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Tag Archives: Geometry
Tilings of 5-Fold Rotational Symmetry: V. Conservation of Area in Substitution
The substitution method fills the inflated parent tile with child tiles having no overlap or gaps. Thus, the total area of the child tiles has to be exactly equal to the area of the parent tile. This allows us to … Continue reading
Posted in Quasiperiodic design, Self-similarity, Tilings, Uncategorized
Tagged Geometry, quasiperiodic Tiling
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Tilings of 5-Fold Rotational Symmetry: IV. Algebraic Integers
In the last post we have seen that inflation factors r for the substitution method are elements of the integer ring ℤ[τ]={h+k τ | h,k ∊ ℤ}, where τ is the golden ratio. Trivially, the sum of two elements of … Continue reading
Tilings of 5-Fold Rotational Symmetry: III. Substitution Method
Trying to extend the rosette does not work, instead one better uses the substitution method. It increases the size of the rhombi by an inflation factor r to get large parent tiles which are then replaced by child tiles of … Continue reading
Posted in Quasiperiodic design, Tilings, Uncategorized
Tagged Geometry, Quasiperiodic design, quasiperiodic Tiling
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Tilings of 5-Fold Rotational Symmetry: I. Ptolemy’s Theorem
You are certainly familiar with quasiperiodic Penrose tilings of 5-fold rotational symmetry. In this series of posts I examine such tilings with rhombic tiles. We will see that they are easy to make and that they are closely related to … Continue reading
Posted in Quasiperiodic design, Tilings, Uncategorized
Tagged Geometry, quasiperiodic Tiling
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Curves
Frank Farris begins his book “Creating Symmetry” with symmetric curves of N-fold rotational symmetry. An example: He uses that we can interpret points (x,y) of the plane as complex numbers z=x+i*y. Thus a complex function f(t) of a real parameter t defines … Continue reading
Posted in Kaleidoscopes, programming
Tagged generative design, Geometry, ornament, Rotational symmetry
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Creating Symmetry
Recently I found a very exciting book: “Creating Symmetry – The Artful Mathematics of Wallpaper Patterns” by Frank A Farris. It has many beautiful images and explains the mathematics behind them very well, such that you could do your own work. His … Continue reading
Posted in Kaleidoscopes, Tilings
Tagged Art, generative design, Geometry, kaleidoscope, symmetry
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Geometry of kaleidoscopes with periodic images
About a year ago I explained “how to program an ideal kaleidoscope” to get the same as three mirrors put together. Often, one gets images that are not periodic. They have cut lines with a mismatch between the two sides, … Continue reading
Posted in Kaleidoscopes
Tagged Geometry, kaleidoscope, mirror symmetry, period doubling, periodic images, Rotational symmetry
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improved class Vector – the code
// a class for two-dimensional vectors, similarly as PVector, with extensions for complex numbers //————————————————————- // // some important values float vectorSmall=0.0001; float vectorDiameter=8; color vectorColor=color(255); // and here’s the class class Vector{ float x,y; // … Continue reading
improving the class Vector
I am not happy with my class Vector. Looking at my posts “Nautilus” and “self-similar fractals …” I realize that complex numbers and vectors should be put together with their methods, which are mostly mappings of the plane. In particular, … Continue reading
high resolution images with off-screen drawing
In an earlier post I have shown how to make smooth images at any scale using the pdf-renderer. But you can do this only with graphics objects such as line, point, shape, ellipse and so on. It won’t work if … Continue reading
Posted in Anamorphosis, Cellular automata, Extra, Fractals, Kaleidoscopes, programming
Tagged Geometry, image resolution, pixel, processing, programming, smoothing images
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Geometry in color 