Tag Archives: Math

Rotations, mirrorsymmetry and the scalar product

In the last post we have seen that scalar products between a pixel’s position in the output image and certain vectors e define periodic and quasi-periodic designs. We want symmetric images and thus we have to see how the scalar product changes … Continue reading

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How to calculate the corner points and the projection method

Again something for the more mathematically minded. I am discussing here some important details on the earlier post “A tiling of 12-fold rotational symmetry …” . Actually, these calculations are similar as in the earlier post “How to find these … Continue reading

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Exhibition of the self-similarity of the tiling

In the last post, I claimed that the tiling one gets from two hexagon-grids is self-similar. But this is not clear from the image I have shown because it has only about 2000 tiles. This is not enough. Just look … Continue reading

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Why these tilings are not periodic

Often, you can put together two periodic patterns of different length and you get a new pattern, which is periodic too. The length of the period of the joint pattern is the least common multiple of the period lengths of … Continue reading

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Another tiling of eight-fold rotational symmetry

I am using the same basic method as for the Amman-Beenker tiling, see my post “How to find these corner points of the tiles“, but now with smaller squares. The distance between the centers of the squares remains equal to 1, … Continue reading

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The basic Amman-Beenker tiling

With the details presented in the earlier posts we can easily get large parts of this tiling. I have tried to choose colors, which are not too contrasting and too hard on the eyes: We see that small patterns are … Continue reading

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How to find these corner points of the tiles.

I am trying to explain all in detail and I hope that I will not be too boring. We first find the centers of the squares of the two square grids. If the length of their sides is equal to … Continue reading

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