Sir Roger Penrose: Non-repeating patterns

Date Published: 24.01.2014

Insights into the ‘forbidden symmetry’ of Sir Roger Penrose’s non-repeating patterns are given in a new film released this month by the Royal Institution.

Penrose paving outside the bar in the Bowra building.

‘Forbidden crystal symmetry in mathematics and architecture’ was filmed at an Royal Institution (RI) event in 2013 where Wadham Fellow and Emeritus Rouse Ball Professor of Mathematics, Sir Roger Penrose, reveals the mathematical underpinnings and origins of these ‘forbidden symmetries’ and other related patterns. His talk is illustrated with 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.

An example of Penrose Paving can be seen outside the bar of the Bowra Building at Wadham. According the to the College history, edited by Jane Garnett and Cliff Davies: “In 1974, Roger Penrose discovered that it was possible to construct a pattern from just two different shapes, each of them a rhombus with angles which are multiples of 36 degrees; the pattern achieves fivefold symmetry and, most remarkably, can be extended to infinity without repeating itself. Extraordinarily, ten years later, chemists discovered a new class of metallic alloys with similar fivefold symmetry in defiance of the previously accepted laws of crystallography.”

Wadham’s Senior Research Fellow and Professor of Mathematics, Nicholas Woodhouse, added: “The familiar tilings by squares, or hexagons, or octagons combined with squares, and so forth, are all periodic.  This means that when the patterns are continued indefinitely, they have translational symmetry. The whole pattern can be picked up and shifted without rotation in various ways without producing any change. For example, a tiling by six inch squares is unchanged when it is shifted six inches parallel to the sides of one of the tiles. The Penrose tilings can also be continued indefinitely, but not in a way that has translational symmetry.  They can have rotational symmetry about a central point, but however the complete pattern is translated without rotation and then overlaid on the original, it will never match it exactly. ‘Continued indefinitely’ is important here: small portions of the pattern will appear over and over again. It is the complete pattern that lacks translational symmetry.”

In their film summary, the RI report: “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 been used in many buildings throughout the world.”

The tiling outside the Oxford Mathematical Institute 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.