The Tübingen triangle is a form of
substitution tiling In geometry, a tile substitution is a method for constructing highly ordered Tessellation, tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational ...
. It is, apart from the
Penrose rhomb tilings and their variations, a classical candidate to model 5-fold (respectively 10-fold)
quasicrystal
A quasiperiodicity, quasiperiodic crystal, or quasicrystal, is a structure that is Order and disorder (physics), ordered but not Bravais lattice, periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks trans ...
s. The inflation factor is – as in the Penrose case – the golden mean,
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
Tübingen (; ) is a traditional college town, university city in central Baden-Württemberg, Germany. It is situated south of the state capital, Stuttgart, and developed on both sides of the Neckar and Ammer (Neckar), Ammer rivers. about one in ...
, Germany, thus the name. They can be obtained by cut-and-project on the
5-cell honeycomb
In Four-dimensional space, four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb (geometry), honeycomb. It is composed of 5-cells and recti ...
.
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.
[E. Harriss (Drawings of 2005-12-01) und D. Frettlöh (Text of 2006-02-27)]
Tuebingen Triangle.
Downloaded on 2015-03-06.
See also
*
Mathematics and art
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art mathematical beauty, motivated by beauty. Mathematics can be discerned in arts such as Music and mathematics, music, dance, painting, Mathema ...
References
{{DEFAULTSORT:Tubingen triangle
Aperiodic tilings