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Monthly Archives: September 2012
A tiling with triangles and rhombs only
We can dissect the rhomb into triangles and rhombs without using squares. This dissection destroys its mirrorsymmetry but leaves the rotational symmetry around its center intact. Together with the dissection of the triangle into rhombs and triangles shown in the … Continue reading
Posted in Tilings
Tagged enantiomorphic, Iterative method, mirror symmetry, quasiperiodic Tiling, Rotational symmetry
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Finding an iteration method for the Stampfli tiling – mission impossible
I have caught a cold. I am not able to do new work and thus I am writing up some old leftovers. It is not possible to find an iteration method for the Stampfli tiling. One finds easily how to … Continue reading
Posted in Tilings
Tagged iteration, Iterative method, quasiperiodic Tiling, Stampfli tiling
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Using the wrong harmonics …
If we combine sinusoidal waves making a square pattern, f(x,y)=cos(kx)+cos(ky) with other waves of higher frequency g(x,y)=cos(a kx)+cos(a ky) we should use an integer ratio a between the frequencies to get again the same periodicity. If the ratio a is … Continue reading
Synthesizing quasiperiodic tilings
Synthesizers for electronic music combine simple waves to create complex sounds. Similarly, we create quasiperiodic structures from simple sinusoidal waves. I presented a first attempt in the post “quasiperiodic designs from superposition of waves“. A more complete method is discussed … Continue reading
Autumn leaves
In autumn find a lot of colorful leaves and we dry them. Later we are disappointed as they lose their color and become brittle. Thus I now prefer to scan the leaves with the scanner on my printer. It is … Continue reading
Making quasiperiodic structures using waves: A simple program
// this is a simple program // it shows more clearly how I get quasiperiodic structures // from waves // feel free to experiment //———————————————— // to run the program you need processing // simply download processing from processing.org … Continue reading
Experiments with waves
I extended the program that mixes waves to get some more decorative and less scientific pictures. Using n=9 and a nonlinear color model gives:
Quasiperiodic pattern from eight waves and the AmmannBeenker tiling
Using four waves at right angles we get a periodic structure of fourfold symmetry. A square grid of the same periodicity is easily fitted to this structure. Now, together with an extra set of four waves rotated by 45 degrees … Continue reading
Patterns of waves with eight and twelvefold rotational symmetry
As discussed in the previous post “Quasiperiodic designs from superposition of waves” we get a quasiperiodic structure with eightfold rotational symmetry using eight waves (n=8). Surprisingly, cosine waves of the same sign or alternating signs give us essentially the same … Continue reading
Quasiperiodic designs from superposition of waves
Quasiperiodic crystals have sharp diffraction patterns with a quasiperiodic structure. Thus their atomic density is made of a corresponding set of sinusoidal waves. Inspired by these ideas I mixed waves to get quasiperiodic designs. Now, if you want to have … Continue reading