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Professor Jost-Hinrich Eschenburg: Self similarity for penrose and isfahan patterns, 1392/12/22

كتابخانه خانه رياضيات اصفهان

پروفسور جاست - هينريچ اشنبرگ

Planar patterns can express the idea of infinity in two different ways: by repetition ( periodicity) or by self -similarity, where the same details appear on different scales. self imilarity is less obvious than periodicity, therefore it does not occur too often in arts. One of the sites it occurs is Isfahan , at Darb-i-Imam Shrine, at one of the Iwans of Friday Mosque and some other places. these 300 years old patterns have much in common with the aperidic patterns discovered only40 years ago by the mathematician Roger Penrose. in fact there is a large coincidence between the two patterns. both have in common the idea of self-similarity and the local pentagonal symmetry. however, there are differences: the Isfahan pattern has a global dekagonal ( 10 -gon ) symmetry wich is not shared by the penrose patterns. however. the concidence of both patterns sheds some light on the penrose patterns, too. it uncovers a new penrose pattern with a hidden approximate 10 - gon- symmetry. the new pattern does not arise by the common construction which uses a projection of part of the 5-dimensional periodic grid onto some plane in 5-space ( " projection method" ) . thus a 300 year old Iranian art work has some influence on todays mathematics.

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