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 Apollonian gasket as a fractal in tiled hyperbolic space
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Monthly Archives: June 2017
Quasiperiodic design with 8fold rotational symmetry from 4dimensional space
Using the recipe of the last post for fourdimensional space (p=4) I got this image of 8fold rotational symmetry: A center of approximate 8fold rotational symmetry is near the lower left corner. Large brown patches appear at roughly equal distances. … Continue reading
Rotational symmetry from space with an even number of dimensions
For an embedding space with an even number of dimensions p=2q we do similarly as for an odd number of dimensions, see the earlier post “Rotational symmetry from…“. Note that now we should not use an angle of 2π/p between … Continue reading
Posted in Anamorphosis, Kaleidoscopes
Tagged anamorphosis, kaleidoscope, Rotational symmetry
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Periodic design with 3fold rotational symmetry from 3dimensional space
Three dimensional space gives a threefold rotational symmetry in the drawing plane. The designs are periodic. Note that if you put a cube on one of its points and look along its spacediagonal from above, then you see an object with … Continue reading
Posted in Anamorphosis, Kaleidoscopes, Tilings
Tagged kaleidoscope, Rotational symmetry, translational symmetry
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Inversion symmetry doubles the rotation symmetry for an odd number of dimensions
We now want to impose inversion symmetry in addition to rotational symmetry on our designs. This means that the mapping functions should not change upon inversion of the position. Thus X(x,y)=X(x,y) and Y(x,y)=Y(x,y). Let’s consider space with an odd number … Continue reading
quasiperiodic patterns of 5fold symmetry from 5 dimensional space
I now want to see some images. Using a photo of a caterpillar as input image I get I used the simplest quasiperiodic mapping functions resulting from the theory of the last post and A pixel at position (x,y) gets … Continue reading
Posted in Anamorphosis, Kaleidoscopes, Quasiperiodic design
Tagged anamorphosis, Art, Quasiperiodic design, Rotational symmetry
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Rotational symmetry from space with an odd number of dimensions
We now look at the easier case for the post “Quasiperiodic and periodic kaleidoscope from higher dimensional space“, where the embedding space has an odd number of dimensions, p=2q+1. The unit vectors lie at equal angles and form a star … Continue reading
Rotations, mirrorsymmetry and the scalar product
In the last post we have seen that scalar products between a pixel’s position in the output image and certain vectors e define periodic and quasiperiodic designs. We want symmetric images and thus we have to see how the scalar product changes … Continue reading
Posted in Anamorphosis, Kaleidoscopes, programming, Tilings
Tagged Math, mirror symmetry, Rotational symmetry
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