[math-fun] Elliptic curve calculator
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales: http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/ I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation. Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/
Cute! Squaring is horribly difficult, though. On Wed, Aug 29, 2012 at 12:28 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Thanks. Hmm, yes, you have to be able to construct a tangent to a curve! Or, alternatively, you can use the identity: (yx)(x/y) = x^2 where y is an arbitrary constant. That takes three line-drawing operations, though, so I suppose it may class as 'horribly difficult'. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/ ----- Original Message ----- From: "Mike Stay" <metaweta@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, August 29, 2012 8:54 PM Subject: Re: [math-fun] Elliptic curve calculator
Cute! Squaring is horribly difficult, though.
On Wed, Aug 29, 2012 at 12:28 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
See: A single scale nomogram, Math. Gaz., 33(1949) 43 or 37(1953) 39. R. On Wed, 29 Aug 2012, Adam P. Goucher wrote:
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Oh, wow, you beat me to it by six decades. Slightly embarrassing on my part... Searching 'nomogram' has returned some related diagrams involving degenerate cubics (unions of three lines, or a line and conic). I seem to recall that the Richter scale involves a similar principle. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/ ----- Original Message ----- From: "Richard Guy" <rkg@cpsc.ucalgary.ca> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, August 29, 2012 9:08 PM Subject: Re: [math-fun] {Spam?} Elliptic curve calculator
See: A single scale nomogram, Math. Gaz., 33(1949) 43 or 37(1953) 39. R.
On Wed, 29 Aug 2012, Adam P. Goucher wrote:
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
_______________________________________________ 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
Is there a URL for the Math Gaz paper(s)? On Wed, Aug 29, 2012 at 4:17 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Oh, wow, you beat me to it by six decades. Slightly embarrassing on my part...
Searching 'nomogram' has returned some related diagrams involving degenerate cubics (unions of three lines, or a line and conic). I seem to recall that the Richter scale involves a similar principle.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
----- Original Message ----- From: "Richard Guy" <rkg@cpsc.ucalgary.ca> To: "math-fun" <math-fun@mailman.xmission.com**> Sent: Wednesday, August 29, 2012 9:08 PM Subject: Re: [math-fun] {Spam?} Elliptic curve calculator
See: A single scale nomogram, Math. Gaz., 33(1949) 43 or
37(1953) 39. R.
On Wed, 29 Aug 2012, Adam P. Goucher wrote:
You may be aware of the Abelian group operation on elliptic curves. We
can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
Dear Neil, Not as far as I know, but you can get at it via JSTOR. R. On Wed, 29 Aug 2012, Neil Sloane wrote:
Is there a URL for the Math Gaz paper(s)?
On Wed, Aug 29, 2012 at 4:17 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Oh, wow, you beat me to it by six decades. Slightly embarrassing on my part...
Searching 'nomogram' has returned some related diagrams involving degenerate cubics (unions of three lines, or a line and conic). I seem to recall that the Richter scale involves a similar principle.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
----- Original Message ----- From: "Richard Guy" <rkg@cpsc.ucalgary.ca> To: "math-fun" <math-fun@mailman.xmission.com**> Sent: Wednesday, August 29, 2012 9:08 PM Subject: Re: [math-fun] {Spam?} Elliptic curve calculator
See: A single scale nomogram, Math. Gaz., 33(1949) 43 or
37(1953) 39. R.
On Wed, 29 Aug 2012, Adam P. Goucher wrote:
You may be aware of the Abelian group operation on elliptic curves. We
can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
______________________________**_________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/**cgi-bin/mailman/listinfo/math-**fun<http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>
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-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Here's a snippet from an interview with Guy in which he explains the concept: Guy: ... OK. My first theorem is a very nice one. If you look in an early issue of the Mathematical Gazette, roughly the British equivalent of the Monthly, you’ll find “A Single Scale Nomogram.” I merely made the observation that a cubic equation with no x^2 term has zero for the sum of its roots. If you draw a cubic curve, y = x^3 + ax + b and put a straight line y = mx + c across it, the sum of the x-coordinates of the intersections is zero. If the curve is symmetrical about the origin (b = 0) and you change the sign of x on the negative half, then one coordinate is equal to the sum or difference of the other two. Combine this with the principle of the slide rule, which simply adds one chunk to another. For example, if the chunks are logs, you have multiplication and division. Anything you can do with a slide rule you can do with any single-scale nomogram. That was my first theorem, I suppose. On Wed, Aug 29, 2012 at 1:08 PM, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
See: A single scale nomogram, Math. Gaz., 33(1949) 43 or 37(1953) 39. R.
