# Post

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

What I really want to know is what you

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

*This post was originally on Google+*