Fred Lunnon wrote:
I presume you intended "equations with rational coefficients".
Equations with constructible coefficients. For example, sqrt(sqrt(sqrt(2))) is constructible with compass and straightedge, even though it is not a quadratic algebraic number. Specifically, the set of origami-constructible numbers is the smallest field containing Q and closed under the operations of square-root and cube-root.
Do you have a reference?
Notice that your class is strictly greater than the one claimed to be strictly constructible elsewhere --- eg. in
Page 130: "The roots of the general complex cubic with constructible coefficients can also be constructed" (I think that my class is strictly identical to the one claimed to be strictly constructible elsewhere. :)
--- since solving a cubic equation via taking (square and) cube roots is only possible in general over |C rather than |R --- when the equation has three real roots, the cube roots turn out to be of complex numbers.
We can work in C, by using the plane R^2 as an Argand diagram. Then, cube-rooting a complex number (represented by the point P) is equivalent to the following operations: 1. constructing a point Q on the line OP, such that OQ^3 = OP (that is to say, cube-rooting the modulus of the complex number, which is necessarily real, and you've already asserted that we can cube-root real numbers); 2. trisecting the angle QOR (where OR is the principal axis), and taking the intersection S with the circle of centre O passing through Q; 3. drawing the equilateral triangle centred on O with a vertex at S; the vertices of the resulting triangle are the three cube-roots. Using an analogous process (by square-rooting the modulus, bisecting the angle and taking the two intersections of the line with the circle centre O and radius OQ), you can also compute square-roots. That leaves addition, multiplication, subtraction and division, which are relatively easy with compass and straightedge (which can be emulated with origami). Sincerely, Adam P. Goucher
WFL
On 7/24/13, Adam P. Goucher <apgoucher@gmx.com> wrote:
You can solve arbitrary linear, quadratic, cubic and quartic equations with origami. You actually get the `and quartic' for free, since Ferrari's formula enables one to solve a general quartic by using a `black box' capable of solving general quadratics and cubics.
Consequently, a regular n-gon is constructible with origami if and only if phi(n) is a 3-smooth number.
Sincerely,
Adam P. Goucher
----- Original Message ----- From: Fred lunnon Sent: 07/24/13 03:53 PM To: math-fun Subject: Re: [math-fun] Origamics
Plenty of relevant hits on the web, once I thought to search for them! Including
http://buzzard.ups.edu/courses/2012spring/projects/engle-origami-ups-434-201...
which says that rationals, square roots and cube roots(!) are possible. Not quite the same thing as roots of cubic equations --- unless the paper is complex? --- but near enough.
WFL
On 7/24/13, Charles Greathouse <charles.greathouse@case.edu> wrote:
I thought all cubic numbers were constructable with origami, including the rational numbers as a subset.
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Wed, Jul 24, 2013 at 9:30 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
I recently acquired an e-copy of this Kazuo Haga monograph, which explores (at considerable and oriental length) geometrical constructions achievable via paper-folding. In particular, he claims to have constructed all prime length ratios up to 31:1 , prompting the following question.
Given a square of paper --- assumed accurately foldable to superpose two existing points such as corners, or a point upon an existing line such as an edge --- are all rational ratios constructible?
Fred Lunnon
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