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May 5th, 2011, 03:16 AM   #1
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Origami constructions

I am reading about this exciting new topic - Origami (new at least to me). I am reading it in Cox, but there is one thing I dont quite understand yet.
In one of his examples, we are suposed to find the real roots of the cubic equation $x^3+ax+b=0$.
As I understand it, The point is that the simultaneous tangent ($y=mx+z$) to the two given parabolas, gives a solution to the origanl cubic equation - namely $m$.
So... I thought there were only one simultaneous tangent, but Cox conclude the example with:
Quote:
 Hence the slopes of the simultaneous tangents to the parabolas are roots of the cubic $m^3+am+b$
.

It would be nice if someone could point out my error(s).

 May 5th, 2011, 05:29 AM #2 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Origami constructions studentx, I hope you don't take offense when I say that I think you err by assuming we know what these parabolas are and moreover what any of the context of this discussion is. I would suggest that you enclose far more detail and/or excerpts from whatever text you have your nose in (it sounds interesting). I can only guess that you are worried that two parabolas (what restrictions placed on them I shall not venture to guess) cannot have two simultaneous tangents. Is this your main concern? If so, take the example of the two straight lines y = ±2x and the two parabolas y? = x²+1 and y? = -(x²+1). By perturbing this example, you can create others. -Ormkärr-
 May 5th, 2011, 06:08 AM #3 Member   Joined: Nov 2010 Posts: 53 Thanks: 0 Re: Origami constructions Thanks, you gueesed right (and thank you for the nice example) Sorry for my assumtions. It was based on my impression (from many of the comments and replies I have read), that some of you know pretty much everything worth knowing about algebra (from my current point of view). But for future topics, I will give more details (at least if it is an exotic topic).
 May 5th, 2011, 06:18 AM #4 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Origami constructions studentx, No worries, I was actually a bit amused by the way you presented your question as if in mid-thought. And maybe a little bit wounded (pride-wise) that I had not heard of this topic! But I am glad the difficulty was cleared up, please report back soon when you come across some attractive result in this field, since I for one would be interested to learn! Also my apologies if I was harsh, I am typing mainly from the medulla oblongata right now! -Ormkärr-
 May 5th, 2011, 07:13 AM #5 Member   Joined: Nov 2010 Posts: 53 Thanks: 0 Re: Origami constructions No worries, no harm done. it is really an intriguing subject. Origami ("ori"=paper and "gami"=folding) is a japanese art concerned with paperfolding. Origami constructions consist only of series of folds on a square paper. The really amazing thing with origami, is that one can do some constructions not possible by straightedge and compass. For example: one can trisect any angle using origami, Duplication of the cube is not a problem with origami and (as I mentioned) one can solve cubic equations with origami! In additon all straightedge and compass constructions can be done using origami. So I think it is a fun topic to study. Here is a link to a talk, which show some interesting aplications of origami: http://www.ted.com/talks/robert_lang_fo ... igami.html
 May 5th, 2011, 07:49 AM #6 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Origami constructions I am familiar with Origami as an art form, just not as a mathematical enterprise. Will look over the talk once I have a bit of time, thanks for the link! -Ormkärr-
 June 14th, 2014, 12:57 AM #7 Banned Camp   Joined: Feb 2013 Posts: 224 Thanks: 6 Neusis They are neusis "solutions". They are practical solutions good only to the resolution to what the human eye can see and the thickness of the physical drawn line. Cube Root https://www.flickr.com/photos/85937466@N02/14119217417/ Angle Trisection https://www.flickr.com/photos/859374...57636438514124 Euclidean geometry is essentially imaginary algebraic geometry. A point is infinitely small (in layman's terms), and a line is infinitely fine. Thus neusis would need an infinite magnifying glass, and it would take forever. Neusis CANNOT trisect an angle in the sense that a segment can be trisected by the Intercept Theorem. https://www.flickr.com/photos/859374...57633525195904 An angle can be trisected by Euclidean method in 3D space by a parabola. https://www.flickr.com/photos/859374...57635099724524 https://www.flickr.com/photos/859374...57635099724524 Neusis (Origami) cannot trisect an angle in the way a segment can be trisected. I'm willing to wager \$10,000 USD on it. https://www.flickr.com/photos/859374...57636438514124

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