Polytans, polyaboloes, whatever you want to call them

Polytans (also polyaboloes, depending on which website you're looking at) are the shapes made from joining isoceles right-angled triangles (45°-45°-90° triangles*) together edge-to-edge. Theres one 1-tan, which is just the triangle on its own, then three 2-tans, four 3-tans (tritans? triaboloes?) and 14 tetratans/tetraboloes. The numbers grow colossally fast compared to the numbers of polyominoes, polyiamonds or just about anything else; probably partly because there's often more than way to append a triangle to an edge.

Here's some pretty pictures of the 1- through 4-tans:

Fig. 1: Here they are, courtesy of a tedious as balls half-hour in Microsoft Paint. Upon completing this I realised I could have just generated them using Peter Esser's solver and took a screen shot. You live and learn.

Above this, there are 30 penta-tans, 107 hexa-tans, 318 hepta-tans and none of those look like real words, I can kinda see why the 'aboloes suffix gets used. Yeah, pentaboloes and hexaboloes rolls off the tongue a lot better.

I have made little acrylic sets of the 1- to 4-aboloes to play with. The larger sets I haven't gotten around to doing yet, partially because coronavirus and lockdown and all that, and also because the place I usually get them made have upped their prices and I've only got so much annual budget for polyomino-related spending. But tetraboloes are more than enough of a fiendish challenge in the mean time.

Surprise, surprise, I still can't take a decent photo for toffee.

The combined area covered by the triaboloes and tetraboloes is 34 which is a bit of an ugly number but it's still workable. For a start, we can do rectangles of area 36 with the corners snipped off, as in the image below. this woks for 6x6, 4x9 and 3x12 rectangles.

Fig. 2: The triaboloes are highlighted in a slightly lighter shade of yellow.

 Difficulty-wise, the thin rectangle doesn't seem noticeably easier or harder than the square, but then again they're all infuriatingly tricky for something so deceptively simple-looking. Best technique seems to be to try and use up pieces with lots of diagonal edges first. But that only gets you so far. Prepare for lots of trial and error.

And when you turn over the tray I made for them there's the following configuration, which is just unfairly difficult. The centre requires the square shaped bit, leaving the remaining 13 pieces to fill the square doughnut around it.

Fig. 3: The design on the other side of the tray. Finding a solution to this is left as an exercise for the reader.

Apparently there are 45 solutions to this. I've sunk literally hours into it by hand and found only one so far.
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* Fun fact: It's insanely hard to describe specific triangles without a diagram.