Advanced geometry of Islamic art - BBC News

Islamic Artisans Constructed Exotic Nonrepeating Pattern 500 Years Before Mathematicians - Scientific American

In an interview with the BBC, the lead author on the paper, Peter Lu, is quoted as saying that the Islamic artists "made tilings that reflect mathematics that were so sophisticated that we didn't figure it out until the last 20 or 30 years."

The authors are referring here to Penrose tiling, discovered in the 1970s by mathematical physicist Sir Roger Penrose. The significance of this class of patterns is that they fill space without repeating themselves, despite having five-fold rotational symmetry. If you had two copies of the pattern on transparent plastic, you would only be able to match the patterns up with the sheets in one position relative to each other, unless, that is, you rotate one of the sheets.

Penrose's discovery led to work by Alan Mackay in the early 1980s, describing for the first time the phenomenon of quasicrystals: three-dimensional structures which fill space in an orderly way, but which do not possess translational symmetry. Quasicrystals are in effect a generalisation of Penrose Tiling into three-dimensions.

In the development of Penrose Tiling, Sir Roger was inspired by the work of Johannes Kepler. One tiling in particular from Kepler's 1619 publication Harmonice Mundi showed that is possible to tile a two-dimensional space completely using shapes which have five-fold symmetry, so long as one also uses the shape now known as "Kepler's Monster". These "fused decagon pairs", as Kepler also called them, are essentially the gaps left in the skeleton formed by the arrangement of five-fold symmetrical tiles.

Many of the patterns which Lu and Steinhardt examine also show sets of "fused decagons", depicted clearly in this image from the BBC website. While these appear to be extensions of Kepler's Monster, one subtle difference is clear. Whereas Kepler's shape is formed by the fusion of two decagons from each of which two faces have been removed, leading to a sixteen-faced Monster, the shapes in Lu and Steinhardt's work are all formed by decagons meeting at a single face. The resulting basic eighteen-faced Monster can be seen clearly in figures S3D and S4C of the aforementioned Supporting Online Material.

Despite these differences, the similarities in these patterns are striking, and beg the question as to whether Kepler was himself inspired by Islamic art. It is well known that Kepler's work was significantly influenced by that of the great Islamic scholar Alhazen, who was himself the first person to discover the laws of refraction. Roger Penrose is quoted in the Daily Telegraph recently as saying that Kepler produced "a true portion" of a Penrose pattern "a great deal closer to my tilings than any of the Islamic patterns I have seen so far". Is it not a possibility, however, that Kepler's work is in effect a stepping-stone between that of the ancient Islamic artisans and that of Penrose?

There is a tantalising possibility that we may one day know more of Kepler's work with such patterns than we do already. Writing here, an anonymous Wolfram employee mentions that Penrose is aware of the possible existence of a letter from Kepler in which he "went into much greater detail about his intentions with the tiling". Penrose has apparently thus far not been able to track down this letter, but the possibility of its existence implies that there may yet be more dots to be joined in tracing the history of quasicrystallographic geometry.

As an aside, I am indebted to the work of Victor Ostromoukhov for first informing me of the connection between the works of Kepler and Penrose. Of his work, this paper outlines a novel recent application of the properties of Penrose tiling, and this work concerns the computer generation of a different class of patterns to be found in Islamic art.

The authors are referring here to Penrose tiling, discovered in the 1970s by mathematical physicist Sir Roger Penrose. The significance of this class of patterns is that they fill space without repeating themselves, despite having five-fold rotational symmetry. If you had two copies of the pattern on transparent plastic, you would only be able to match the patterns up with the sheets in one position relative to each other, unless, that is, you rotate one of the sheets.

Penrose's discovery led to work by Alan Mackay in the early 1980s, describing for the first time the phenomenon of quasicrystals: three-dimensional structures which fill space in an orderly way, but which do not possess translational symmetry. Quasicrystals are in effect a generalisation of Penrose Tiling into three-dimensions.

In the development of Penrose Tiling, Sir Roger was inspired by the work of Johannes Kepler. One tiling in particular from Kepler's 1619 publication Harmonice Mundi showed that is possible to tile a two-dimensional space completely using shapes which have five-fold symmetry, so long as one also uses the shape now known as "Kepler's Monster". These "fused decagon pairs", as Kepler also called them, are essentially the gaps left in the skeleton formed by the arrangement of five-fold symmetrical tiles.

Many of the patterns which Lu and Steinhardt examine also show sets of "fused decagons", depicted clearly in this image from the BBC website. While these appear to be extensions of Kepler's Monster, one subtle difference is clear. Whereas Kepler's shape is formed by the fusion of two decagons from each of which two faces have been removed, leading to a sixteen-faced Monster, the shapes in Lu and Steinhardt's work are all formed by decagons meeting at a single face. The resulting basic eighteen-faced Monster can be seen clearly in figures S3D and S4C of the aforementioned Supporting Online Material.

Despite these differences, the similarities in these patterns are striking, and beg the question as to whether Kepler was himself inspired by Islamic art. It is well known that Kepler's work was significantly influenced by that of the great Islamic scholar Alhazen, who was himself the first person to discover the laws of refraction. Roger Penrose is quoted in the Daily Telegraph recently as saying that Kepler produced "a true portion" of a Penrose pattern "a great deal closer to my tilings than any of the Islamic patterns I have seen so far". Is it not a possibility, however, that Kepler's work is in effect a stepping-stone between that of the ancient Islamic artisans and that of Penrose?

There is a tantalising possibility that we may one day know more of Kepler's work with such patterns than we do already. Writing here, an anonymous Wolfram employee mentions that Penrose is aware of the possible existence of a letter from Kepler in which he "went into much greater detail about his intentions with the tiling". Penrose has apparently thus far not been able to track down this letter, but the possibility of its existence implies that there may yet be more dots to be joined in tracing the history of quasicrystallographic geometry.

As an aside, I am indebted to the work of Victor Ostromoukhov for first informing me of the connection between the works of Kepler and Penrose. Of his work, this paper outlines a novel recent application of the properties of Penrose tiling, and this work concerns the computer generation of a different class of patterns to be found in Islamic art.