No Pentagons

January 14, 2015 11:58 AM

Imperfect Congruence - It is a curious fact that no edge-to-edge regular polygon tiling of the plane can include a pentagon ... This website explains the basic mathematics of a particular class of tilings of the plane, those involving regular polygons such as triangles or hexagons. As will be shown, certain combinations of regular polygons cannot be extended to a full tiling of the plane without involving additional shapes, such as rhombs. The site contains some commentary on Renaissance research on this subject carried out by two renowned figures, the mathematician-astronomer Johannes Kepler and the artist Albrecht Dürer.

Bonus link: The Trouble with Five (by Craig Kaplan, at Plus magazine - a short, tantalizing article suitable for school-age readers...)

posted by Wolfdog (16 comments total)

Is it wrong that I'm so charmed by the reference to monsters in the the second link?

[The substitution rule] eliminates the rhombi of the original tiling, but introduces monsters

It's like feeding Gremlins after midnight.
posted by sparklemotion at 12:39 PM on January 14, 2015

Thread saved for later enjoyment. Thanks, Wolfdog.
posted by benito.strauss at 12:45 PM on January 14, 2015

Just wow.
posted by Wolfdog at 1:23 PM on January 14, 2015 [1 favorite]

Jeez Wolfdog, how about a warning before you post a Basilisk on the blue?
posted by X-Himy at 2:24 PM on January 14, 2015 [1 favorite]

Well, not regular pentagons, but did you know that there are now fourteen different types of pentagons known to tile the plane? Whether there are more is currently not known. But here are their pictures and the story of Marjorie Rice, the woman who discovered them.
posted by Obscure Reference at 2:44 PM on January 14, 2015 [9 favorites]

(Well, she only discovered some of them, but it's an interesting story.)
posted by Obscure Reference at 2:50 PM on January 14, 2015

True fact: I know Kaplan well. So any disrespect and I'll report directly to him. And he has tiling powers.
posted by clvrmnky at 2:54 PM on January 14, 2015

Is it wrong that I'm so charmed by the reference to monsters in the the second link?

I've had it with these mothafuckin' rhombi, on this mothafuckin' plane!—Johannes Kepler

posted by XMLicious at 4:10 PM on January 14, 2015 [3 favorites]

What am I supposed to do with all of these ceramic pentagon tiles I was going to use to tile my bathroom floor?
posted by double block and bleed at 4:50 PM on January 14, 2015 [2 favorites]

If they're the right sort of pentagon, you can make a very nice floor tiling. In high school, I tried to convince my guitar-playing friend to repaint his guitar with just such a Cairo tiling, but that never transpired.
posted by Wolfdog at 5:12 PM on January 14, 2015 [1 favorite]

Oh, and if this sort of thing turns you on, there is lots more on Craig's UW page: http://www.cgl.uwaterloo.ca/~csk/
posted by clvrmnky at 6:23 PM on January 14, 2015

After reading this today, I was disporportionately pleased with myself for figuring out how to fit hexagons together into a large pentagon [small PDF]. I feel like it ought to be the board for some sort of connection game in the vein of Hex.
posted by Wolfdog at 6:27 PM on January 14, 2015

That link goes to a Wikipedia page, Wolfdog.
posted by benito.strauss at 6:42 PM on January 14, 2015

So it does. Let's try again: How to fit hexagons together into a large pentagon [small PDF]
posted by Wolfdog at 6:50 PM on January 14, 2015 [1 favorite]

How many kinds of hexagons do you have in there? I could see two for sure---are there more? If not, that'd be especially nice.
posted by leahwrenn at 8:07 PM on January 14, 2015

Just two kinds.
posted by Wolfdog at 1:45 AM on January 15, 2015