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Author Archives: Peter Stampfli
Antialiasing for improving image quality
About a year ago I have briefly shown in my post “smoothing images” that averaging can be important to get good images without pixel noise. For my kaleidoscope app, see http://geometricolor.ch/sphericalKaleidoscopeApp.html, I have improved on these ideas and that’s what … Continue reading
Posted in Kaleidoscopes, programming
Tagged antialiasing, image processing, smoothing images
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Bridges 2018 Stockholm
I have been at the Bridges 2018 conference in Stockholm to present my work on kaleidoscopes. My paper “Kaleidoscopes for NonEuclidean Space” has more details than this blog and is more coherent. The Bridges Organization, which promotes connections between mathematics … Continue reading
Straight lines in elliptic and hyperbolic space
A straight line is the shortest path between two points. Discussing curved space we would better call them geodesic lines to avoid confusion. I want to discuss these geodesic lines for surfaces of a sphere, elliptic space and hyperbolic space. … Continue reading
Posted in Kaleidoscopes, programming
Tagged elliptic geometry, geodesic line, hyperbolic geometry, Poincaré disc, straight line
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Decorations of semiregular tessellations
In the last posts I have shown kaleidoscopes that make repeating images in Euclidean, spherical and hyperbolic spaces. They are decorations of regular tilings. But what about semiregular tilings? Could we decorate them too using mirrors? This would give us … Continue reading
Posted in Kaleidoscopes, Tilings
Tagged hexagonal lattice, kaleidoscope, mirror symmetry, Tessellation
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further wallpapers for hyperbolic space
An equilateral triangle gives us a kaleidoscope of threefold rotational symmetry. With a square we get twofold rotational symmetry. Would reflection at the sides of other regular polygons too give periodic images with rotational symmetry ? To get an hfold … Continue reading
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
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 mfold 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