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 Winter 1 / 20 www.tu-chemnitz.de
Symmetries of Polytopes
Symmetries of Polytopes Regular 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 Regular 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 Regular 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 Vertex-transitive polytopes Theorem. ( Babai , 1977; Ladisch , 2014) 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 Vertex-transitive polytopes Theorem. ( Babai , 1977; Ladisch , 2014) 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 Vertex-transitive polytopes Theorem. ( Babai , 1977; Ladisch , 2014) 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
Edge-transitive polytopes Edge-transitivity in R 3 Theorem. ( Gr¨ unbaum & Shephard , 1987) There are nine edge-transitive polyhedra. DiscMath · 08. November, 2019 · Martin Winter 4 / 20 www.tu-chemnitz.de
Edge-transitive polytopes Starting a classification ... 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 Just edge-transitive polytopes rhombic dodecahedron rhombic triacontahedron DiscMath · 08. November, 2019 · Martin Winter 6 / 20 www.tu-chemnitz.de
Edge-transitive polytopes Just 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 Just edge-transitive polytopes Some thoughts: ◮ Edge graph must be bipartite = ⇒ 2-faces are 2 n -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 Just edge-transitive polytopes Some thoughts: ◮ Edge graph must be bipartite = ⇒ 2-faces are 2 n -gons. ◮ Zonotopes might be a good place to start looking. Theorem. ( W. , 2019+) 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
Arc-transitive polytopes Arc-transitivity in R 3 There are seven arc-transitive polyhedra: DiscMath · 08. November, 2019 · Martin Winter 8 / 20 www.tu-chemnitz.de
Arc-transitive polytopes Arc-transitivity in R 4 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 Arc-transitivity in R 4 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 Arc-transitivity in R 4 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 Number of Wythoffian arc-transitive polytopes dim 1 2 3 4 5 6 7 8 9 10 11 12 13 ... irred. 1 0 7 15 11 19 22 25 19 21 23 25 27 ... prod. 0 0 0 0 0 6 0 14 6 10 0 38 0 ... prisms 0 ∞ 0 ∞ 0 ∞ 0 ∞ 0 ∞ 0 ∞ 0 ... ∞ ∞ ∞ ∞ ∞ ∞ � 1 0 7 15 11 25 22 39 25 31 23 63 27 ... #irred. ( n ) = 2 n + 1 , for n ≥ 9 . DiscMath · 08. November, 2019 · Martin Winter 13 / 20 www.tu-chemnitz.de
Arc-transitive polytopes Non-Wythoffian 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
Spectral methods Eigenpolytopes 0 1 0 · · · | | 1 0 ... u 1 · · · G = ⇒ = ⇒ θ 1 ≥ θ 2 ≥ · · · ≥ θ m = ⇒ u d 0 | | . ↑ . . u 1 , ..., u d ∈ R n DiscMath · 08. November, 2019 · Martin Winter 15 / 20 www.tu-chemnitz.de
Spectral methods Arc-transitive polytopes as eigenpolytopes Conjecture. An arc-transitive polytope P is the θ 2 -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 deg( G P ) ℓ = L ... Laplacian. 2 λ 2 ( L ) ◮ 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
Distance-transitive polytopes Distance-transitive graphs Definition. Distance-transitivity 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 Distance-transitive graphs DiscMath · 08. November, 2019 · Martin Winter 18 / 20 www.tu-chemnitz.de
Distance-transitive polytopes Distance-transitive polytopes Theorem. A distance-transitive polytope P is the θ 2 -eigenpolytope of its edge-graph. Consequences: Theorem. If P ⊂ R d 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 Classification of distance-transitive polytopes Theorem. ( Godsil , 1997; W. , 2019+) If P ⊂ R d 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 2 21 -polytope, ◮ the 7 -dimensional 3 21 -polytope. DiscMath · 08. November, 2019 · Martin Winter 20 / 20 www.tu-chemnitz.de
The End Questions?
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