[math-fun] unusual things
Hello math-funsters, there is an interesting site here, http://www.futilitycloset.com/?new=true a representation of 1/7 in decimal on an ellipse. the other pages are interesting as well, quite amusing, in the same vein , http://www.laboiteverte.fr/ that one is original, very original, I counted at least 100000 original images and pages. http://www.laboiteverte.fr/?s=math like this 'etch-a-sketch' on a sphere : http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique-spherique... and with Pi digits here ??: http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique-spherique... This is what I call original. one drawback : the whole site is in french, not the images. Have fun. Cheers. Simon Plouffe
there is an interesting site here,
http://www.futilitycloset.com/?new=true
a representation of 1/7 in decimal on an ellipse.
the other pages are interesting as well,
quite amusing,
I agree it's fun. But isn't the 1/7 thing less surprising than it sounds? There are three pairs of pairs each of which (not by coincidence) adds up to (9,9). So what we really have here is three points lying on an ellipse with centre (9/2,9/2). That's not quite content-free -- e.g., there is no ellipse with centre (0,0) containing points (1,1), (100,1), (1,100) -- but it's close. Given a centre which wlog is (0,0) and three points, finding a *conic* with that centre passing through those points means finding A,B,C,F such that Axx+Bxy+Cyy=F for all three, and in general that has solutions. (I guess they're hyperbolae about as often as ellipses, and parabolae with probability zero.) The two-digit version is essentially the same thing: three pairs each adding up to (99,99). -- g
If you plot pairs of successive digits occurring in 1/n (zero-extended for finite decimals) as points, there are actually many n for which the points uniquely determine an ellipse. I found these: 7, 16, 39, 63, 88, 91, 143, 160, 202, 240, 264, 273, 369, 429, 505, 540, 606, 675, 693, 740, 819, 1355, ... For the above n, I plotted the points and determined by inspection that the point set consists of either - 5 points at the vertices of a convex pentagon - 6 points at the vertices of a convex hexagon with 180-degree rotational symmetry or symmetry about a line. I did not look at n generating more than 6 points that might happen by chance to lie on an ellipse. I assume that 4 or fewer points cannot determine a unique ellipse (is this true?) I assume that 5 points at vertices of a convex pentagon always determines a unique ellipse (is this true?) Call these (base-10) ellipse numbers. To accurately decide which integers are ellipse numbers, requires an algorithm to decide if n distinct points in Z^2 lie on a unique ellipse. I wouldn't know how to write such an algorithm. There are also hyperbola numbers, e.g. 13 is a base-10 hyperbola number.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Simon Plouffe Sent: Wednesday, June 08, 2016 2:37 PM To: math-fun; Paul Simon; Sylvain Lambert; Louis Plouffe Subject: [math-fun] unusual things
Hello math-funsters,
there is an interesting site here,
http://www.futilitycloset.com/?new=true
a representation of 1/7 in decimal on an ellipse.
the other pages are interesting as well,
quite amusing,
in the same vein ,
that one is original, very original, I counted at least 100000 original images and pages.
http://www.laboiteverte.fr/?s=math
like this 'etch-a-sketch' on a sphere :
http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique- spherique/
and with Pi digits here ??:
http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique- spherique/#jp-carousel-68129
This is what I call original.
one drawback : the whole site is in french, not the images.
Have fun.
Cheers.
Simon Plouffe
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Call these (base-10) ellipse numbers. To accurately decide which integers are ellipse numbers, requires an algorithm to decide if n distinct points in Z^2 lie on a unique ellipse. I wouldn't know how to write such an algorithm.
Consider the n-by-6 matrix whose rows are of the form: (1, x_i, y_i, x_i^2, y_i^2, x_i y_i) for i in {1, 2, ..., n}. Now perform Gaussian elimination via row operations. At the end of the process, there should be k linearly-independent rows together with n-k zero rows. The points determine a unique *conic* provided that k = 5 (i.e. the rank of the original matrix is 5). In that case, drop the zero rows to give a 5-by-6 matrix and compute each of the six minors (determinants of 5-by-5 submatrices), labelling them as follows: (A, B, C, D, E, F) Then the equation of the conic is given by: A + B x + C y + D x^2 + E y^2 + F x y = 0 For this to be an ellipse, we require the 'elliptic' constraint: 4 D F > E^2 which is necessary and sufficient for the conic to be an ellipse.
There are also hyperbola numbers, e.g. 13 is a base-10 hyperbola number.
Yes, follow precisely the same algorithm but replace the 'elliptic' constraint with the 'hyperbolic' constraint: 4 D F < E^2 Best wishes, Adam P. Goucher
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Simon Plouffe Sent: Wednesday, June 08, 2016 2:37 PM To: math-fun; Paul Simon; Sylvain Lambert; Louis Plouffe Subject: [math-fun] unusual things
Hello math-funsters,
there is an interesting site here,
http://www.futilitycloset.com/?new=true
a representation of 1/7 in decimal on an ellipse.
the other pages are interesting as well,
quite amusing,
in the same vein ,
that one is original, very original, I counted at least 100000 original images and pages.
http://www.laboiteverte.fr/?s=math
like this 'etch-a-sketch' on a sphere :
http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique- spherique/
and with Pi digits here ??:
http://www.laboiteverte.fr/le-doodle-dome-de-tyco-lardoise-magique- spherique/#jp-carousel-68129
This is what I call original.
one drawback : the whole site is in french, not the images.
Have fun.
Cheers.
Simon Plouffe
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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Adam P. Goucher -
David Wilson -
Gareth McCaughan -
Simon Plouffe