# Tag Archives: Ammann–Beenker tiling

## Beautifying the double grid

The grids for quasiperiodic tilings do not look good because a lot of irregular shapes arise from superimposing two simple grids, see the article “Doubling the tessellation of triangles“. But we can distort these double grids and get new interesting … Continue reading

## Periodic approximations of quasiperiodic tilings

In the earlier post “Why these tilings are not periodic” I showed that the Ammann-Beenker tiling is quasiperiodic because it arises from superimposing two square grids with rotated by 45 degrees. As seen in the same direction we have the periodic … Continue reading

## Stampfli-tiling and related designs from waves

Three sinusoidal waves make a hexagonal pattern for f(x,y)=cos(x)+cos(x/2+sqrt(3) y/2)+ cos(x/2+sqrt(3) y/2), see the figure at left. Using this and the same pattern rotated by 90 degrees we get patterns of 12-fold rotational symmetry. I found it interesting to draw … Continue reading

## Periodic and quasiperiodic images from cross-sections of 4-dimensional space

In the earlier post “Quasiperiodic designs from waves and higher dimensional space” I have shown that the quasiperiodic wave pattern with 8-fold rotational symmetry is a special cross-section of a periodic pattern in  4-dimensional space. Here I will rotate the … Continue reading

## Quasiperiodic designs from waves and higher dimensional space

I am doing it again – a rather mathematical post. Well, in “An easy way to quasiperiodic tilings” I have shown how to make the Ammann-Beenker tiling using two square grids. Then in “How to find these corner points of … Continue reading

## Quasiperiodic pattern from eight waves and the Ammann-Beenker 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 twelve-fold rotational symmetry

As discussed in the previous post “Quasiperiodic designs from superposition of waves” we get a quasiperiodic structure with eight-fold rotational symmetry using eight waves (n=8). Surprisingly, cosine waves of the same sign or alternating signs give us essentially the same … Continue reading