Tiling the non-Euclidian Plane

I’ve always been intrigued by the soccer ball with its hexagons and pentagons, and with the mind-bending artwork of M.C. Escher.  The fact that the two were mathematically related totally escaped me until recently.  I do OK with basic arithmetic, but quickly get lost when they start mixing non-Roman script with the variables.  So I approached the VizMath MOOC with an expectation that I was going to be totally lost.  I am lost, but wander in amazement at the genius and the beauty.  I am not particularly worried about finding my way.

Daina Taimina’s crocheted creations are a good example.  While not even pretending to understand how these might represent (or fail to represent) the shape of the universe, they drew me in, made me believe I might copy them, if never understand them.  It wasn’t until she showed us the rough approximations of her work in hexagons though, that I found my inspiration to act – and blog.

positive vs. negative curvature

In an attempt to share some of the wonder with my ABE math students, I created a sheet of pentagons, hexagons, and heptagons to cut out and tape together.  There are enough polygons on one sheet to make a complete circle of both curved and negatively curved (hyperbolic) planes.  Extension of the planes, either to a complete sphere or complete insanity will require additional sheets, and probably collaborative effort.

download linkClick the image at the left to download a PDF version of my worksheet.

(And yes, I do know there are extra hexagons. I couldn’t bring myself to waste the white space when it’s so easy to crowd them.  Just be thankful my childhood training in frugality wasn’t so overpowering that I had to tile them.  I was tempted…)


About Jim

Community Adult Educator & Adult Literacy Instructor
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3 Responses to Tiling the non-Euclidian Plane

  1. Pingback: Hey! Something weird is happening | WayUpNorth's Blog

  2. OOooooh, lovely. How did your students like it? I would have loved that in my math class.

    • Jim says:

      Giulia, they liked it, but I think I liked their response even more. They quickly realized that combining hexagons and pentagons made “a soccer ball”. But when they combined heptagons and hexagons and the hyperbolic plane started to take shape one said, “Hey, something weird is happening. Look. Is it supposed to be like this?” It was priceless serendipity that he was playing Pink Floyd’s “Another Brick in the Wall” on YouTube at the time. I’ll have to write another blog post about it with photos.

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