Last update: 2012/07/01

[Japanese] [English]
I discovered new 13 uniform tessellations on the Euclid plane (more detail).

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 (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.**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.**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.

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

**|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.

- Convex (general)
**|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**.**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.**|p (q r|)**

The vertex figure is [p,4,2*q,2*r,4]. This is one kind of special snub tessellations.**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.

- Convex (general)

since 2002/10/18

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