Category Archives: Self-similarity

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 … Continue reading

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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 … Continue reading

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Color symmetry using the length scale of the inflated lattice

I have shown some images with 2-color symmetry upon rotation shown in “images of 10-fold rotational …“. But the fast color changes they hacked them into small pieces. We can get better images if we use a color changing function with … Continue reading

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Self-similarity and color modification

The Penrose tiling is self-similar as many other quasi-periodic tilings. It matches a copy of itself inflated by the golden ratio τ=(1+√5)/2≅1.618, see “Penrose tiling tied up in ribbons“. Noting that our quasi-periodic designs of 5-fold symmetry are closely related to … Continue reading

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Coloring the Julia set

The Julia set of a function f(z) in the complex plane has all points z that remain finite upon iterations of the function. In the last posts I have used expanding functions to get fractal images from iteration, as discussed … Continue reading

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Rainbow colors

We can define a continuous number x of iterations needed to reach the critical radius R. Note that if the n-th iteration of f(z) equals R then x=n, and if the (n-1)th iteration equals R then x=n-1. For values in-between … Continue reading

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self-similar fractals with rotational symmetry from function iteration

I was looking at my posts of march 2013 on complex function iterations, see in particular “fractal surprise from complex function iteration” and “self-similar images from iterated mappings of the plane“, and I got some new ideas I want to … Continue reading

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