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I've seen a paper 3x3 on the internet some months ago and it really fascinated me. So I build one and it went pretty well (one axis broke when I tried to scramble it the first time though:/ ). After I've built it I thought about if one could build other puzzles out of paper and I ended up trying a Megaminx. It doesn't have internal hidden pieces, so it's not that hard to build. After several hours of designing the pieces, they finally fitted and I went on to build all the pieces. I'm still in the process of building them, but I'm far enough now to show first photos of it.

You can see the Megaminx in his current state, in half-turn and the internal mechanism.

I searched on the internet and it seems like I'm the first one to build a Megaminx out of paper. Please correct me if I'm wrong.

I know this thread doesn't really fit into this category, but it's the best category I was able to find.

Are there any screws or springs inside the center stalks/core? What actually allows it to turn?
I really want to see a video of this when you finish; that's amazing.

No it's 100% made out of paper (and superglue of course). In the centerpieces there is a kind of "disc" which is glued to the axis. So the centers can turn freely without falling out.

Nice work, sir. Nice to see something different for a change. I'm a casual Origami enthusiast, and among my many books on the subject is this one, "Mathematical Origami": https://www.amazon.co.uk/Mathematical-Origami-Geometrical-Shapes-Folding/dp/189961818X
If you can obtain it at a reasonable price, it's worth buying. These days, it's likely much of the contents are available in some way free on the web, I haven't looked. I bought my copy before I got online, probably about 2001.
Edit: Had a quick look, here's the authors website: http://www.origamiheaven.com/diagramsindex.htm

Nice work, sir. Nice to see something different for a change. I'm a casual Origami enthusiast, and among my many books on the subject is this one, "Mathematical Origami": https://www.amazon.co.uk/Mathematical-Origami-Geometrical-Shapes-Folding/dp/189961818X
If you can obtain it at a reasonable price, it's worth buying. These days, it's likely much of the contents are available in some way free on the web, I haven't looked. I bought my copy before I got online, probably about 2001.
Edit: Had a quick look, here's the authors website: http://www.origamiheaven.com/diagramsindex.htm

No it's 100% made out of paper (and superglue of course). In the centerpieces there is a kind of "disc" which is glued to the axis. So the centers can turn freely without falling out.

I think you mentioned wanting to make a "double bandaged" square-1 in one of your Q&As (even more bandaged than your "square-0", I guess you could call it a "square-negative 1".)
You should try to make that out of paper

A 4x4 would be way more challenging because it has hidden internal parts and I guess it would be really hard to build them.
Maybe I'll try a Square-1 after I finish the Megaminx.

So after working on it for about 12 hours today, I finally finished it.
Thank you all for your very positive feedback, it really gave me the motivation continue it after it's been sitting on my desk for over a month before I posted it on here.
I can only share photos of the Megaminx by now, I guess a video will be following in a few days, but I was just happy to finish it today and I wanted to let you know.

You can see it solved, in half-turn and scrambled (not a very good scramble, I just handscrambled it a bit).

Ripping isn't a problem at all, since the pieces are pretty stable because of the closed 3D shape. I'm planning on taking this with me to Munich Open 2017, and I don't have that many concerns about it crumpling.

The cube itself is pretty stable. No pops at all. But sadly there do occur many lockups because the pieces are not rounded of. I concidered doing that, but it would have been way more difficult to build the pieces then.

I used regular printer paper, but I'll use thicker paper in my next build.

It would be quite a bit of work for me to work out a good instruction sheet because all the pieces are drawn by hand and I haven't used dotted lines for the cuts and usual lines for the folds so I would have to draw them all again to be able to make understandable instructions.
But if a few people would be interested in instructions then I'll make them.

And just so you have an idea of what you have to build for it, I attached a pdf of the core dodecahedron. It should have an edge-length of about 5 millimeters.