A Few Parallelograms with Heptominoes

Edges with gradient 1:2 are surprisingly forgiving. When I started building this I had no idea just how long and skinny the resulting construction would be; I kind of pictured it in my head looking like the 45-degree parallelogram in this truly ancient blog post until I started building and it sprawled right out across the table.

Fig. 1: A 23x33 parallelogram

 

It turns out that the gentler slope here permits a much wider variety of pieces so it was easier than I'd anticipated. Solve order was like this: Started at the top-right and worked down the right-hand side, then continued around the perimeter of the shape until I reached the top of the left-hand side. This left a long thin internal section to be filled up last, finishing up at the top edge.

(A random sad fact: If some utter maverick decided to solve a heptomino construction the hard way, by using up all the blocky nice pieces at the start and finishing with the most hideous bits imaginable, everyone would just assume they'd solved it the regular way starting at the opposite end. All that effort for nothing.)

Edges with slope 3 should be easier still, since they're a step closer again to straight edges, and therefore even less restricting as to which pieces can be used to build them. But at this point the parallelograms themselves become loooong, and there's not a surface in my house that can comfortably accommodate them, short of the floor (although there's an idea...)

There's also another consideration here - the flatter the gradient, the fewer pieces can fit into the very thin ends of the construction. This limitation clearly rules out parallelograms with a 1:7 or greater gradient, but could make others impossible too. There aren't too many heptominoes kicking around that can fill a six-cell well, that's for sure.

And of course the sloping edges can go the other way, two squares up for one square along, as in the example below:

Fig. 2: A 19x40 parallelogram that doesn't lean quite as much as the other one.

I know, I know, the central holes aren't perfectly aligned with the long edges, but they're about as good as you can get I think. Holes positioned to match the gradient of the edges tend to like odd-number length vertical spacing between them, which doesn't play nicely with the (even) total height of the construction.

I could find more. I really should do. But I didn't want to go more than a month without a blog post so here. Have this lazy half-baked one.