MathHappens intern Asa Grumdahl is interested in Heesch numbers. 

Unfortunately, a formal presentation of Heesch numbers and questions surrounding them is too complicated as a starting place to engage the general public. The question then was “How might we engage novices in this sophisticated bit of mathematics?”

Step one was to name this activity we were designing. “The Surrounding Game” gives you a sense of the basic idea, even before you’ve gotten started. 

We designed a playing surface—a single tile glued in the center of a large regular polygon.

Then we provide you with a bunch of replicas of the shape. Can you surround it completely with copies of itself?

Yes!

Can you surround that whole arrangement?

Notice that there are two colors of tiles, a design choice that makes it easier to distinguish the layers from each other as you work. We also chose to vary the thickness with the color. This further distinguishes the two tile types visually, and increases accessibility for puzzlers with visual impairments. We also chose to darken the boundary of the central tile to further orient you as you play.

This particular tile is from the MathHappens set of Bubble Tiles, and it is impossible to do a second surrounding. You can only surround it once, so its Heesch number is 1.

In more formal language, a surrounding is called a corona. The largest possible number of coronas is the tile’s Heesch Number. 

You cannot surround a circle with copies of itself without leaving gaps, so a circle’s Heesch number is 0. You can surround a square with copies of itself infinitely many times, so a square’s Heesch number is ∞.

It takes some work to convince yourself that you can surround this first shape exactly once, but it is possible with some time and attention. Importantly, it is easy to get started. You just need the question “Can you surround this tile with these copies of itself?” We can talk about Heesch numbers later on if you’re interested.

After you’ve done that, you can swap in a new shape.

This one is much more challenging, but the question is the same (although the answer is different!)

Before we had a physical cardboard prototype in our hands, we wondered whether the underlying task would be “Surround this shape two times,” because we know its Heesch number is 2, or “How many times are possible?”

Our initial explorations have made a convincing case that—for most people—it is plenty challenging to make two coronas with a complicated shape; we don’t need to add mystery into the mix.

A few design choices

• Gluing the tile in the center of the surface keeps everything from sliding around as you play.
• In order to have multiple levels of challenge with multiple tiles—while still conserving space—we designed a cutout in the large surface so you can swap out new tiles.
• We designed a little shelf system so that you can tile and pick up the whole play surface to show your family and friends, and to pose for photos.

We are still play-testing the whole setup, including various tiles with interesting Heesch numbers. We’ll post files to this blog entry once we have a closer-to-final version.

Final notes:

• If you have a set of Bubble Tiles, you can play the Surrounding Game with them. What is each Bubble Tile’s Heesch number?
• Hat tiles also make an interesting investigation. You can play with digital hat tiles on Polypad. Two subquestions here: (1) What if you’re not allowed to flip the hats over? (2) What if the central tile is flipped, but none of the others is allowed to be? What Heesch numbers result in each of these cases?
• Much of our work was supported by Dr. Casey Mann’s 2004 paper “Heesch’s Tiling Problem” which is accessible to many people who are not research mathematicians, and which contains extensive references. Note that this paper predates the discovery of the hat tile, and references the then-unanswered question about aperiodic monotiles. Finally, Dr. Mann is part of the research team that discovered the 15th (and final) type of convex tiling pentagon.
• We do a lot of this kind of prototyping in cardboard. It is recycled, recyclable, inexpensive, and easy to laser cut. The final versions will likely be in 1/8″ and 1/4″ thick Baltic birch. Here is Asa after cutting a large collection of test shapes for this protoyping project.