On Wed, 29 Aug 2012, Adam P. Goucher wrote:
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
_______________________________________________ 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
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
On 8/29/2012 2:15 PM, Mike Stay wrote:
Here's a snippet from an interview with Guy in which he explains the concept:
Guy: ... OK. My first theorem is a very nice one. If you look in an early issue of the Mathematical Gazette, roughly the British equivalent of the Monthly, you’ll find “A Single Scale Nomogram.” I merely made the observation that a cubic equation with no x^2 term has zero for the sum of its roots. If you draw a cubic curve, y = x^3 + ax + b and put a straight line y = mx + c across it, the sum of the x-coordinates of the intersections is zero. If the curve is symmetrical about the origin (b = 0) and you change the sign of x on the negative half, then one coordinate is equal to the sum or difference of the other two. Combine this with the principle of the slide rule, which simply adds one chunk to another. For example, if the chunks are logs, you have multiplication and division.
What is the base of the logarithms on a slide rule. :-) Brent Meeker
Thanks for that description. Oh, my elliptic curve calculator is definitely not equivalent to Guy's nomogram. Whereas Guy's nomogram is isomorphic to the real line, my version is isomorphic to the circle group. However, Guy's nomogram generalises very nicely to a quartic with no cubic term, such as y = x^4 + ax^2 + bx + c. We can then draw a couple of scales on that quartic, as before Assuming that this is positive-definite (which we can do just by increasing the value of c), we can then hit it with a projective transformation to result in a closed curve, avoiding all of the infinity issues my elliptic curve calculator suffers from. Unfortunately, the quartic nomogram doesn't really do anything useful: essentially, if you give it two points A and B, it will find two other points C and D such that A + B + C + D = 0. I doubt that this can be used for practical computation. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/ ----- Original Message ----- From: "Mike Stay" <metaweta@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, August 29, 2012 10:15 PM Subject: Re: [math-fun] {Spam?} Elliptic curve calculator Here's a snippet from an interview with Guy in which he explains the concept: Guy: ... OK. My first theorem is a very nice one. If you look in an early issue of the Mathematical Gazette, roughly the British equivalent of the Monthly, you’ll find “A Single Scale Nomogram.” I merely made the observation that a cubic equation with no x^2 term has zero for the sum of its roots. If you draw a cubic curve, y = x^3 + ax + b and put a straight line y = mx + c across it, the sum of the x-coordinates of the intersections is zero. If the curve is symmetrical about the origin (b = 0) and you change the sign of x on the negative half, then one coordinate is equal to the sum or difference of the other two. Combine this with the principle of the slide rule, which simply adds one chunk to another. For example, if the chunks are logs, you have multiplication and division. Anything you can do with a slide rule you can do with any single-scale nomogram. That was my first theorem, I suppose. On Wed, Aug 29, 2012 at 1:08 PM, Richard Guy <rkg@cpsc.ucalgary.ca> wrote:
See: A single scale nomogram, Math. Gaz., 33(1949) 43 or 37(1953) 39. R.
On Wed, 29 Aug 2012, Adam P. Goucher wrote:
You may be aware of the Abelian group operation on elliptic curves. We can exploit this to multiply, divide and square-root real numbers just by drawing a straight line on an elliptic curve marked with two logarithmic scales:
http://cp4space.wordpress.com/2012/08/29/elliptic-curve-calculator/
I've tried it on a printed version, and can achieve somewhere between 2 and 3 significant figures of precision, depending on the calculation.
Obviously it is of no practical use today, since we have electronic calculators and so forth. But in theory, this could have been made way back in 1830, when it could have been used for calculations in navigation and ballistics.
Sincerely,
Adam P. Goucher
http://cp4space.wordpress.com/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (5)
-
Adam P. Goucher -
meekerdb -
Mike Stay -
Neil Sloane -
Richard Guy