Tübingen triangle explained

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,

\varphi = (a)/(b) = (1+ \sqrt{ 5)/() } \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

• Mathematics and art

References

1. Baake, M and Kramer, P and Schlottmann, M and Zeidler, DPlanar 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. Downloaded on 2015-03-06.