Regular Polyhedra in the 3-Torus.
Antonio Montero
Daniel Pellicer
Centro de Ciencias Matemáticas
Universidad Nacional Autonoma de México
Workshop on Symmetries In Graphs, Maps and Polytopes
West Malvern, U.K.
July, 2014
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Polyhedra
A polyhedron P is geometric realization of a connected graph Sk(P)
(called the 1-skeleton of P) together with a family of subgraphs of Sk(P)
(called faces) such that:
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
2 / 26
Polyhedra
A polyhedron P is geometric realization of a connected graph Sk(P)
(called the 1-skeleton of P) together with a family of subgraphs of Sk(P)
(called faces) such that:
Every face is (isomorphic to) a cycle or an infinite path.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
2 / 26
Polyhedra
A polyhedron P is geometric realization of a connected graph Sk(P)
(called the 1-skeleton of P) together with a family of subgraphs of Sk(P)
(called faces) such that:
Every face is (isomorphic to) a cycle or an infinite path.
The vertex-figures are (isomorphic to) cycles.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
2 / 26
Polyhedra
A polyhedron P is geometric realization of a connected graph Sk(P)
(called the 1-skeleton of P) together with a family of subgraphs of Sk(P)
(called faces) such that:
Every face is (isomorphic to) a cycle or an infinite path.
The vertex-figures are (isomorphic to) cycles.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
2 / 26
Polyhedra
A polyhedron P is geometric realization of a connected graph Sk(P)
(called the 1-skeleton of P) together with a family of subgraphs of Sk(P)
(called faces) such that:
Every face is (isomorphic to) a cycle or an infinite path.
The vertex-figures are (isomorphic to) cycles.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
2 / 26
Polyhedra
A polyhedron P is geometric realization of a connected graph Sk(P)
(called the 1-skeleton of P) together with a family of subgraphs of Sk(P)
(called faces) such that:
Every face is (isomorphic to) a cycle or an infinite path.
The vertex-figures are (isomorphic to) cycles.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
2 / 26
Polyhedra
A polyhedron P is geometric realization of a connected graph Sk(P)
(called the 1-skeleton of P) together with a family of subgraphs of Sk(P)
(called faces) such that:
Every face is (isomorphic to) a cycle or an infinite path.
The vertex-figures are (isomorphic to) cycles.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
2 / 26
Polyhedra
Symmetries
A flag of P is an incident triplet (vertex, edge, face).
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Polyhedra
Symmetries
A flag of P is an incident triplet (vertex, edge, face).
A symmetry of P is an isometry (of the ambient space) that preserves
P.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
3 / 26
Polyhedra
Symmetries
A flag of P is an incident triplet (vertex, edge, face).
A symmetry of P is an isometry (of the ambient space) that preserves
P.
P is regular if its group of symmetries acts transitively on flags.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Regular Polyhedra
Platonic Solids
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Regular Polyhedra
Kepler-Poinsot Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Regular Polyhedra
Petrie-Coxeter Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Regular Polyhedra
Plane Tessellations
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
8 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
8 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
8 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
8 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
8 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
8 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
9 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
9 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
9 / 26
Regular Polyhedra
Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
9 / 26
Regular Polyhedra
Classification Theorem
Theorem (Grünbaum-Dress (70’s - 80’s); McMullen-Schulte (1997))
There exists 48 regular polyhedra in euclidean space E3 .
18 finite polyhedra
I
I
I
2 with tetrahedral symmetry.
4 with octahedral symmetry.
12 with icosahedral symmetry.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
10 / 26
Regular Polyhedra
Classification Theorem
Theorem (Grünbaum-Dress (70’s - 80’s); McMullen-Schulte (1997))
There exists 48 regular polyhedra in euclidean space E3 .
18 finite polyhedra
I
I
I
2 with tetrahedral symmetry.
4 with octahedral symmetry.
12 with icosahedral symmetry.
