Category Archives: Anamorphosis

further wallpapers for hyperbolic space

An equilateral triangle gives us a kaleidoscope of three-fold rotational symmetry. With a square we get two-fold rotational symmetry. Would reflection at the sides of other regular polygons too give periodic images with rotational symmetry ? To get an h-fold … Continue reading

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How to program fast kaleidoscopes

This post repeats parts of earlier posts but I am trying to expand the ideas and explain them better. First, I am showing you how to make rosettes with rotational symmetry and mirror symmetry. This is easier than making kaleidoscopic images, … Continue reading

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Elliptic kaleidoscopes

In “further hyperbolic kaleidoscopes” I used two straight lines and a circle to make a triangle that defines a kaleidoscope. For k,n and m-fold rotational symmetries at its corners, the sum of its three angles is π(1/k+1/n+1/m). If this sum is … Continue reading

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The rotational and mirror symmetry at the center

In the last post I used mirror symmetry at two crossing straight lines and the related inversion at a circle. The mirror symmetries generate a k-rotational symmetry for an angle of intersection of π/k. With these symmetries I map any point … Continue reading

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A hyperbolic kaleidoscope

In “creating symmetry” Frank Farris shows a wallpaper for hyperbolic space. He uses the Poincaré plane to project the hyperbolic space to our Euclidean drawing surface. The wallpaper then results from mirror symmetries at vertical lines at x=0 and x=0.5 … Continue reading

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Five fold rotational symmetry: Tuning the harmonics

In “better images from higher harmonics ?” I have replaced the basic sine and cosine functions by Fourier series approaching a symmetric triangular wave. This gave images with more details and somewhat smaller bulls-eyes. Here I want to show similar results … Continue reading

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Morphing between square symmetry and eight fold rotational symmetry

A long time ago in “Crazy graph paper” I have shown a morphing between the square lattice and the quasiperiodic Ammann-Beenker tiling of eight-fold rotational symmetry. We can do similar morphs with mapping functions using waves. The wave vectors (1,0) … Continue reading

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