Robinson Triangles

Robinson triangles, also known as golden triangles due to their close relation with the golden ratio discussed previously, make up the spikes of regular pentagrams.

Pentagram

Wikipedia

A golden triangle is an isosceles triangle where a ratio between one of the identical sides $ a $ and the base $ b $ is the golden ratio $ \phi $.

$ \frac {a}{b} = \phi $
Golden triangle

Wikipedia

A similar triangle to the Robinson triangle is the golden gnomon:

Golden gnomon

Wikipedia

The golden gnomon is another isosceles triangle where a ratio between one of the identical sides $ a $ and the base $ b $ is the reciprocal of the golden ratio $ \frac{1}{\phi} $.

Regarding Penrose tiling, a golden triangle and two golden gnomons make up a regular pentagon, which ties in with the golden ratio local pentagon symmetry discussed earlier. P2 Penrose tiling are made from kites and darts. A kite is made from two golden triangles, and a dart is made from two gnomons.

Further Readings:

One final note – if you like proofs, you will enjoy this site: http://mrbertman.com/penroseTilings.html. It contains many definitions and theorems that deal with how to place a tile correctly (the site uses the term “legally”) and why those rules exist. You can even create your own Penrose tiling!


Penrose Tiling Home

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Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood