 - 2013-06-02 17:23:09+0000 - Updated: 2013-06-02 17:23:09+0000
Originally shared by John BaezI asked Greg Egan: what happens when you try to tile the plane with regular pentagons and decagons (10-sided shapes)?   And he sent me this.

You can get two regular pentagons and a regular decagon to meet snugly at a point: if you do the math, this is because

1/5 + 1/5 + 1/10 = 1/2

But you can't tile the plane so that two regular pentagons and a regular decagon  meet at every vertex!

These are all the other failed tilings of this sort:

1/4 + 1/5 +1/20 = 1/2
1/3 + 1/10 + 1/15 = 1/2
1/3 + 1/9 + 1/18 = 1/2
1/3 + 1/8 + 1/24 = 1/2

and my favorite:

1/3 + 1/7 + 1/42 = 1/2

Puzzle: Why do none of these work?  For example, why can't you tile the plane so that a regular triangle, octagon and 24-gon meet at each vertex?

What I really want to know is what you can do with these failed tilings.  Egan's picture here is a hint.  Another is here:

http://gruze.org/tilings/fathauer

#geometry #tilings Shared with: Public
Reshared by: Ivan Zvolsky

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