Edge-Transitive Polytopes - TU Chemnitz wimart/slides/ ¢  Symmetries of Polytopes Regular

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  • 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

    www.tu-chemnitz.de

  • Symmetries of Polytopes

  • Symmetries of Polytopes

    Regular polytopes (classification: Schläfli, 1852)

    dim 2 3 4 5 6 7 8 9 · · · # ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s

    Definition. I regular := flag-transitive

    flag := (vertex ⊂ edge ⊂ face).

    DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de

    www.tu-chemnitz.de

  • Symmetries of Polytopes

    Regular polytopes (classification: Schläfli, 1852)

    dim 2 3 4 5 6 7 8 9 · · · # ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s

    Definition. I regular := flag-transitive

    flag := (vertex ⊂ edge ⊂ face).

    DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de

    www.tu-chemnitz.de

  • Symmetries of Polytopes

    Regular polytopes (classification: Schläfli, 1852)

    dim 2 3 4 5 6 7 8 9 · · · # ∞ 5 6 3 3 3 3 3 · · · ← only more 3-s

    Definition. I regular := flag-transitive

    flag := (vertex ⊂ edge ⊂ · · · ⊂ facet).

    DiscMath · 08. November, 2019 · Martin Winter 2 / 20 www.tu-chemnitz.de

    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

    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

    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

    www.tu-chemnitz.de

  • Edge-transitive polytopes

  • Edge-transitive polytopes

    Edge-transitivity in R3

    Theorem. (Grünbaum & Shephard, 1987) There are nine edge-transitive polyhedra.

    DiscMath · 08. November, 2019 · Martin Winter 4 / 20 www.tu-chemnitz.de

    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

    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

    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

    www.tu-chemnitz.de

  • Edge-transitive polytopes

    Just edge-transitive polytopes

    Some thoughts:

    I Edge graph must be bipartite =⇒ 2-faces are 2n-gons.

    I 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

    www.tu-chemnitz.de

  • Edge-transitive polytopes

    Just edge-transitive polytopes

    Some thoughts:

    I Edge graph must be bipartite =⇒ 2-faces are 2n-gons.

    I 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

    www.tu-chemnitz.de

  • Arc-transitive polytopes

  • Arc-transitive polytopes

    Arc-transitivity in R3

    There are seven arc-transitive polyhedra:

    DiscMath · 08. November, 2019 · Martin Winter 8 / 20 www.tu-chemnitz.de

    www.tu-chemnitz.de

  • Arc-transitive polytopes

    Arc-transitivity in R4

    There are 15 known edge-transitive 4-polytopes:

    I six regular 4-polytopes → 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,

    I five rectifications → of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),

    I two bitruncations → of 4-simplex and 24-cell,

    I 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

    www.tu-chemnitz.de

  • Arc-transitive polytopes

    Arc-transitivity in R4

    There are 15 known edge-transitive 4-polytopes:

    I six regular 4-polytopes → 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,

    I five rectifications → of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),

    I two bitruncations → of 4-simplex and 24-cell,

    I 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

    www.tu-chemnitz.de

  • Arc-transitive polytopes

    Arc-transitivity in R4

    There are 15 known edge-transitive 4-polytopes:

    I six regular 4-polytopes → 4-simplex, 4-cube, 4-crosspolytope, 24-, 120- and 600-cell,

    I five rectifications → of 4-simplex, 4-cube, 24-, 120- and 600-cell (rect. 4-crosspolytop = 24-cell),

    I two bitruncations → of 4-simplex and 24-cell,

    I 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

    www.tu-chemnitz.de

  • Arc-transitive polytopes

    DiscMath · 08. November, 2019 · Martin Winter 10 / 20 www.tu-chemnitz.de

    www.tu-chemnitz.de

  • Arc-transitive polytopes

    DiscMath · 08. November, 2019 · Martin Winter 11 / 20 www.tu-chemnitz.de

    www.tu-chemnitz.de

  • Arc-transitive polytopes

    DiscMath · 08. November, 2019 · Martin Winter 12 / 20 www.tu-chemnitz.de

    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) = 2n+ 1, for n ≥ 9.

    DiscMath · 08. November, 2019 · Martin Winter 13 / 20 www.tu-chemnitz.de

    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

    www.tu-chemnitz.de

  • Spectral methods

  • Spectral methods

    Eigenpolytopes

    G =⇒

     0 1 0 · · · 1 0 0

    . . . ...

     =⇒ θ1 ≥ θ2 ↑

    u1, ..., ud ∈ Rn

    ≥ · · · ≥ θm =⇒

     | |u1 · · · ud | |

    

    DiscMath · 08. November, 2019 · Martin Winter 15 / 20 www.tu-chemnitz.de

    www.tu-chemnitz.de

  • Spectral methods

    Arc-transitive polytopes as eigenpolytopes

    Conjecture.

    An arc-transitive polytope P is the θ2-eigenpolytope of its edge-graph.

    Consequences:

    I P is uniquely determined by its edge-graph.

    I P realizes all the symmetries of its edge-graph.

    I For edge-length ` and circumradius r holds

    r

    ` =

    √ deg(GP )

    2λ2(L) L ... Laplacian.

    I P is a perfect polytope.

    I Every projection of P is either not arc-transitive