Concentric Rectangles

I'd like to blame the fact I'm stuck indoors avoiding the Coronavirus for this one, but realistically I'd have probably found the time to solve this anyway, given that I lead such an exciting life:

It's an awful photo, but here's the gist of it. Imagine a 7x9 pentomino rectangle (with a little 3-cell hole) inside a 13x21 hexomino rectangle, inside a heptomino rectangle I can't remember the dimensions of and can't be arsed to count, all surrounded by the octominoes (plus 8 holes) in a whopping 47x85 rectangle. And that's what I spent an entire afternoon doing. Well, two sittings with a break for dinner in the middle.

It would maybe have been a bit nicer if I'd started with a central monomino and worked my way up through all polyomino sizes, a la Karl Wilk's Polyominium, but there didn't seem to be a way of doing it that yielded such nice symmetrical layers like this. It's difficult to wrap a pentomino rectangle around a hole big enough for just the tetrominoes, let alone anything else.
In fact, when I had the original idea that became this, it was born out of the fact I'd built a hexomino pattern that just happened to be able to fit the 7x9 rectangle inside.

A sort of precursor solution found way back.

Peeps with a keen eye will have spotted that the pent-, hex- and heptomino sections of that early solution are totally different to the ones this time round - more a testament to how bored I was than anything else.

Fun fact about the 7x9 pentomino solution: when the triomino hole is vertical there are 360 possible solutions, whereas with the horizontal hole there are a mere 150 (excluding rotations, reflections and all that jazz.)

Here's the full solution drawn up so you can actually see where one piece ends and another begins:

And there's a little voice inside me saying "What about a layer of nonominoes?" but realistically I'm not going to be able to do that without a physical set of them and that ain't gonna be cheap. Besides, there's not a flat surface in my house big enough to hold all those pieces.

Octiamonds

Polyiamonds are tricky. Even the smaller sets seem harder than their similarly-sized polyomino counterparts. Hexiamonds, for example. There's 12 of them, same as the pentominoes, and in a geometric sense they don't seem any more jagged or otherwise unruly, but for whatever reason solving anything with them seems a lot harder.
I mean, it could just be that I'm more used to polyominoes and that a sort of intuition for other polyforms would build naturally with time. And with polyominoes I know all the handy tips and tricks, which pieces (or types of pieces) are most useful in which situations, whereas with polyiamonds I don't have that (yet).
With larger sets of polyominoes, (i.e. hexominoes and above) a technique emerges of saving the more cooperative shapes for the end game, and as the sets get larger this pool of 'nice' pieces increases rapidly in size. But with (say) octiamonds it's not so obvious which shapes are the most useful. They're all pretty hideous, actually, at least to the untrained eye. The little hexagon made of six triangles is the closest analogue I can think of to the 2x2 square block that makes for nicely-behaved polyominoes. But there are only 4 out of 66 octiamonds which contain it and even these 4 pieces don't play especially nicely together with each other.

So the solution below was the result of about an hour and a half of stumbling about cluelessly followed by a flash of pure luck.

Fig. 1: The 66 octiamonds in a 12x22 parallelogram.

 Another fun fact: Drawing these out neatly is really hard too. Pixel art and triangles don't mix too well.

Finally! Truncated 55x55 Square with Octominoes

Third time lucky, eh?
Getting that central 13-hole configuration to work was surprisingly tricky - they're too close together to just treat as individual holes, sling a holey octomino around some of them and be done with it.*

Then throughout the rest of the solve I had in the back of my mind a little nagging concern that maybe some issue like parity would render this solution impossible anyway. The octominoes as a full set have no glaring issues the way the tetrominoes and hexominoes do, but the 363 non-holey ones are imbalanced when checkerboard-coloured. And since I'd used the 6 holey pieces first I was in effect left with this imbalanced set. I assumed (well, hoped) that since the construction's dimensions were odd x odd this might negate the issue; I use this as a rule of thumb for hexomino constructions because it usually means that the overall structure is sufficiently unbalanced and therefore solvable.
Whether this makes any sense mathematically I have no idea.

Total solve time was approximately 5 hours spread over a few days. Total time drawing up the digitised image of the solution probably took another hour on top of that, come to think of it.

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* This construction by David Bird does something similar, there's probably only a handful of ways of accommodating those holes in that shape.