Tübingen triangle

Non-periodic tiling of the plane

Tübingen triangle Tübingen triangles

The Tübingen triangle is a form of substitution tiling. It is, apart from the Penrose rhomb tilings and their variations, a classical candidate to model 5-fold (respectively 10-fold) quasicrystals. The inflation factor is – as in the Penrose case – the golden mean,  φ = a b = 1 + 5 2 ≈ 1.618. {\displaystyle \varphi ={\frac {a}{b}}={\frac {1+{\sqrt {5}}}{2}}\approx 1.618.}

The prototiles are Robinson triangles, but the relationship is different: The Penrose rhomb tilings are locally derivable from the Tübingen triangle tilings.

These tilings were discovered and studied thoroughly by a group in Tübingen, Germany, thus the name. They can be obtained by cut-and-project on the 5-cell honeycomb.[1]

Since the prototiles are mirror symmetric, but their substitutions are not, left-handed and right-handed tiles need to be distinguished. This is indicated by the colours in the substitution rule and the patches of the relevant figures.[2]

## See also

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* Mathematics and art

## References

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1. ^ Baake, M and Kramer, P and Schlottmann, M and Zeidler, D Planar patterns with fivefold symmetry as sections of periodic structures in 4-space Internat. J. Modern Phys. B, 1990, 4, 15–16, pp. 2217–2268, 92b:52041
2. ^ E. Harriss (Drawings of 2005-12-01) und D. Frettlöh (Text of 2006-02-27): Tuebingen Triangle. Archived 2015-04-02 at the Wayback Machine Downloaded on 2015-03-06.