Penrose tiles

[Curious layman]

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

https://en.m.wikipedia.org/wiki/Penrose_tiling

[joigus]

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:

https://www.youtube.com/watch?v=2oPGmxDua2U

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.

[Ghideon]

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?

https://en.m.wikipedia.org/wiki/Hexagonal_tiling

 

[joigus]

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

[Curious layman]

Thanks Joigus,  +1