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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I presume you intended "equations with rational coefficients". Do you have a reference? Notice that your class is strictly greater than the one claimed to be strictly constructible elsewhere --- eg. in http://nyjm.albany.edu/j/2000/6-8.pdf --- 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. 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
-
Adam P. Goucher -
Fred lunnon