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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
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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