__Infinite and Semi-infinite tessellations__

In addition to the 'traditional' eleven tessellations of the plane. Seven further tessellations can be generated if infinite polygons (or 'apeirogons') are permitted. An apeirogon has a dihedral angle of pi and appears as a straight line.

If the apeirogons are regarded as being normal to the plane, they appear to be synonymous with the hemi-spherical polygons in uniform polyhedra.

Note: A number of these tessellations involve retrograde polygons. All apeirogons are regarded as prograde and are listed in the vertex figures as "Inf".

__Infinite tessellation 1__

Inf, Inf.

**K**:-, **B**:-, **C**:-

This is a trivial case of two apeirogons joined at their edges. It is actually a fourth regular tessellation. This and the next two tessellations could be regarded as families of tessellations if it is taken that an apeirogon can cycle its centre more than once, meaning there are an infinite series of apeirogons each of which could form one of these tessellations.

__Infinite tessellation 2__

Inf, 4, 4

**K**:-, **B**:59, **C**: 2 oo | 2 = t{2,oo}

The apeirogons in case 1 can be separated by a band of squares to form a semi-regular tessellation. With the apeirogons normal to the plane this becomes an infinite prism.

__Infinite tessellation 3__

Inf, 3, 3, 3

**K**:-, **B**:60, **C**: | 2 2 oo = s{2 : oo}

Separating the apeirogons by a band of triangles forms another semi-regular tessellation. With the apeirogons normal to the plane this becomes an infinite anti-prism.

__Infinite tessellation ____4__

Inf, 4, Inf, ^{4}/_{3}

**K**:E, **B**:-, **C**: 4/3 4 | oo

From this tessellation on, we start to introduce retrograde polygons. The edge/vertex network of the 4,4,4,4 tessellation forms the basis for the above semi-regular tessellation. Only the coloured squares are present.

__Infinite tessellation ____5__

( Inf, 3, Inf, 3, Inf, 3 ) / 2

**K**:D, **B**:-, **C**: 3/2 | 3 oo

The edge/vertex network of the 3,3,3,3,3,3 tessellation forms the basis for the above semi-regular tessellation. Only the coloured triangles are present. This is the only tessellation to date which requires the '2' divisor as the vertex figure encircles the vertex twice.

__Infinite tessellation ____6__

Inf, 6, Inf, ^{6}/_{5}

**K**:P, **B**:62, **C**: 6/5 6 | oo

The edge/vertex network of the 6,3,6,3 tessellation forms the basis for the above semi-regular tessellation. Only the coloured hexagons are present.

__Infinite tessellation ____7__

Inf, 3, Inf, ^{3}/_{2}

**K**:P, **B**:62, **C**: 3/2 3 | oo

Finally the edge/vertex network of the 6,3,6,3 tessellation also forms the basis for the above semi-regular tessellation. Only the coloured triangles are present.

Apeirogons also occur in a number of the 'star' tessellations.

**Next: 'Star' and
'Retrograde' Tessellations
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to tessellations
**