Pellagons

Some New Tattoo Designs and Bad Names

One of my best recent impulse purchases was a notebook of hexagonal graph paper.¹ I would expect that the majority of hex grid paper purchases are made by chemistry folks², and would bet that the remainder is swallowed up by tabletop gamers. As an engineering undergraduate, I was already infected with a lifelong need to use the regulation yellow grid paper for everything.³ The square grid is great for square patterns but as even the bees know, the hexagon is the Best Shape. I didn’t have much of a thread to chase down with my new doodling medium until I was exploring (so-called) Pell Tiles and asked the natural and very cool⁴ question “but what if they were hexagons?“

Instant Icons

The Pell Tiles post might give a better introduction to this concept, but the rules of this geometric construction are pretty simple. Begin with one hexagon at the center, and wind incrementally longer strips of hexes around it. Here’s a look⁵ at the first six strips:

The first two non-trivial convex⁶ shapes that fall out are easy to spot. the second strip completes a simple but neat trifecta, and the fourth strip completes a sort of hexagonal oval. The next one, though, gleefully defies (my) intuition.⁷

The Sixth Hexangular Form

A wonderful conflagration of symmetry and lopsidedness

When I was sketching this out the first time, I almost skipped right past this shape. It doesn’t appear on first blush to meet the qualifications the first two do, but it turns out to be a sneaky entry into the “convex” pantheon. We’re going to have to get into definitions after all.

Consider the boundaries of each of the complete “convex” collections of hexagonal strips (from now on I’ll just call these hexangles) we’ve looked at so far. Each one is straight and uninterrupted. Taking away or adding a hexagon to any of these boundaries causes a jagged edge that ruins the convexity of the hexagon set. The concept is a bit more foreign, especially compared to the familiarity of rectangular convexity.⁸

Some more examples of convex hexangles, unrelated to the incremental strip construction.

Here are a few non-convex (aka concave) examples, since sometimes that goes a longer way in helping a concept stick:

Note the caves.

So you’re convinced about the sixth strip. Now we’ve got the rules to make a pattern we can start hunting.

Bounding Bags

Hand drawing, I can find convex hexangles at strips 1, 2, 4, 6, 13, 15, and 24. These each contain total hex areas of 1, 3, 10, 21, 91, 120, and 300 respectively. The latter set is necessarily a subset of the triangular numbers, but this filter doesn’t seem to be much used. Neither of these sequences are anywhere to be found in the OEIS.⁹ When we looked at rectangular Pell Tiles, this was how we found generating functions for new tiles, how big they were, and which strip completed them. Now we’re at a bit of a dead end (or, phrased slightly more optimistically, a new frontier).

Part of the new wrinkle is that the “bounding box” we used for Pell Tiles followed a relatively easy to follow pattern that is called the quarter-squares. That processional pace (1+1+2+2+3+3…) has a hex analog that takes a bit funkier form, because adding a “side” to a hex honeycomb isn’t so neat and tidy. The first few “bounding bags” we care about aren’t as interesting since they’re made by just adding a single hex, so I’ll skip to 5 hexes.

Looks like our sixth hexangular form wasn’t such an exception.

This is an even more herky-jerky pattern than the quarter-squares. Starting from the beginning, the marginal number of hexes added to make the next largest bounding bag in this scheme looks like

\(1+1+1+1+1+2+1+2+2+2+2+3+2+3+3\ldots\)

which generates a sequence nobody has any reason to know on sight…

\(\{1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21,\ldots\}\)

but one that can be found in the integer sequence bible. These represent the set of possible areas our convex hexangles can have. When intersected with triangular numbers (recall the nth triangular number is the sum of 1 to n), we have all of the sizes of convex hexangles constructed by winding incrementally larger strips of hexagons around a unit center.¹⁰

Here are the first dozen members of this set. The first few I found by hand and was relieved to see pop up when running a quick script. The rest just look like big fuzzy hexagons. I’ll take it.

\(\{1, 3, 10, 21, 91, 120, 300, 990, 2080, 8911, 11781, 29403,\ldots\}\)

The associated strip numbers:

\(\{1, 2 , 4 , 6 , 13 , 15 , 24 , 44 , 64 , 133 , 153 , 242 , \ldots\}\)

Looks like the herky-jerkiness survives into the final sets. The gaps are all over the place. Keep your eyes peeled for this OEIS newcomer! Here’s up to the 24th strip:

1. This might be the most swirly-deserving sentence I’ve ever typed.

2. And they are advertised this way. Lots of carbon chain diagrams to draw!

3. Not even exaggerating here. Work notes, doodles, even letters to loved ones go on this pad. It’s an unimprovable product.

4. Don’t swirly me.

5. As always, huge shouts to manim.

6. There’s a better word here I suspect. No collection of hexagons is literally convex, but the concept of rectangularity that Pell Tiles leans on doesn’t translate 1:1 into the hexagonal tiling. I get more into this in a bit.

7. I think the groups formed by the fourth and sixth hex strips are both really pleasing graphically. I’m between one of these and the eighth-strip Pell Tile for my next tattoo.

8. I wouldn’t die on this hill, but I think a more formal definition for a convex hexangle would require that the set of centroids of the included hexagons were convex (i.e. the boundary hexes’ centroids represent all hexes’ centroids’ convex hull).

9. Yet!

10. What is all this loose hair doing on my desk?