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Category Archives: Tilings
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 … Continue reading
Posted in Kaleidoscopes, Tilings
Tagged hyperbolic space, mirror symmetry, periodic images, regular polygons, Rotational symmetry
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A rosette in a roman mosaic is an exponential transform of a periodic tiling
In the depot of the museum of Avenches (Switzerland) lies this interesting fragment of a large roman mosaic :You see immediately that this is a rosette with rotational symmetry, except for the fruit at the center. Looking closer we see an … Continue reading
Fractal tiling of a sphere with octahedral twocolour symmetry
The octahedron can have a nice twocolour symmetry. We get it from putting two tetrahedrons together, making a stellated octahedron. It is an eightpointed star and has already been discussed by Pacioli in his book “de divina proportione” in the … Continue reading
Posted in Fractals, Kaleidoscopes, Selfsimilarity, Tilings
Tagged color symmetry, fractal, kaleidoscope, octahedron, spherical tiling
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A fractal tiling of both octahedral and icosahedral symmetry
I want to show you a fractal tiling which can be seen as a decoration of a sphere with octahedral symmetry and at the same time as another decoration of a sphere with icosahedral symmetry. It arises as the limit … Continue reading
Posted in Fractals, Tilings
Tagged fractal, icosahedron, octahedron, Poincaré disc, Selfsimilarity, Tiling
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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
Posted in Kaleidoscopes, Selfsimilarity, Tilings
Tagged Apollonian gasket, fractal, icosahedral symmetry, spherical geometry, Tiling
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Apollonian gasket as a fractal in tiled hyperbolic space
Reading the fascinating book « Indra`s Pearls », written by David Mumford, Caroline Series and David Wright, you discover that the Apollonian gasket can be created by multiple inversions at four touching circles. Three of the circles are of equal … Continue reading
Posted in Fractals, Kaleidoscopes, Tilings
Tagged Apollonian gasket, fractal, hyperbolic geometry, kaleidoscope, Tiling
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Apollonian gasket as a spherical fractal with tetrahedral symmetry
Before discussing the relation between the Apollonian gasket and tilings of the sphere, I want to present briefly the spherical kaleidoscope with tetrahedral symmetry. A tetrahedron has three different kinds of points with rotational symmetry. Four equilateral triangles make up … Continue reading
Posted in Anamorphosis, Fractals, Kaleidoscopes, Tilings
Tagged Apollonian gasket, fractal, kaleidoscope, spherical geometry, tetrahedral symmetry, Tiling
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waves – a browser app for creating quasiperiodic wallpapers
I have made a browser app that lets you create quasiperiodic wallpapers. You find it at http://geometricolor.ch/waves.html . It uses a symmetric superposition of waves as proposed by Frank Farris and presented by Erica Klarreich in the Quantamagazin in “How … Continue reading
Inversion in a single circle
You might think that discussing the inversion in a circle is somewhat underwhelming. But, as I am using multiple inversion in many circles to create fractal images, I found that there are some important ideas you will not find so … Continue reading
Posted in Anamorphosis, Kaleidoscopes, programming, Tilings
Tagged inversion, inversion in a circle, mirror symmetry
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Various projections of hyperbolic kaleidoscopic images
Similarly to the earlier post “Different projections of spherical kaleidoscopic images” I am now showing the same kaleidoscopic image using different projections you can use in my kaleidoscope browser app http://geometricolor.ch/sphericalKaleidoscopeApp.html. It primarily generates images as Poincaré discs. A typical result … Continue reading