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Category Archives: Tilings
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: II. Rhombic Rosette
As a child I was often drawing rosettes of rhombi. Something like this: And then I tried to extend the rosette to get a large tiling of the same symmetry. Obviously without success. The border of such rosettes is always … Continue reading
Posted in Quasiperiodic design, Tilings, Uncategorized
Tagged quasiperiodic Tiling, Rotational symmetry
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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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Generating Quasiperiodic Tilings and Fractals VI: The Ammann-Beenker Tiling
The Ammann-Beenker tiling is a quasiperiodic tiling of eight-fold rotational symmetry. It has squares and rhombi with an acute angle of 45 degrees as tiles. This is a small patch: The square tiles are divided into two triangles. The substitution … Continue reading
Posted in programming, Quasiperiodic design, Tilings
Tagged programming, quasiperiodic Tiling, Rotational symmetry
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Generating Quasiperiodic Tilings and Fractals IV: A Fractal Tree
With my browser app http://geometricolor.ch/qpg/quasiperiodicGenerator/quasiperiodicAndFractal.html you can create many of the popular fractals using line segments, such as the Dragon curve. I am showing you how to define a fractal tree and presenting some additional options of the program. The … Continue reading
Posted in Fractals, programming, Tilings
Tagged fractal, programming, Quasiperiodic design, quasiperiodic Tiling
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Generating Quasiperiodic Tilings and Fractals III: The Sierpinsky Triangle
Now I am showing how to define the Sierpinsky triangle for my browser app at http://geometricolor.ch/qpg/quasiperiodicGenerator/quasiperiodicAndFractal.html I will do it step by step. The Sierpinsky triangle and its substitution rule look like this: As mentioned previously, the browser app reads … Continue reading
Posted in Fractals, programming, Tilings
Tagged fractal, Rotational symmetry, Tiling
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Generating Quasiperiodic Tilings and Fractals II: Javascript Object Notation
With my browser app at http://geometricolor.ch/qpg/quasiperiodicGenerator/quasiperiodicAndFractal.html you can make your own tilings and fractals. You simply have to define them in textfiles using Javascript Object Notation (JSON), which is a small subset of Javascript for defining objects. In this post … Continue reading
Posted in Fractals, programming, Tilings
Tagged fractal, programming, quasiperiodic Tiling
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Generating Quasiperiodic Tilings and Fractals I: The Browser App
I have done many different quasi-periodic tilings using the substitution method. You can see some of them in this blog. Examples are: https://geometricolor.wordpress.com/2012/06/05/iteration-of-rhombs-filling-the-gap-2/, https://geometricolor.wordpress.com/2012/05/20/a-tiling-of-octagons-squares-and-rhombs/, and https://geometricolor.wordpress.com/2021/03/04/yet-another-tiling-with-12-fold-rotational-symmetry/. Others are published elsewhere, see https://archive.bridgesmathart.org/2021/bridges2021-315.html, or there are browser apps generating them, see … Continue reading
Posted in Fractals, programming, Quasiperiodic design, Tilings
Tagged fractal, programming, quasiperiodic Tiling
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Geometry in color 