Quasiperiodic tiling with pentagons – the Penrose connection

The last post “A quasiperiodic tiling with pentagons” is close to the original reasoning of Penrose, see the article “Penrose tiling” in Wikipedia. He dissected the Pentagon into six smaller ones and filled the gaps with other tiles. He took care to leave no gaps and to have no overlapping tiles. His work is fantastic. Note that he has not used a computer !

Iteration scheme for pentagons. The smaller blue pentagons fill up the gaps.

Now, similarly as for the pentagrams, we find a simplified iteration for pentagons without gaps. We simply use the pentagons that surround the pentagrams. Then this scheme results in many overlapping pentagons.

First I have drawn first the larger pentagons in yellow and then the smaller ones in semitransparent blue on top. This leaves the black borders of the large pentagons still visible.

Pentagon after four iterations. Fractal pentagram shapes appear.

The corresponding quasiperiodic tiling.

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2 Responses to Quasiperiodic tiling with pentagons – the Penrose connection

  1. The artwork I produce starts as spreadsheets such as excel. Is there hope for more graphical mathematics programs using multiple geometries such as pentagrams?

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