No, the title isn't a list of random technical terms. I'm putting this here as a fast way to say "I found it first."

[And you'll really see why I need to post pictures and learn to type symbols!]

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The smallest unit of the tessellation is a pentagon with side lengths and angles in the order: 1, 120 degrees, 2, 120 degrees, 1, 90 degrees, square root of 3, 120 degrees, square root of 3, 90 degrees.

This particular pentagon tiles in more than one way, one being a subset of the previously described Type 3. Another can be non-repetitive; this tiling consists of 2 parts. Part 1 is the "medallion." The medallion is formed by uniting three of the pentagons at one vertex with sides of length square root of 3 meeting, giving a hexagon of side length 2. Part 2 is the "hexagonal annulus" created by adding one pentagon to each side of the medallion (side of length 2 facing the medallion), creating a second hexagon of side length 2x square root of 3.

The annuli can be combined into a regular hexagonal tessellation and, if all the medallions are oriented in the same direction, the whole becomes a regular tessellation of pentagonal units. It is possible to rotate any of the medallions 60 degrees and still tile the plane, so it is possible to create tessellations that are of wallpaper groups not described in the 14 types of convex pentagons that tile the plane. In addition, it is possible to orient the medallions randomly, which creates a non-repeating tiling using one basic unit, something not previously described.

Whew!

2 days ago

## 9 comments:

So all the edges of the medallions are the same length? Otherwise how could they be oriented randomly? Yes, I definitely need a picture.

I also need Glaven to make a sexually pervese yet funny comment. Glaven...

Oops. Meant perverse. But pervese is also an interesting word.

Dude, I am biology, the "fluffy" science. I think I get how the sides can be the different lengths, but I need a picture.

I guess I could try to draw it... Where is that goddamn protractor?

I highly admire Escher's work. I was fortunate enough to find a rare out of print book that depicts many of his tessellated blueprints. Fascinating stuff. Although I will say of his tessellated work I enjoy the challenge and originality of his animal Mosaics. Abnormally tessellating creatures and figurines no two alike and not one of the 3 shades ever touching each other side by side. I'd run another 100miler again if I could run along side you and Escher.

Don't let the artisans of Spain or Italy get your tessellation. Two Mecca's of the tile and tessellation. There are some cool looking tessellations on some of the old facades downtown Duluth if you have a keen eye for art and look up rather than through the windows.

wynn

Ultrarunners are often asked what they think about during long runs.

I guess that this is exhibit "A".

I'm confused...

RBR- I adore fluffy science teachers, whatever they teach! Sociology's where I draw the line on hard science - they all sound like they got their training from Dr. Phil and Oprah.

Oh... and Anonymous, I would never think about this stuff while running - no visual memory!

So you're denying the allegations of Tessellation Enhanced Running (TEH)!

sdf

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