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I'm doing a project on the relationship between math and origami and this page is very useful, except that it doesn't ever say what "s" is in the equations (the very first one is F(s) = p1 + s(p2 - p1) and they never tell you what the F or the s stand for. The s is bothering me more.) I'm so frustrated. Could anyone answer this as soon as possible please? Thanks!!
See http://www.nytimes.com/2004/06/22/science/22orig.html about non straight-fold origami -- Anon
This page talks about the Huzita-Hatori axioms, including a 7th axiomatic fold discovered by Koshiro Hatori in 2002 that is independent of Huzita's 6 established axioms.
I'm not in a position to judge the validity of this page, and haven't found any other online references. I hope someone in a better position to judge will update the article as appropriate. Hv 03:02, 2 September 2005 (UTC)
Here's the link to Hatori's own discussion. Hv 03:09, 2 September 2005 (UTC)
This page is seriously broken. The Axioms don't specify a model, so one should not assume that one is working with an inner product space or a space where parametric equations make sense. For example, if a piece of paper is not convex, one cannot apply the algorithm in the exposition of Axiom 2 to find folds mapping any two points to each other. Other algorithms can be broken easily as well.
I don't know enough about this field to fix the article myself, but it is either missing hypotheses used in the field or is taking mathematical liberties without justification. It is not obvious that every model for these axioms is convex. Indeed, that is false. I also take issue with the assumption that every model of the axioms is an inner product space. -- poopdeville
Intuitively, if you take a star-shaped sheet of paper, you'll see that you can (1) fold it so that any two points coincide, (2) fold it so that there is a single line going between them, (3) fold it so that any line coincides with any other, and (4) so on. But you can't use the algorithms presented in the article to find these folds because there is no guarantee that the parametric function F(s) exists everywhere.
If Axiom 5 and 6 can have 0 solutions, how are they true?-- Kuciwalker 16:30, 17 January 2006 (UTC)
Kuciwalker, I had a similar feeling of 'how can they be axioms if they're not always true?' However, on Robert Lang's origami page ( http://www.langorigami.com/science/hha/hha.php4) it reads,
"Humiaki Huzita and Benedetto Scimemi presented a paper [...] in which they identified six distinctly different ways one could create a single crease by aligning one or more combinations of points and lines [...] on a sheet of paper. Those six operations became known as the Huzita axioms. The Huzita axioms provided the first formal description of what types of geometric constructions were possible with origami."
So I think the axioms describe the ways in which folds may be made, but do not necessarily imply that they can be made in all situations. I agree with you that the way the axioms are presented, as unequivocal statements, is confusing.
What does this mean?
In what sense are the 7 axioms complete?
And in what sense is the seventh axiom independent? As far as I know, the first 6 axioms can construct all points that you get by (iteratively) taking square roots and cube roots. Does axiom 7 add anything to that?
194.24.138.3 ( talk) 23:26, 18 March 2010 (UTC)
I would like to see it mentioned that a 360-gon can be created through paper folding while it cannot be created by compass and straight edge. Since the 360 gon is used for the division of 1 degree it might be nice to show that paper folding provides a potential for constructing it. Also, if anyone has a link to someone constructing 1 degree with paper folding... I'd love to see it. â Preceding unsigned comment added by Peawormsworth ( talk ⢠contribs) 19:35, 4 September 2012 (UTC)