12 infinte pure polyhedra (which include Petrie-Coxeter polyhedra).
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
10 / 26
Regular Polyhedra
Classification Theorem
Theorem (Grünbaum-Dress (70’s - 80’s); McMullen-Schulte (1997))
There exists 48 regular polyhedra in euclidean space E3 .
18 finite polyhedra
I
I
I
2 with tetrahedral symmetry.
4 with octahedral symmetry.
12 with icosahedral symmetry.
12 infinte pure polyhedra (which include Petrie-Coxeter polyhedra).
6 planar polyhedra.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
10 / 26
Regular Polyhedra
Classification Theorem
Theorem (Grünbaum-Dress (70’s - 80’s); McMullen-Schulte (1997))
There exists 48 regular polyhedra in euclidean space E3 .
18 finite polyhedra
I
I
I
2 with tetrahedral symmetry.
4 with octahedral symmetry.
12 with icosahedral symmetry.
12 infinte pure polyhedra (which include Petrie-Coxeter polyhedra).
6 planar polyhedra.
12 blended polyhedra.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
10 / 26
What is next?
Higher dimension (rank).
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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What is next?
Higher dimension (rank).
Less symmetry.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
11 / 26
What is next?
Higher dimension (rank).
Less symmetry.
Change the ambient space.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
11 / 26
The 3-torus
Consider a group Λ generated by 3 linearly independent translations of E3 .
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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The 3-torus
Consider a group Λ generated by 3 linearly independent translations of E3 .
The 3-torus associated to Λ (denoted by T3 (Λ)) is the quotient space
E3 /Λ.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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The problem
Determine the groups Λ such that a regular polyhedron in E3 induces a
regular polyhedron in T3 (Λ).
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
13 / 26
Let Λ be a group generated by 3 linearly independent translations by the
vectors v1 , v2 and v3 . Let o the origin of E3 , we define de lattice Λ
associated to Λ as the set
Λ = oΛ = {n1 v1 + n2 v2 + n3 v3 : ni ∈ Z}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
14 / 26
Let Λ be a group generated by 3 linearly independent translations by the
vectors v1 , v2 and v3 . Let o the origin of E3 , we define de lattice Λ
associated to Λ as the set
Λ = oΛ = {n1 v1 + n2 v2 + n3 v3 : ni ∈ Z}
Some examples:
Cubic Lattice: Λ(1,0,0) .
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
14 / 26
Let Λ be a group generated by 3 linearly independent translations by the
vectors v1 , v2 and v3 . Let o the origin of E3 , we define de lattice Λ
associated to Λ as the set
Λ = oΛ = {n1 v1 + n2 v2 + n3 v3 : ni ∈ Z}
Some examples:
Cubic Lattice: Λ(1,0,0) .
Body-centred Lattice: Λ(1,1,1) .
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
14 / 26
Let Λ be a group generated by 3 linearly independent translations by the
vectors v1 , v2 and v3 . Let o the origin of E3 , we define de lattice Λ
associated to Λ as the set
Λ = oΛ = {n1 v1 + n2 v2 + n3 v3 : ni ∈ Z}
Some examples:
Cubic Lattice: Λ(1,0,0) .
Body-centred Lattice: Λ(1,1,1) .
Face-centred Lattice: Λ(1,1,0) .
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
14 / 26
Lemma
Let Λ a group generated by 3 linearly independent translations. Let g = ts
an isometry of E3 (with s ∈ O(3) and t translation). The following are
equivalent:
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
15 / 26
Lemma
Let Λ a group generated by 3 linearly independent translations. Let g = ts
an isometry of E3 (with s ∈ O(3) and t translation). The following are
equivalent:
g induces an isometry ḡ : T3 (Λ) → T3 (Λ).
