### John Baez

Shared publicly -**The joy of tilings**

Ever since I was a kid, I've loved the ways you could tile the plane with regular polygons. Some are used for floor tiles - pondering these is a great way to stay entertained while sitting in public restrooms. But unfortunately, a lot of the fancier ones have not come into wide use.

There are 3

**regular**tilings: you can use equal-sized regular triangles, squares or hexagons to tile the plane. If you let yourself use several kinds of regular polygons in the same tiling but demand that every vertex look alike, you get 8 more choices: the

**uniform**tilings.

Only recently did I learn about the

**n-uniform**tilings, where you relax a bit and let there be n different kinds of vertices.

The picture here supposedly shows a 4-uniform tiling. But I must be sleepy or something because I'm only seeing 3 kinds of vertices. I see one kind where:

a blue dodecagon, a green hexagon and a red square meet

one where:

a red square, a green hexagon, a red square and a yellow triangle meet

and another where:

a red square, a green hexagon, a red square and a yellow triangle meet.

The last two sound the same! But they're different in this way: no symmetry of the whole tiling can carry the first to the second. You see, one is closer to a blue dodecagon than the other!

I just see these 3 kinds. I'm allowing reflections as symmetries. Otherwise I could get one more kind... but I'm pretty sure reflections are allowed in this game!

**Puzzle 1:**what's the 4th kind of vertex?

According to the experts, there are 20 2-uniform tilings. There are 61 3-uniform tilings. There are 151 4-uniform tilings. There are 332 5-uniform tilings. There are 673 6-uniform tilings. And I guess the list stops there only because people got tired!

Of course I'd be delighted if I had spotted an error in this list. But I probably just need more coffee! That's how it often works in math.

This picture was drawn by +Tom Ruen. You can find it, along with lots more, here:

https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons

I think some of these should be deployed as bathroom tiles in public restrooms. We supposedly have this great, high-tech civilization, yet we're not taking full advantage of math in the decorative arts!

**Puzzle 2:**what uniform tiling is this 4-uniform tiling based on, and how?

#geometry

56

12

18 comments

Steve Wenner

+

1

2

1

2

1

The subject of plane tilings is indeed huge. I find tilings with regular polygons too restrictive and prefer the richer possibilities of marked isohedral tilings (any tile can be mapped to any other by a symmetry of the tiling that also preserves the interior markings on each tile). M. C. Escher explored many of the 93 types of marked isohedral tilings in his fanciful tessellations. These possibilities are further enriched by considering how color symmetry can be folded in. The definitive work on this stuff is “Tilings and Patterns” by Branko Grunbaum and G. C. Shephard. I had a lot of fun 25 years ago classifying Escher’s tessellations according to Grunbaum’s & Shephard’s scheme and creating animations to morph similar types back and forth. (I linked to my video before; but, at the risk of being a repetitive boor, here it is again: https://www.youtube.com/watch?v=EymU_8ZVEzA

Add a comment...