Why certain symmetries are mathematically forbidden in crystals
In short: In this Q&A, Sir Roger Penrose explores the fascinating world of non-repeating patterns and the mathematical rules that govern what can and cannot exist in crystal structures.
While classical crystallography dictates that only 2, 3, 4, and 6-fold symmetries are possible, Penrose discusses the existence of 'forbidden' symmetries. Since the 1970s, patterns exhibiting 5, 8, 10, and 12-fold 'almost' symmetric properties have been observed, deviating from exact symmetry by only an infinitesimal amount.
The discussion covers the practical application of these patterns in architecture, such as a granite and stainless steel tiling composed of thousands of diamond-shaped tiles. These tiles use specific matching rules to create a non-repeating pattern with 5-fold and 10-fold symmetry. This phenomenon is also present in the atomic structures of quasi-crystalline materials, a discovery that led to Dan Shectman receiving the 2011 Nobel Prize in Chemistry.
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Sources
- Summary by Intellect · source: Roger Penrose: Forbidden crystal symmetry - Event Q&A
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