Fractal tiling of a sphere with octahedral two-colour symmetry

The octahedron can have a nice two-colour symmetry. We get it from putting two tetrahedrons together, making a stellated octahedron. It is an eight-pointed star and has already been discussed by Pacioli in his book “de divina proportione” in the 16th century. Leonardo da Vinci did the drawings for this book. Here is his image for the stellated octahedron with some additional colouring:

Drawing of the stellated octahedron by Leonardo da Vinci with additional colouring.

One of the tetrahedrons is coloured green and the other blue to show you how they are put together. You can see that each point of the star is surrounded by points of the other colour. The points are obviously smaller tetrahedrons. Cutting them away you get an octahedron. Give its sides the same colours as the points you have cut away from them. Then, the octahedron has a two-colour symmetry as a green triangle side now shares its edges only with blue triangles, and vice versa. Can we use inversion in circles and reflection at straight lines to obtain a spherical tiling with a similar two-colour symmetry?

We start with a spherical triangle that generates a tiling of the sphere with tetrahedral symmetry using reflection at its sides. It has an angle of 90 degrees and two angles of 60 degrees, which makes it isosceles. We add an inverting circle that crosses the two legs of the triangle at angles of 60 degrees and touches the hypothenuse. Then, multiple reflections at these elements gives us this:

Fractal tiling with two-colour symmetry obtained from multiple reflection at the yellow circles and lines. The dotted line shows the projection of the equator of a matching sphere.

Here, the triangle and circle are shown in yellow. You can see that the right angle of the basic triangle lies at the center of the image. The additional circle covers most of it, except for two small triangles. Multiple reflection at these four elements maps every point of the plane into one of these small triangles, which is discussed more in detail in “Fractal Images from Multiple Inversion in Circles”. In this image I have used green colours for points that get mapped to one triangle and blue colours for points that go to the other one.

The entire plane is covered by blue and green discs. They are Poincaré representations of hyperbolic space tiled by images of these triangles. Every disc touches only discs of the other colour and none of the same colour. Two green and two blue discs can form a closed chain with a gap inside. The gap contains two green and two blue discs that touch its sides. Together with the surrounding four discs we get five chains of discs and five gaps,  which again will contain four discs. Thus arises a fractal self-similar structure with a two colour symmetry, as a rotation by 90 degrees exchanges green and blue colours.

The octahedral symmetry becomes obvious if we make an inverse stereographic projection to a sphere. Its equator is shown in the image above as a dotted yellow line. Matching two input images to the two tiling triangles we get a fractal kaleidoscopic image, such as this one:

Sphere with a kaleidoscopic decoration of octahedral two-colour symmetry.

On this hemisphere you see four large discs. Their borders are incircles of the triangular sides of an octahedron with two-colour symmetry. The gaps in-between are filled by images of these circles, making a fractal decoration of the sphere with octahedral two-colour symmetry. The opposite hemisphere looks alike except for an exchange of the blue and green discs.

This is an especially symmetric result of multiple reflections. But it is quite similar to other less symmetric configurations of the reflecting elements. You can create similar images with my public browser app at http://geometricolor.ch/circleInTriangle.html and GIMP.

This entry was posted in Fractals, Kaleidoscopes, Self-similarity, Tilings and tagged , , , , . Bookmark the permalink.

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