# Penrose tiles

## Recommended Posts

Can someone explain to a layman what they do, Why are they important?

##### Share on other sites

One important reason why 5-fold approximate symmetry is interesting is that you cannot tile the plane with regular pentagons for a very special reason. It's some kind of peculiar geometrical frustration.

If N is the number of sides of a regular polygon. You have,

Triangles (N=3) --> You can tile the plane

Squares (N=4) --> You can tile the plane

N-gones, N>5 --> You cannot tile the plane because angle is too big

N=5 is special because you still have angle left, there's no angular "deficit", but there is a mismatch. Penrose re-discovered this tiling, which appears in some mosques and other religious buildings. The idea is that it creates the illusion of symmetry, but the pattern does not really repeat itself.

Here's an interesting lecture by John Baez on number 5, and why it is an amazing number:

He mentions Penrose tilings, but it's about number 5 in general.

I'm not aware of any practical use, but approximate 5-fold symmetry does appear in Nature. Baez mentions diffraction patterns in some crystals as another example.

##### Share on other sites
45 minutes ago, joigus said:

N-gones, N>5 --> You cannot tile the plane because angle is too big

What happens at N=6, a hexagon?

##### Share on other sites

Sorry, my mistake. N=6 is the last one. +1

##### Share on other sites

Thanks Joigus,  +1

## Create an account

Register a new account