The {6,4} sponge

In this vertex star, 4 hexagons, or 6-gons, are arranged regularly (all dihedral angles the same) around the central vertex. Since all of the polygons are the same and arranged similarly, this vertex star is highly symmetric. As a result, it can be labeled in 10 different ways, much more than most vertex stars. And, also, as one might expect due to such a high degree of symmetry, each vertex star can only be connected in one way (ignoring the markings) so that only one geometric shape can be created. This shape is one of the three regular Coxeter-Petrie sponges. (The other two are the regular {4,6} sponge and the {6,6} sponge.) The 10 vertex symbol labelings combine to produce 29 different labeled versions.

The regular {6,4} sponge   (

  1.    [a a^ a a^;  a]
  2.    [a+ a^+ a+ a^+;  a+]
  3.    [a+ a^+ a+ a^+;  a-]
  4.    [a+ a^- a+ a^-;  a+]
  5.    [a+ a^- a+ a^-;  a-]
  6.    [a b a b;  a b]
  7.    [a b a b;  b^ a^]
  8.    [a+ b+ a+ b+;  a+ b+]
  9.    [a+ b+ a+ b+;  a- b-]
  10.    [a+ b+ a+ b+;  b^+ a^+]
  11.    [a+ b+ a+ b+;  b^- a^-]
  12.    [a+ b+ b^+ a^+;  a+ b-]
  13.    [a+ b+ b^+ a^+;  a- b-]
  14.    [a+ b+ b^+ a^+;  b^- a^-]
  15.    [a+ a^+ b+ b^+;  a+ a^+ b-]
  16.    [a+ a^+ b+ b^+;  a- a^- b-]
  17.    [a+ a^+ b+ b^+;  b- b^- a-]
  18.    [a b+ c b-;  a b- c]
  19.    [a b+ c b-;  c b+ a]
  20.    [a+ b a- c;  a- b a+ c]
  21.    [a+ b a- c;  a+ c a- b]
  22.    [a+ b+ c+ d+;  a+ b+ c- d-]
  23.    [a+ b+ c+ d+;  a- b- c- d-]
  24.    [a+ b+ c+ d+;  a+ b+ d^- c^-]
  25.    [a+ b+ c+ d+;  a- b- d^+ c^+]
  26.    [a+ b+ c+ d+;  a+ d+ c+ b+]
  27.    [a+ b+ c+ d+;  b^+ a^+ d^- c^-]
  28.    [a+ b+ c+ d+;  b^- a^- d^- c^-]
  29.    [a+ b+ c+ d+;  c- d- a- b-]

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Last updated: April 18, 2019