Okay, so it's not even really a puzzle but I found it at MOMANY and I think it's fun to play with (and related to geometry at least...). Basically it's one of those magic photo cubes (with the hinges and stuff) but it opens up to make 2 stellated rhombic dodecahedra (kind of hard to explain, you might just have to watch the video). I posted it on Reddit (my first Reddit post, would be nice if you would bump it up for me ) and on YouTube so you can see, but I'm wondering if you could make something similar (but perhaps less interesting) with some other shape that's related to the cube, like an octahedron. Any thoughts?

My mind is blown into tiny pieces.....oh my gosh haha

Two truncated octahedra can be hinged and combined in exactly the same way as the Yoshimoto cube. This is merely a different way of bisecting the eight small cubes, but is probably quite nice too. It should actually work better because the hinges of the two parts will be side by side instead of one inside the other.

Edit:
As for completely different shapes, you could look into hexaflexagons (which are 2d) and their 3d counterpart called kaleidocycles.

I actually just won a Yoshimoto cube auction. Yay! I've wanted one of these for so long, but didn't want to pay Moma's price. Saved 25 bucks.

Yeeeaars ago, when I was only 4 or so, I had a version of the Yoshimoto Cube that was called the Shinsei Mystery. I had picked it up at a little church yard sale/flea market and my mom bought it for me. It was missing its identical half, but I didn't know at the time. I thought it was the coolest thing in the world. Tragically I lost it only a few days later because I brought it with me everywhere and must have left it lying somewhere... Oh well.

Anyway, Moma sells a set of all three forms of the Yoshimoto cubes for $250(!). They look cool, but that price tag is ridiculous. link

Aslo, here's a cool page about Kaleidocycles (where you can also see the Yoshimoto Cube no.2/Shinsei Miracle): link

Hi MDW,
I have that one, the Shinsei Miracle, it's really fun, but mine is old and if I use it too much I'm afraid of wearing it out.

Thanks guys! I didn't even realize that there were two other variants of the Yoshimoto Cube, and I'm especially excited about the Shinsei Miracle variant (hooray for rhombic triacontahedra)!! Interesting, though, that they are selling the whole shebang for $240 when the one costs $70...

EDIT: or is it rhombic dodecahedra? I looked at a video and couldn't quite tell...