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/po/ - Papercraft & Origami

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File: origami_solar_panels.gif (1.42 MB, 500x281)
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I figure it might be a cool idea to have a thread dedicated to all things math and science related to papercraft and origami.

Feel free to share anything inherently sciency or mathematical about our shared hobbies, and even include science inspired patterns and models!

To kick us off we have the (pic related) origami-inspired solar panels that one day might be on a satellite.

File: doubling the cube.png (44 KB, 1197x293)
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Another thing to kick off the thread:

Ancient Greeks were especially interested in constructing all of Euclidean geometry solely using the restricted tool set of a straight-edge (unmarked ruler) and compass. They soon had three main problems they couldn't seem to solve:
> trisecting angles
> squaring the circle (given a circle, construct a square with equal area to the circle)
> doubling the cube (given a cube, create another larger cube whose volume was twice that of the original).

It was eventually shown that these problems were actually impossible to do with straight-edge/compass constructions, but in 1934 an Italian mathematician Margherita Beloch
Piazzolla discovered and proved that using the power of folding paper, one could indeed double the cube (those familiar with origami also know that you can trisect an angle!).

The construction is as in the picture. First, it's important to acknowledge that doubling the cube relies on constructing a segment of length cuberoot(2). If BC is given as a segment of length 1, then to construct a segment of length cuberoot(2), one can fold the square into thirds (lines PQ and RS, you can do this using Haga's theorem with origami as well!) and then fold the points C and S so that they lie respectively on the line segments AB and PQ. The resulting fold will now create a segment AC of length cuberoot(2)!

Surprisingly, the fold, known as the Beloch fold, went under the radar until 1986 when Peter Messer independently found it and presented it as his own.
File: pyramid_durer.png (174 KB, 655x675)
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Interestingly enough, one of the first methodical paper folders in European history was Albrecht Dürer (21 May 1471 - 6 April 1528), who was a famous artist and art theorist.

In the fourth book of his saga, Underweysung der Messung (I believe it is German for something like "understanding measurement"? maybe a German /po/ster can help), Dürer explores 3D shapes via unfolding them (pic. related, a tetrahedron with two projections and its unfolded form).

A translated quote:
>[m]any other solids can be constructed which touch a hollow sphere with all their corners but have irregular surfaces [ungleyche felder]. Some of these I shall draw below, opened up, so that anyone can fold them together [selbs zamen mu e g legen] by using two layers of paper, glued together, and then cut to the depth of one with a sharp knife. It is then easy to fold it along the edges. Use this method for the following figures. They can be utilized in many ways.

If anyone is interested I am following along the book here: https://www.springer.com/us/book/9783319724867
File: 21076-pdf.pdf (1.8 MB, PDF)
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While we're at it, here's a copy of Euclid's Elements. Was a standard textbook throughout the history of western classical education. I think it's worth flipping through for anyone, but especially for origami doers, where it's useful to formalise some of one's knowledge.
File: hartmann_sundials.png (1015 KB, 873x913)
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It definitely is a very cool historical treatise on geometry as well! It has the precursor to Dedekind cuts in it, using Eudoxes' theory.

The attached picture is of paper sundials that were crafted and sold by Georg Hartmann, a friend of Durer's. This was one of the possible inspirations for Durer's nets, as shown in >>572870 .
Japanese use this type of fold for their space program
Do you have a link? A preliminary google search only brings up stuff about NASA and it is all hypothetical stuff.
There is a fun theorem, the "fold and cut" theorem.

Take an ideal sheet of paper, fold it as you wish so it lies flat at the end of the fold and then cut all the layers allong the same straight line. You will obtain at least two parts, unfold them and look at the shape you obtained.

The fold and cut theorem states that using this procedure you can obtain every shape you want providing it has polygonal borders. You can even obtain something which is not convex, connected or simply connected. You can find some crease patterns on the internet for different fold and cuts designs. I once made one for a dick and a svastika (welcome to 4chan), i'll post them if i can find them on my computer.

The proof of this theorem is, however, not fun at all.
I like that theorem! Unfortunately I don't like the exposition of any of the Demaines' work. :(
any interesting application?

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