • Regular Polytopes, (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} (page 280, Goursat's tetrahedra) [https://books.google.com/books?id=iWvXsVInpgMC&dq=Goursat%27s+tetrahedra&pg=PA284]
  • Norman Johnson The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) He proved the enumeration of the Goursat tetrahedra by Coxeter is complete
  • 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)
  • {{KlitzingPolytopes|../explain/goursat.htm|Dynkin Diagrams|Goursat tetrahedra}}
  • Norman Johnson, Geometries and Transformations (2018), Chapters 11,12,13
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups 1999, Volume 4, Issue 4, pp 329–353 [https://link.springer.com/article/10.1007%2FBF01238563]

Category:Tetrahedra