Why certain symmetries are mathematically forbidden yet physically possible
In short: In classical crystallography, certain patterns are strictly impossible. Yet, since the 1970s, scientists have observed structures that defy these rules. By using just two specific shapes, mathematician Roger Penrose discovered a way to create infinite, non-repeating patterns that challenge our fundamental understanding of order and symmetry.
For many years, crystallographic laws dictated a strict constraint: periodic crystals could only possess 2-fold, 3-fold, 4-fold, or 6-fold symmetries. These patterns possess translational symmetry, meaning the entire arrangement can be shifted in various directions without changing its appearance. In a standard tiling of squares, for instance, shifting the grid by the width of one square leaves the pattern looking identical.
In 1974, Sir Roger Penrose identified a mathematical loophole. He proved it was possible to construct a pattern using only two different rhombus shapes—specifically with angles that are multiples of 36 degrees. This construction achieves 5-fold and 10-fold symmetry, but with a profound twist: the pattern lacks translational symmetry. While small portions of the design may appear to repeat, the complete, infinite pattern will never match itself exactly when shifted.
This mathematical curiosity transitioned from theory to physical reality when chemists discovered a new class of metallic alloys exhibiting similar fivefold symmetry. This discovery, which earned Dan Shechtman the 2011 Nobel Prize for Chemistry, proved that these 'forbidden symmetries' are mathematically possible by deviating from exact symmetry by an arbitrarily small amount. Today, these complex arrangements are not just theoretical; they are used in architectural designs, such as the granite and stainless steel tiling at the Mathematics Institute in Oxford.
These non-repeating patterns, known as Penrose tilings, represent a bridge between pure mathematics and the material world. They demonstrate how structures can exhibit rotational symmetry around a central point while remaining fundamentally non-periodic, forever defying the traditional expectation of repeating, predictable patterns.
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Sources
- Summary by Intellect · source: Roger Penrose - Forbidden crystal symmetry in mathematics and architecture
- Also: Sir Roger Penrose: Non-repeating patterns
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