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Solving cubic equations with origami

PostPosted: August 19th, 2018, 1:46 am
by Merlyngami
(WARNING: this will not make much sense if you do not know what cubic equations are)
Not sure if this is the correct place to post this, but I would like to share a fascinating application of origami to math that I have just learnt of. Basically, folding can be used in conjunction with a pen and a ruler to solve cubic equations (3rd degree polynomials) of the form ax³ + bx² + cx + d = 0

Here is the video I learnt it from:


I find it interesting that using folding is the only way cubic equations can be solved geometrically on paper. A compass and straight edge cannot do this. I highly recommend you watch the above video and give it a go yourself if you are interested by both math and origami. It takes a bit of getting the hang of to do yourself, but the video explains it pretty well. This one also helps: https://www.youtube.com/watch?v=6MZEBB24Irg

Apparently, through origami is also the only way you can trisect an angle. I wonder what other applications origami has to math?

Re: Solving cubic equations with origami

PostPosted: August 24th, 2018, 1:25 am
by Baltorigamist
Is it considered a cubic equation if you need to reflect two points onto a line simultaneously?

I don't know if you're familiar with the Huzita-Hatori (Huzita-Justin) axioms, but they're pretty fascinating and also relevant to this video.

Re: Solving cubic equations with origami

PostPosted: August 27th, 2018, 5:40 pm
by OriGenes
Baltorigamist wrote:I don't know if you're familiar with the Huzita-Hatori (Huzita-Justin) axioms, but they're pretty fascinating and also relevant to this video.

It is Axiom 6, which I just implemented in Origami Draw, my app for iPhone and iPad (see this thread). Here is an animated GIF showing the construction of all three roots:

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and some static images:

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It's easier than drawing by hand! :wink:

Re: Solving cubic equations with origami

PostPosted: August 28th, 2018, 4:49 pm
by Baltorigamist
Here is an interesting paper I just found which covers an analogous topic in simpler(?) terms. It's definitely an interesting read.

Re: Solving cubic equations with origami

PostPosted: August 29th, 2018, 4:34 am
by Merlyngami
Wow, this is fascinating. Thanks Baltorigamist, I tried to find a paper on the beloch square but couldn't (until now). Sorry if this has been asked origenes, but will oridraw ever be ported to android?