## A variant of the Apollonian gasket with icosahedral symmetry

We modify the Apollonian gasket presented in the earlier post Apollonian gasket as a spherical fractal with tetrahedral symmetry. In an icosahedron, five triangles meet at their corners, which gives us a fivefold rotational symmetry. At the centers of the triangles we have a threefold rotational symmetry and at the middle of their edges a twofold one. The corresponding kaleidoscopic triangle thus has corner angles of 36, 60 and 9’0 degrees. Their sum is 186 degrees, just enough to make the triangle elliptic. Multiple reflection at the sides of this triangle makes a stereographic projection of a sphere with an icosahedral tiling :

Spherical tiling with icosahedral symmetry generated by reflections at the blue lines and circle shown in stereographic projection. The dashed line marks the projection of the equator.

Adding an inverting circle we get a fractal. It is a decoration of the icosahedral tiling :

Icosahedral fractal, shown in black, resulting from reflection at two lines and two circles. It decorates the icosahedral tiling, which is shown in yellow. This is a stereographic projection.

The icosahedral symmetry becomes evident using an inverse stereographic projection to a sphere. The normal projections of the upper and lower hemispheres are the same, except for a rotation by 36 degrees. This is a view of the lower hemisphere :

Normal view of a hemisphere with an icosahedral tiling (yellow) decorated by a variant of the Apollonian gasket (black).

Drawing both hemispheres of the gasket together we get :

Superposition of both hemispheres of a variant of the Apollonian gasket with icosahedral symmetry.

You can see how they nicely fit together.

The stars of five-fold rotational symmetry contain small copies of themselves. This results from a recursive packing of discs, quite similar to the Apollonian gasket. But now it is based on ideal pentagons instead of triangles:

The five discs, shown in green, leave a gap in the shape of a pentagon. Putting fifteen discs, which are coloured blue, in this gap we get eleven gaps. They are again pentagons and are filled in the same way.

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