## Friday, June 17, 2016

### World Tessellation Day One

Emily Grosvenor came up with the idea of a World Tessellation Day in connection with her charming children's book, Tessalation! June 17th is M. C. Escher's birthday (1898) and there could be no more fitting day.

Tessellations are definitely my favorite topic in mathematics. The intersection of history, art, geometry (shape and transformation), algebra, and even analysis... what could be better. Some of the greatest surprises in math have come from tilings (quasicrystals, pentagon 15) and some of the greatest mathart. I've seen them engage students of all ages.

For my post, I've been thinking about so many things, but that coalesced into a 'My Favorites' post:

My Favorite Tessellations

HM: pattern blocks.

From a recent class, Hannah made this neat dodecagon and octagon tiling. They remind me a lot of these from Simon Gregg and Daaniel Ruiz Aguilera.

10. Non-Euclidean Tilings

Hyperbolic, especially. Here's a beauty from John Baez's Google+ page.

9. Pythagorean Tiling

A tessellation that demonstrates the most famousest of theorems? That's saying a lot, that is.

8. Archimedean (Semi-Regular) Tilings

So what combinations are possible? Is this all of them? Could the semi-regular tilings be the first of these kind of problems?

And then you add the delicious topological feature of dual tessellation relationships...  The gif on the right is from thinking about a Sam Shah prompt on this idea. (On GeoGebraTube)

7. Pentagon 15

How deep are tessellations? They still surprise us. Every quadrilateral tessellates.

A monohedral tiling is a tiling where all the tiles are congruent. An isohedral tiling is a monohedral tiling where for any two tiles there is a symmetry of the tiling that maps one tile to the other. There are exactly three types of convex hexagon monohedral tilings. (Here's a good NRICH problem with one.) Every convex quadrilateral has a monohedral tiling. And we knew all 14 convex pentagon monohedral tilings. Several by one of my favorite mathematicians, Marjorie Rice. (Her website.)  And then they found the 15th. (GeoGebraTube

6. Pinwheel Tiling

Straight from the mind of John Conway.
5. Spiraling Polygrins

I went from fond of these to berserk when Christopher Danielson started making them. (On GeoGebraTube or TMWYK store.)

4. Rep-Tiling

When a tile can be composed to make a larger similar image of itself. Then it makes a tessellation by either deflating each tile into smaller images. Or inflating by composing larger and larger similar arrangements.

3. Penrose Tiling

These were my exposure to quasiperiodic tilings. There properties are many and wonderful. At one point I was stuck on my thesis and my advisor (Nigel Higson) gave me these to work on. My best ever Mathematica program generated them by projecting n-dimensional integral lattices onto an intersecting plane. For part my thesis I then made quasiperiodic integral operators out of them.

2. Islamic Tilings

Most recently, Daniel Ruiz Aguilera got me working on the Qarawiyyin Mosque Tiling. (GeoGebraTube) Endless riches with new work still being done. As a bonus, these are often interspersed with knotting, another favorite.

1. Escherized Tiling

Instead of mine, let me show some ooooold student work from a couple of preservice art teachers in one of my first courses taught at Grand Valley. I still keep these in my office.

Current: Self-tiling. Since Math Munch unveiled this great Lee Sallows self-tiling I've been curious. They deflate in only one way, but inflate in four ways - I can't figure out what that means about the structure. (GeoGebraTube)

So many types that didn't make the list. And despite the numbering, I'm just crushing on them all.

I hope one of these pave the way for you, or maybe showed you a new kind, or just reminded you of old favorites.

And happy first World Tessellation Day! Tile on, brothers and sisters.