I found Project Origami: Activities for Exploring Mathematics in my university's library the other day and quickly FUBAR'd (folded-up beyond all recognition) the couple sheets of paper I had with me at the time.
I showed the book to a couple friends who have studied some topology, and we looked at one project with the goal of creating a double torus out of paper. Apparently, it's possible to make one out of an octagon of the infinitely stretchable material of topology-land following a diagram like this:
Just connect the edges labelled with the same letters so that the arrows point the same way, and there's a double torus.
I had trouble conceptualizing how the surface contorted in 3-space, even though I could understand why joining an octagon's edges like that makes a double torus (two single torus diagrams joined at a corner, stretched into a diagonal of the octagon).
My question is: what can I do to improve my conceptual understanding of what's going on? If I didn't know what a single torus diagram looked like, I'd have no clue what the octagon diagram was creating.
Is there some kind of software that will let me play around with infinitely-stretchable material and join edges?
Anywhere I can order really stretchy origami paper?
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I'm no topologist, I just like to sketch things.
We know that the $ca$ loop can be pinched apart like that because the endpoints of the other $a$ and $c$ segments are identified already.
Now it just remains to attach two handles to connect $a$ to $a$ and $c$ to $c$.