E3
T3 (Λ)
Tero, Daniel (CCM - UNAM)
g
ḡ
/ E3
/ T3 (Λ)
Regular Polyhedra in T3
SIGMAP 2014
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Lemma
Let Λ a group generated by 3 linearly independent translations. Let g = ts
an isometry of E3 (with s ∈ O(3) and t translation). The following are
equivalent:
g induces an isometry ḡ : T3 (Λ) → T3 (Λ).
E3
T3 (Λ)
g
ḡ
/ E3
/ T3 (Λ)
s preserves Λ.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
15 / 26
The results
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Tetrahedral Symmetry
Theorem
Let Λ be a group generated by 3 linearly independent translations. If P is
a regular polyhedron in E3 with tetrahedral or octahedral symmetry, then
P induces a regular polyhedron in T3 (Λ) if and only if
Λ ∈ {aΛ(1,0,0) , bΛ(1,1,0) , cΛ(1,1,1) }.
for some parameters a, b, c.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
17 / 26
Icosahedral Symmetry
Theorem (Crystallographic Restriction )
If G is a group of isometries in E3 that preserves a lattice, then G does
not contain rotations of periods other than 2, 3, 4 and 6.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
18 / 26
Icosahedral Symmetry
Theorem (Crystallographic Restriction )
If G is a group of isometries in E3 that preserves a lattice, then G does
not contain rotations of periods other than 2, 3, 4 and 6.
Theorem
Let P be a regular polyhedron in E3 with icosahedral symmetry. There is
not group Λ generated by 3 linearly independent translations such that P
is a regular polyhedron in T3 (Λ).
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Infinite Polyhedra
. . . too many vertices . . .
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Infinite Polyhedra
. . . too many vertices . . .
Λ 6 T(P)
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Pure Polyhedra
Theorem
Let Λ be a group generated by 3 linearly independent translations. If P is
an infinite pure regular polyhedron in E3 , then P induces a regular
polyhedron in Λ if and only if
Λ ∈ {aΛ(1,0,0) , bΛ(1,1,0) , cΛ(1,1,1) }.
for some discrete parameters a, b, c.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Planar and Blended Polyhedra
There is a distinguished plane with a planar tessellation associated.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
{4, 4}, {4, 4}#{} and {4, 4}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Planar and Blended Polyhedra
{4, 4}, {4, 4}#{} and {4, 4}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
{4, 4}, {4, 4}#{} and {4, 4}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
{4, 4}, {4, 4}#{} and {4, 4}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
{3, 6}, {3, 6}#{}, {3, 6}#{∞}, {6, 3}, {6, 3}#{} and {6, 3}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
{3, 6}, {3, 6}#{}, {3, 6}#{∞}, {6, 3}, {6, 3}#{} and {6, 3}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
{3, 6}, {3, 6}#{}, {3, 6}#{∞}, {6, 3}, {6, 3}#{} and {6, 3}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
23 / 26
Planar and Blended Polyhedra
{3, 6}, {3, 6}#{}, {3, 6}#{∞}, {6, 3}, {6, 3}#{} and {6, 3}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar and Blended Polyhedra
{3, 6}, {3, 6}#{}, {3, 6}#{∞}, {6, 3}, {6, 3}#{} and {6, 3}#{∞}
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Planar Polyhedra
Coxeter classified the regular maps of the 2-torus.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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Planar Polyhedra
Coxeter classified the regular maps of the 2-torus.
Our results generalize Coxeter’s, in the sense that every planar
polyhedron in T3 is an embedding of a regular toroidal map.
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
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24 / 26
What have we done and what’s next?
We classify the regular polyhedra in T3 that come from regular
polyhedra in E3 .
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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What have we done and what’s next?
We classify the regular polyhedra in T3 that come from regular
polyhedra in E3 .
Is the list complete?
Tero, Daniel (CCM - UNAM)
Regular Polyhedra in T3
SIGMAP 2014
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Thank you!
Figure: {4, 6|4}/Λ(1,1,1)
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