Roger von Oech

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      CJ Millisock

      In 9th grade, I learned that it was once thought impossible to tessellate pentagons, but that a tessellation had been discovered, and that the discovery was relatively recent compared to most geometrical discoveries.

      This page shows some pentagon tessellations, and cites 1918 as the earliest discovery (of the examples shown).

      Check it out!
      http://www.mathpuzzle.com/tilepent.html

      Nedra Weinreich

      Pentagons and the number five are related to my very favorite ratio phi (the golden mean, etc).

      Here is some info on how that works:
      http://www.championtrees.org/yarrow/phi/phi1.htm

      And where you can find it cropping up in nature:
      http://www.unitone.org/naturesword/sacred_geometry/phi/in_nature/

      But I have a feeling that you might already know about that.

      Pavel Sokolov

      You can find a lot of info on Islamic Art, if you search for "quasi periodic +ornament +pentagons". E.g. an article in BBC with second headline "A study of medieval Islamic art has shown some of its geometric patterns use principles established centuries later by modern mathematicians." http://news.bbc.co.uk/2/hi/middle_east/6389157.stm

      Roger von Oech

      Chet: Thanks for the tiling link. I encourage other readers to check it out. I especially liked "Type 9." It seemed the most alive and organic. I could see it on my tile floor.

      Nedra: Great link to the Golden Ratio (1.1618). Thanks. I'm a fan as well. As you no doubt know, the individual pieces in the amazing "Ball of Whacks" also are defined by Phi. The ratio of the rhombus base's long axis to its short axis is 1.618. So, if you want a does of Gold Ratio, play with your BOW!

      Pavel: Good to hear from hear you. Islamic tile patterns are quite stunning. Interestingly, physicists have determined that there are thirty-two different ways in which atoms and molecules in a crystal can be symmetrically arranged in a pattern, and these are all represented in the tilings of the fourteenth century Moorish palaces of the Alcazar and Alhambra palaces in Spain.

      Randy

      Take a look at:
      http://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html a site on geometry in art & architecture, for its pentagon/pentagram section.

      Randy

      And, the Giant's Causeway in County Antrim Ireland has, among its basaltic columns, a remarkable number that are pentagonal - often co-existing adjacent to those that are hexagonal or otherwise polygonal. The proximity of natural geometric shapes of a differing number of sides is thought provoking!

      Peter Hoh

      I like how connecting the points of a pentagon creates an internal pentagram. And inside the pentagram, there's a smaller pentagon.

      Roger von Oech

      Randy: Thanks for the Dartmouth link — a lot of good stuff. Also, I wasn't familiar with the basaltic columns — interesting stuff!

      Peter: Me too.

      Matthew Cornell

      Interesting! I'm a little surprised you're sticking with fives (I'm enjoying my ball of whacks, BTW). Have hexagons "been done" already? A bit off-topic, but I liked these patterns: http://local.wasp.uwa.edu.au/~pbourke/texture_colour/hexagon/ . This one's pretty, I think: http://local.wasp.uwa.edu.au/~pbourke/texture_colour/displayimage.cgi?hexagon/pattern6.jpg

      Roger von Oech

      Matthew: That's a fabulous link. I'll be going back there for more stimulation!

      Bart

      These are or are similar in function to window tracery of Gothic Architecture. I would like to know what other languages call these grates so a search provides better results.

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