Mandara

- The World of Uniform Tessellations -

There are Uniform Tessellations on the Hyperbolic plane which are similar to Uniform Polyhedra or Uniform Tessellations on the Euclid plane. I classified them into the following types.

Based on (p,q,r)

• Based on (2,2,p)
  The figures based on (2,2,p) are Prisms or Antiprisms.

  

• Based on (2,3,p)
  The figures based on (2,3,3), (2,3,4) or (2,3,5) are Uniform Polyhedra. The figures based on (2,3,6) are Uniform Tessellations on the Euclid plane. The figures based on (2,3,p>=7) are Uniform Tessellations on the Hyperbolic plane.
  • Convex (general)
    
  • Non-convex (general)
    
  • Non-convex (based on (3,7,7/2))
    
  • Non-convex (special)
    

• Based on (2,p,q)
  The figures based on (2,4,4) are Uniform tessellations on the Euclid plane. The others are Uniform Tessellations on the Hyperbolic plane.
  • Convex (general)
    
  • Non-convex (general)
    
  • Non-convex (special)
    

• Based on (p,q,r)
  The figures based on (3,3,3) are Uniform tessellations on the Euclid plane. The others are Uniform Tessellations on the Hyperbolic plane.
  • Convex (general)
    
  • Non-convex (special)
    

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

• Based on (p,q,r,s)
  The figures based on (2,2,2,2) are Uniform Tessellations on the Euclid plane. The others are Uniform Tessellations on the Hyperbolic plane.
  • Convex (general)
    
  • Non-convex (special)
    

• Based on (p,q,r,s,...,t)
  • Convex (general)
    

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)
    
  • Non-convex (special)
    There are non-convex "complex" tessellations on the Euclid plane.
    
    

    More detail

• |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)
    

• 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)
    

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

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

  • Non-convex (special)
    There are non-convex "complex" tessellations on the Euclid plane.
    
    

    More detail