Forbidden crystal symmetry in mathematics and architecture

With Sir Roger Penrose

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Revealing forbidden symmetry

Sir Roger Penrose provides a unique insight into the "forbidden symmetry" of his famous penrose tiles and the use of non-repeating patterns in design and architecture. 

It is a rigorous mathematical theorem that the only crystallographic symmetries are 2-fold, 3-fold, 4-fold, and 6-fold symmetries.

Yet, since the 1970s 5-fold, 8-fold, 10-fold and 12-fold "almost" symmetric patterns have been exhibited, showing that such crystallographically "forbidden symmetries" are mathematically possible and deviate from exact symmetry by an arbitrarily small amount. Such patterns are often beautiful to behold and designs based on these arrangements have now been used in many buildings throughout the world.

In this Ri event Sir Roger Penrose reveals the mathematical underpinnings and origins of these "forbidden symmetries" and other related patterns. His talk is illustrated with numerous examples of their use in architectural design including a novel version of "Penrose tiling" that appears in the approach to the main entrance of the new Mathematics Institute in Oxford, officially opened in late 2013.

The tiling is constructed from several thousand diamond-shaped granite tiles of just two different shapes, decorated simply with circular arcs of stainless steel. The matching of the tiles forces them into an overall pattern which never repeats itself and exhibits remarkable aspects of 5-fold and 10-fold symmetry.

Similar features have been found also in the atomic structures of quasi-crystalline materials. The initial discovery of such material earned Dan Shectman the 2011 Nobel Prize for chemistry, his work having launched a completely novel area of crystallography. 

Images of the completed Mathematics Institute in Oxford courtesy of Vanesa Penrose.

The filming and production of this event was supported by the Science and Technology Facilities Council

Production by Edward Prosser. Additional camera operation by Mark Billy Svensson. 

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Engineering, Maths


Sir Roger Penrose
London, UK
Filmed in:
The Theatre

Royal Institution

Collections with this video:
Ri Talks, The Crystallography Collection

Licence: © The Royal Institution

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