Uniform Tessellations - Complex

Based on (p,q,r)

Based on (p,q,r,...,s)

Complex

• |p q (r|s)
  The vertex figure is [p,3,q,3,2*s,r,2*s,3]. Two types of tessellations |p q * and * r|s are combined.
  • Convex (general)

    |2 2 (2|2)

    |2 2 (2|3)

    |2 2 (3|2)

    |2 3 (2|2)

  • Non-convex (special)

    [3,3,3,4/3,4/3]	[3/2,Inf,3/2,Inf, 3/2,4,4]	[3/2,Inf,3/2,Inf, 3/2,4/3,4/3]
    [3/2,Inf,3/2,Inf, 3/2,12/5,6,12/5]	[3/2,Inf,3/2,Inf, 3/2,12/7,6/5,12/7]	[3/2,Inf,3/2,Inf, 3/2,12,6/5,12]	[3/2,Inf,3/2,Inf, 3/2,12/11,6,12/11]
    [3/2,Inf,3/2,4/3,4/3, 3/2,4/3,4/3]	[3/2,Inf,3/2,4,4, 3/2,4,4]

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• |p q r|s
  The vertex figure is [p,3*s,q,3*s,r,3*s]. They are snub tessellations which snub faces are 3*s-gon.
  • Convex (general)

    |2 2 3|2

    |2 2 4|2

    |2 3 3|2

    |2 3 4|2

• p|(|q r s)
  The vertex figure is p*[q,3,r,3,s,3]. The snub tessellation |p q r - vertex figure is [q,3,r,3,s,3] - is repeated p times around the vertex.
  • Convex (general)

    2|(|2 2 4)

    2|(|2 3 4)

    3|(|2 2 4)

• |p (q r|)
  The vertex figure is [p,4,2*q,2*r,4]. This is one kind of special snub tessellations.
  • Convex (general)

    (|2 (2 2|)) => 4|2 4

    |2 (2 3|)

    (|3 (2 2|)) => 2 3|2 2

    |3 (2 3|)

• More complex
  The above complex types can be combined, then more complex uniform tessellations are generated.
  • Convex (general)

  • Non-convex (special)

    [3/2,Inf,3/2,4,4, 3/2,4/3,4/3]

    [4,Inf,4/3, 8/3,4,8/3]

    [4,Inf,4/3, 8,4/3,8]

    [4,Inf,4/3, 8/3,8]

    [4,Inf,4/3, 8,8/3]

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