Islamic Penrose Tiles - for Sam

My point was not that European painters could not paint the human form. For the vast majority of them their subject matter was limited and so was the style. And the society around them was 1) totalitarian 2) highly restrictive 3) very dangerous to be 'different' in. The inquisition, persecution of jews, and the systematic abuse of everyone by the noble classes and the church. These patterns only became vastly more clear when medieval european cultures colonized or otherwise came in contact with non-europeans, slavery included in the mix.

Your first post seems as silly to me to feel sad for the architects of some of beautiful churches of Europe, or Medieval painters who focused on Jesus or other Christian themes. Or to assume that these tile and mosaicmakers REALLY wanted to be doing something else. I would guess some did. But it just seemed like you needed the tiles to be part of bad Islam.

I like the painting though.

 

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It's true that European society was very strictly religious, but at least they could paint people.

 

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Here is a study on the math of Islamic architecture:

http://www.sciencedaily.com/releases/2007/02/070222155706.htm

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Direct strapwork and girih-tile construction of 10/3 decagonal patterns.

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Periodic girih pattern from the Seljuk Mama Hatun Mausoleum in Tercan, Turkey (~1200 C.E.), where all lines are parallel to the sides of a regular pentagon, even though no decagon star is present

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Girih-tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-i Imam shrine, Isfahan, Iran (1453 C.E.).

Ref: Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture
Science 23 February 2007

 

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Its fascinating stuff,
i remember watching a neat open university program about the Alhambra and the mathematics and patterns of the tiles.
http://findarticles.com/p/articles/mi_qn4158/is_19990401/ai_n14223717/pg_2

http://www.catnaps.org/islamic/geometry.html

 

Quoting from my text book, Symmetry, Shape, and Space by Kinsey and Moore:

"The most well-known set of aperiodic tiles are the Penrose tiles. The set was introduced by Roger Penrose in 1974 and contains only two tiles along with a set of rules for how these tiles must be put together. Penrose began his search for an aperiodic set of tiles by looking at pentagons. While it is true that the plane cannot be tiled by regular pentagons, Penrose studied the gaps left when one tried. He then took smaller pentagons and tried to fill in the holes. After several subdivisions, he found that the holes could have only a few shapes. These he called diamonds, paper boats, and stars. Significant insight and refinement led to Penrose's first aperiodic set of six tiles and finally to the set of two tiles most commonly known as the kite and dart, names suggested by John Conway, another mathematician who has contributed a great deal to what is known about Penrose tilings.

"Anyone who has studied stars and pentagons should expect to find the golden ratio somewhere in a situation involving these two shapes, and indeed it makes several appearances here. In any infinite tiling of the plane by kites and darts, the number of kites used is the golden ratio times the number of darts. In any tiling of a finite section of the plane, the ratio of kites to darts will approximate the golden ratio. The approximation improves as the area tiled increases. The area of a kite is the golden ratio times the area of a dart. With these facts in mind, it is probably not surprising that the golden ratio is part of the actual measurements of the tiles."

I think this defines the measurements of the angles in the tiles.

"The kite and dart are cut from a rhombus with side lengths equal to the golden ratio and main diagonal length equal to the golden ratio + 1. Connect the vertices at the obtuse angles to the main diagonal at a distance of the golden ratio from an acute angle (and hence a distance of one from the other acute angle). The large piece is the kite; the small piece is the dart. Notice that the kite is made of two isosceles triangles with the golden ratio for the length of the equal sides. The length of the third side of these triangles is equal to one. These triangles are sometimes called golden triangles. Consequently, the kite is also made of two isosceles triangles with equal sides of length one and a third side of length equal to the golden ratio.

"The rules for constructing a nonperiodic tiling with the Penrose tiles are simple. However, following them consistently is not necessarily easy. First, as with many tilings, only sides of the same length can be put together. This rule ensures that no vertex of one tile can occur in the middle of the side of another tile. The second rule requires a certain direction along the sides of the tiles, To enforce the direction rule, some people have put notches and bumps on the tiles, some use dots or holes, and some reshape the tiles to fit only in the correct way. Inspired by some of Escher's prints, Penrose reshaped the tiles as chickens. John Conway puts arcs on the tiles and requires that the arcs of the same color must meet to form continuous curves. Thus, in constructing a nonperiodic tiling, dark arcs must join dark arcs, and light arcs must join to light. Conway went on to prove a number of results involving the way the arcs connect."

There are 7 ways to put these tiles around a vertex. The star and the sun have fivefold rotational symmetry, and all of the groupings have at least one line of reflectional symmetry.

"As you work out from a vertex, sometimes there is only one way to place a tile at a certain point. Sometimes there are several choices. Sometimes it seems as if you have a choice, but farther out in your tiling you will find a place where no tile will fit and you will have to go back and change your original choice. If there is only one choice for fitting a tile around a vertex grouping, we say that tile is in the empire of the vertex."

The book goes on. To make things more simple for you, there is a Penrose tile applet: http://www.geocities.com/SiliconValley/Pines/1684/Penrose.html