## Regular polygons as kaleidoscopes

We are using reflection at the sides of a regular polygon to get a space filling periodic image. Its symmetries depend on the symmetry of the image, which lies inside the polygon. As an example, let us look at the equilateral triangle. Patching an image without any particular symmetry on the triangle we get this: The black lines show the sides of the triangle. We see that the reflections generate three-fold rotational symmetries at each corner of the triangle. Each corner has a different appearance. Thus the symmetry of the image has the orbifold symbol *333. A square gives a similar result, which you can work out yourselves.

In general, we want that the reflections at the sides of the polygon generate dihedral groups with n-fold rotational symmetry. The intersection angles of the sides then has to be an integer fraction of 180 degrees. This means that the corner angles cannot be larger than 90 degrees. This makes it impossible to use polygons with more than 4 corners, if they have straight lines as sides and we can only have a three-fold rotational symmetry at the corner of triangles. Can we get more ?

We can use circle arcs as sides of polygons and inversion in these circles as reflections. This allows us to use polygons with any number m of corners getting any n-fold rotational symmetry at the corners. An example is a regular triangle with corner angles of 45 degrees. Again patching an image without symmetry on the triangle gives us : The inversions reduce the size of the basic triangle and distort its image. Thus the resulting image is limited to the inside of a disc, which is actually a representation of a periodic decoration of bent hyperbolic space. The reflections now make four-fold rotational symmetries at the three corners. Each is different and thus this image has the symmetry *444.

For a more symmetric image we can put a rosette with three-fold rotational symmetry on the basic triangle. This is such a result : We see that the image two different centers of symmetry : A three-fold rotational symmetry at the center of the triangle and a three-fold dihedral symmetry at its corners, which have all the same design. The orbifold symbol is thus 3*4.

As a last possibility we can use an image with a three-fold dihedral symmetry the center of the triangle : Now we have points of eight-fold, three-fold rotational symmetry and two-fold rotational symmetry. They all have mirror symmetry too and thus the orbifold symbol is now *832.

Note that regular polygons always result in periodic images. Most belong to hyperbolic space, but there are two in flat Euclidean and one in elliptic space. Using an irregular polygon we can get more interesting fractal images. As in this post, we can add symmetries compatible with the polygon. This results in images of different appearance, although they all use the same tiling.

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