**Hyperbolic Star Tessellations **

Starred tessellations also exist on the Hyperbolic plane. Regular, semi-regular and uniform tessellations can all be generated.

An excellent introduction to the Hyperbolic Plane and beautiful images of regular (non-starred) tessellations has been generated by Don Hatch (one of the co-authors of Tyler) - it is linked here. See also Hironori Sakamoto's World of Uniform Tessellations.

**Regular Star Tessellations**

*(click on the image for a larger version)*

The above image shows a {7/2,
7} tessellaton. The notation {p,q} meaning that q p-gons are arranged
around each vertex. A regular tessellation of the form {^{p}/_{2}, p} exists for
all p>=7 where p is odd. It may not be easy to see the pattern of ^{7}/_{2}-gons from the above image, in which case a second image is presented here
which has one vertex and its surrounding ^{7}/_{2}-gons highlighted. Note that
as the dual of a {p,q} is a {q,p} then tessellations also exist of the form {p, ^{p}/_{2}} i.e. p p-gons wrapped twice around each vertex - the grid for the {p,
^{p}/_{2}}
is the same as that for {3, p} - e.g. this {3,7} contains
{7, ^{7}/_{2}}..

For p>=8 and p even, ^{p}/_{2} does not generate a
star polygon, however if ^{p}/_{2} is regarded as two '^{p}/_{2}'-gons with one gyrated by
2*pi/p with regard to their common centre, (think of a ^{6}/_{2}-gon as a Star
of David - examples of 6/2, 8/2 and 10/2), then a family of 'pseudo'-regular tessellations can be generated, for
example this {8/2, 8}

Interestingly, the form {^{p}/_{2}, p} also extends for
n<7. For n=6, **{ ^{6}/_{2},6}** is a uniform (planar) tessellation of

**Semi-regular Star Tessellations**

*(click on the image for a larger version)*

The above image shows a tessellation with a vertex figure of 7/2, 3, 7/2, 3, 7/2, 3. It is one of an infinite number of such figures.

Other examples of such tessellations are:

7/2, 7, 7/2, 7

7/3, 7, 7/3, 7, 7/3,
7

7/2, 4, 7/2, 4,
7/2, 4, 7/2, 4

and to show that not all such tessellations have
to have heptagons:

5/2, 4, 5/2, 4,
5/2, 4, 5/2, 4

**Uniform Star Tessellations**

*(click on the image for a larger version)*

The above image shows a tessellation with a
vertex figure of 4, 7/3, 4, inf. It is formed by taking a {7, 3}
tessellation and replacing the heptagons with ^{7}/_{3}-gons sized such that their
edges can be joined by squares.

Thanks again to Melinda Green and Don Hatch's Tyler applet which was used to generate all of these images.