Just Triangles (Polyforms III)

In a former, more optimistic life, I wanted to write a book for elementary school children that would get them excited about math and proofs. This would of course go against the grain. Proofs have essentially been eliminated from all education until the beginning of graduate school. With good & evil reason: Not because they are too difficult or not important enough, but because it could possibly induce the children to come to their own conclusions.

I also was ignorant about who controls public education: Neither the students, nor their parents, nor the teachers, and not even the text book authors. It is solely those people who are making money with it.

Before I get the reputation to be yet another hopeless conspirationist, here is another message in a bottle, in multiple parts. It is once again about polyforms. I need to say what the shapes are that we are allowed to use, and what we want to do with them. In the simplest, we are using four shapes, which are deflated/inflated triangles like so:

They already have received names, which count the number of edges that have been inflated. We (you) are going to tile shapes like these that have no corners:

We call these circular regions, because they consist essentially of a few touching circles with a bit of filling to avoid holes or corners. The circular regions above consist of two, three, and seven circles, respectively, and they already have been tiled. We can start asking questions: What shapes without corners can you come up with? Are they all circular regions?

Then there is time for exploration: Find all ways to tile a circle (the circular region with just one circle) with the curved triangles. Find all ways to tile the bone (the circular region with just two circles) using only two different kinds of triangles:

Now, upon experimenting, the number of curved triangles to be used to tile a circular region is not quite arbitrary.
The first, not so trivial, observation is that for a circular with N circles, we will need 8N-2 triangles. That is because each circle contributes 6 triangles, and for adding a circle we have to use 2 more triangles. This is not quite a proof yet, but at least an argument. There are also interesting problems when the domain is allowed to contain holes…

Because the different curved triangles contribute a different amount of area each, there is a second formula.
Let’s denote by N(0), N(1), N(2), and N(3) the number of curved triangles of each type (zero, one, two, three) that appear in a tiling of a circular region with N circles. Then N(1)+2N(2)+3N(3) = 12N. This formula counts on the left hand side how many triangle edges are inflated and hence contribute extra area. On the right hand side, we count the same, using say the pattern we see in the tiling of the circular region with 7 circles in the second image above.

The two formulas together allow you to determine how many triangles of each kind you need in an N-circle region, if you are only using two different triangles.

To be continued?