Pentaboloes

Think of this as an unofficial continuation of this blog post from way earlier.

Polyaboloes (or polytans sometimes, depending on who you ask) are the set of shapes made from joining together edge to edge the triangles that result from chopping a unit square in half. The sets where n triangles are joined together for n = 1, 2, 3 and 4 have 1, 3, 4 and 14 pieces respectively, and these are discussed in detail at the above blog post link. And also at this website link. Which is just the same stuff as the blog post but with a few extra bits snuck in.

Pentaboloes are the next set in the sequence - the shapes made by attaching five of those triangles:

Notice how I was too lazy to trim the edges off the bottom of the tray. And too lazy to paint or varnish or do anything with the pieces.

There are 30 of these in total. Mine are made of MDF, lightly toasted by the laser that cut them. 30 multiplied by 2.5 gives a total area of 75 unit squares, which feels like it should be a good thing; it's not prime or anything. But it means that for rectangles (the most obvious shape to try and pack a set of pieces into) we've only got one option - 5x15 - and it just doesn't seem to be possible. I've tried by hand, and I've let Peter Esser's mops solver churn away at it in the background while writing this, and it seems to get a maximum of 29 pieces placed but no higher. I'm wondering if parity has a filthy triangular cousin, or maybe there's a lack of available orthogonal edge sections within the set of pieces to fully build a rectangle with 40 units of edge.

(This is the point where a proper polyform website would launch into an investigation of just why the 5x15 doesn't seem to be possible, but this is Polyominoes The Blog so don't kid yourselves, all you're getting from me is a shrug and a suggestion to look into it yourself.)

You get slightly better luck with a 7x11 rectangle with the corners snipped off. I have a solution saved from ages ago and I can't remember if it was found by hand or by computer. And there's no obvious way to tell either. With sets of pieces like the n-ominoes for n > 5, I generally save certain pieces for last, so there's a visible gradient across a human-found solution going from wiggly nasty pieces all the way to big blocky square pieces. But with these there are only like three or four really nice pieces - the three consisting of a domino with a half-square attached to it somewhere. And they all end up properly mixed in with the other pieces. Finishing a solution with these is much more perseverance than technique.

One other thing you can do with these that's fractionally easier than solving just using the pure pentaboloes set is to introduce the triaboloes (or tri-tans if you call them that) into the mix, and solve things with an area of 81 units². Here's one of many solutions to the 9x9; this was solved by hand, and it took an absolute age... I just left it unfinished on my desk then ever so often took a punt, desperately rearranging a few pieces. And then one day, after several cumulative hours of trying, the bits I happened to have left fell perfectly into place. This solution has the four triaboloes separated from each other, which looks nice but was entirely unintentional.

There'll be a solution where all four triaboloes are bunched together in one clump, but finding that can be a job for someone else.

I have been toying with the idea of getting the hexaboloes laser cut at some point, but I get the feeling they're going to be even more of an endurance test than these, which doesn't fill me with a lot of motivation.