Edge- Transitive Polytopes Martin Winter Professorship for - - PowerPoint PPT Presentation
Edge- Transitive Polytopes Professorship for Algorithmic and Discrete Mathematics Edge- Transitive Polytopes Martin Winter Professorship for Algorithmic and Discrete Mathematics 08. November, 2019 DiscMath 08. November, 2019 Martin
Edge- Transitive Polytopes Professorship for Algorithmic and Discrete Mathematics
Martin Winter
Professorship for Algorithmic and Discrete Mathematics
DiscMath · 08. November, 2019 · Martin Winter 1 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
(classification: Schl¨ afli, 1852)
dim 2 3 4 5 6 7 8 9 · · · # ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s
DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
(classification: Schl¨ afli, 1852)
dim 2 3 4 5 6 7 8 9 · · · # ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s
Definition.
◮ regular := flag-transitive flag := (vertex ⊂ edge ⊂ face).
DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
(classification: Schl¨ afli, 1852)
dim 2 3 4 5 6 7 8 9 · · · # ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s
Definition.
◮ regular := flag-transitive flag := (vertex ⊂ edge ⊂ · · · ⊂ facet).
DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Almost every finite group is the symmetry group of a vertex-transitive polytope.
DiscMath · 08. November, 2019 · Martin Winter 3 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Almost every finite group is the symmetry group of a vertex-transitive polytope. Examples: Birkhoff polytope, TSP polytopes, ...
DiscMath · 08. November, 2019 · Martin Winter 3 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Almost every finite group is the symmetry group of a vertex-transitive polytope. Examples: Birkhoff polytope, TSP polytopes, ... Keywords: orbit polytopes, representation polytopes, ...
DiscMath · 08. November, 2019 · Martin Winter 3 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
unbaum & Shephard, 1987) There are nine edge-transitive polyhedra.
DiscMath · 08. November, 2019 · Martin Winter 4 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Question.
Are there half-transitive polytopes?
What about half-transitive abstract polytopes?
DiscMath · 08. November, 2019 · Martin Winter 5 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
rhombic dodecahedron rhombic triacontahedron
DiscMath · 08. November, 2019 · Martin Winter 6 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
rhombic dodecahedron rhombic triacontahedron
Question.
Are there just edge-transitive polytopes in d ≥ 4 dimensions?
DiscMath · 08. November, 2019 · Martin Winter 6 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Some thoughts:
◮ Edge graph must be bipartite = ⇒ 2-faces are 2n-gons. ◮ Zonotopes might be a good place to start looking.
DiscMath · 08. November, 2019 · Martin Winter 7 / 20 www.tu-chemnitz.de
Edge-transitive polytopes
Some thoughts:
◮ Edge graph must be bipartite = ⇒ 2-faces are 2n-gons. ◮ Zonotopes might be a good place to start looking.
There are no just edge-transitive zonotopes in ≥ 4 dimensions.
DiscMath · 08. November, 2019 · Martin Winter 7 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
There are seven arc-transitive polyhedra:
DiscMath · 08. November, 2019 · Martin Winter 8 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
There are 15 known edge-transitive 4-polytopes:
◮ six regular 4-polytopes
→ 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,
◮ five rectifications
→ of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),
◮ two bitruncations
→ of 4-simplex and 24-cell,
◮ two runcinations
→ of 4-simplex and 24-cell,
+ an infinite family of (p, p)-duoprisms.
DiscMath · 08. November, 2019 · Martin Winter 9 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
There are 15 known edge-transitive 4-polytopes:
◮ six regular 4-polytopes
→ 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,
◮ five rectifications
→ of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),
◮ two bitruncations
→ of 4-simplex and 24-cell,
◮ two runcinations
→ of 4-simplex and 24-cell,
+ an infinite family of (p, p)-duoprisms. All of these are uniform polytopes.
DiscMath · 08. November, 2019 · Martin Winter 9 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
There are 15 known edge-transitive 4-polytopes:
◮ six regular 4-polytopes
→ 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,
◮ five rectifications
→ of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),
◮ two bitruncations
→ of 4-simplex and 24-cell,
◮ two runcinations
→ of 4-simplex and 24-cell,
+ an infinite family of (p, p)-duoprisms. All of these are uniform polytopes. (in fact, Wythoffian)
DiscMath · 08. November, 2019 · Martin Winter 9 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
DiscMath · 08. November, 2019 · Martin Winter 10 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
DiscMath · 08. November, 2019 · Martin Winter 11 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
DiscMath · 08. November, 2019 · Martin Winter 12 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
dim 1 2 3 4 5 6 7 8 9 10 11 12 13 ... irred. 1 7 15 11 19 22 25 19 21 23 25 27 ... prod. 6 14 6 10 38 ... prisms ∞ ∞ ∞ ∞ ∞ ∞ ...
∞
7
∞
15 11
∞
25 22
∞
39 25
∞
31 23
∞
63 27 ...
#irred.(n) = 2n + 1, for n ≥ 9.
DiscMath · 08. November, 2019 · Martin Winter 13 / 20 www.tu-chemnitz.de
Arc-transitive polytopes
Are these lists complete?
Question.
Are there non-uniform arc-transitive polytopes. ... or stronger ...
Question.
Are there non-Wythoffian arc-transitive polytopes.
DiscMath · 08. November, 2019 · Martin Winter 14 / 20 www.tu-chemnitz.de
Spectral methods
G = ⇒ 1 · · · 1 ... . . . = ⇒ θ1 ≥ θ2 ↑ u1, ..., ud ∈ Rn ≥ · · · ≥ θm = ⇒ | | u1 · · · ud | |
DiscMath · 08. November, 2019 · Martin Winter 15 / 20 www.tu-chemnitz.de
Spectral methods
Conjecture.
An arc-transitive polytope P is the θ2-eigenpolytope of its edge-graph.
Consequences:
◮ P is uniquely determined by its edge-graph. ◮ P realizes all the symmetries of its edge-graph. ◮ For edge-length ℓ and circumradius r holds
r ℓ =
2λ2(L)
L ... Laplacian. ◮ P is a perfect polytope. ◮ Every projection of P is either not arc-transitive, or has a different edge-graph.
DiscMath · 08. November, 2019 · Martin Winter 16 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
A graph G is distance-transitive, if for any i, j,ˆ ı, ˆ ∈ V (G) with dist(i, j) = dist(ˆ ı, ˆ ) there is a φ ∈ Aut(G) with φ(i) = ˆ ı and φ(j) = ˆ .
DiscMath · 08. November, 2019 · Martin Winter 17 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
DiscMath · 08. November, 2019 · Martin Winter 18 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
Theorem.
A distance-transitive polytope P is the θ2-eigenpolytope of its edge-graph.
Consequences:
Theorem.
If P ⊂ Rd is a distance-transitive polytope, then ◮ P is the unique distance-transitive polytope with this edge-graph. ◮ P realizes all the symmetries of its edge-graph.
DiscMath · 08. November, 2019 · Martin Winter 19 / 20 www.tu-chemnitz.de
Distance-transitive polytopes
If P ⊂ Rd is a distance-transitive polytope, then P is one of the following: ◮ a regular polygon, ◮ the icosahedron, ◮ the dodecahedron, ◮ a cross-polytope, ◮ a hyper-simplex (this includes regular simplices), ◮ a demi-cube, ◮ a cartesian power of a simplex (this includes hypercubes), ◮ the 6-dimensional 221-polytope, ◮ the 7-dimensional 321-polytope.
DiscMath · 08. November, 2019 · Martin Winter 20 / 20 www.tu-chemnitz.de
Questions?