February 2018 - mathfun@mailman.xmission.com

Re: [math-fun] Fourier Transform video
by Keith F. Lynch 06 Feb '18

06 Feb '18

Henry Baker <hbaker1(a)pipeline.com> > I'm curious as to what others think about this Fourier Transform > video (1/26/2018). > http://www.3blue1brown.com/ > https://www.youtube.com/c/3blue1brown Interesting. I've been playing with Fourier transforms for decades, but I had never thought of them that way before. Another visual representation is the waterfall display, in which time is the vertical axis, frequency is on the horizontal axis, and power is represented by brightness. In this representation, a Fourier transform can be represented by a 90-degree rotation. That makes it clear why doing a Fourier transform twice results in a time reversal. (It also results in a frequency reversal, but since negative frequencies are equivalent to positive, this makes no difference.) That representation also raises the question of what the math is behind smaller rotations. For instance what's the name of, and formula for, the 45 degree rotation, i.e. the transform that gives the Fourier transform if done twice. But mostly I'm not very visually oriented. I often get insights that make no sense to visual thinkers. For instance when an engineer friend tried to explain Laplace transforms to me, I was puzzled for a while, until I suddenly realized that he was talking about the Fourier transform of imaginary frequencies. Or, equivalently, of imaginary times. He was rather disturbed by that. But he later confirmed that the math comes out right -- which made him even more disturbed. Years later, I used a Fourier or Laplace transform of both real and imaginary frequencies to predict who would attend the next meeting of a science fiction club for which I had decades of attendance data. (As club secretary, I prepared the single-page sign-in sheet, and I wanted to list the names of those likely to show up, and leave blanks for others to fill in with their names.) It worked quite well. For instance compare the attendance of the April 1st meeting that I printed in the club journal distributed at that meeting as a prank to the actual attendance of that meeting that I printed in the following month's journal, <http://wsfa.org/journal/j05/4/#a1m> and <http://wsfa.org/journal/j05/5/#ffm>.

1 0

[math-fun] Two random intersection puzzles
by James Propp 05 Feb '18

05 Feb '18

1. Given a convex quadrilateral, consider the four bounding line segments (i.e. the sides) and the two segments obtained by joining the midpoints of opposite sides — six line segments in all. Let L be a line not parallel to any of the six line segments. Show that a random translate of L, conditioned to intersect at least one of the six segments, can be expected to intersect 3 of them on average. 2. Consider the twelve line segments in a planar projection of the 1-skeleton of a parallelepiped. Let L be a line not parallel to any of the twelve line segments. Show that a random translate of L, conditioned to intersect at least one of the twelve segments, can be expected to intersect 4 of them on average. Jim Propp

2 4

[math-fun] Outside In written accompaniement
by Andy Latto 04 Feb '18

04 Feb '18

I was recommending the video Outside In, about turning the sphere inside out to a friend. The VCR tape of it I have would be of limited usefulness even if I could find it, but it was pretty easy to find on youTube at https://www.youtube.com/watch?v=wO61D9x6lNY But the VCR tape was accompanied by a wonderful written guide, which had marked sections so that it could be read at 3 or 4 different levels of mathematical sophistication, and I can't find my copy. Does anyone know if this is available somewhere, online or otherwise? -- Andy.Latto(a)pobox.com

3 3

[math-fun] Locker puzzle
by Dan Asimov 04 Feb '18

04 Feb '18

Puzzle: ------- Suppose there is a school with an infinity of lockers, numbered 1, 2, 3, .... Initially all lockers are closed. Definition: A locker is "switched" if it is changed from shut to open, or vice versa. At 1/2 minute to noon, every 2nd locker 2, 4, 6, ... is switched. at 1/3 minute to noon, every 3rd locker 3, 6, 9, ... is switched. Or in general: For each positive integer n >= 2: * At 1/n of a minute to noon, every nth locker n, 2n, 3n, ... is switched. No other changes are made to the lockers. Question: --------- Which lockers are open at noon? —Dan

2 1

Re: [math-fun] The Orchard Problem
by Dan Asimov 02 Feb '18

02 Feb '18

That writeup seems really, really nice! Very cool. —Dan Ed Pegg, Jr. wrote: ----- I wrote a blog on the Orchard Problem, with many new solutions. http://blog.wolfram.com/2018/02/02/cultivating-new-solutions-for-the-orchar… -----

2 1

[math-fun] The basketball scoring approximation.
by Ed Pegg Jr 02 Feb '18

02 Feb '18

https://oeis.org/A077835 is the basketball scoring sequence. Something special happens between the combined scores equaling 39 and 40. It's often said that by then, you can be 67% sure of who will win the game. Using math explored at http://community.wolfram.com/groups/-/m/t/1277291 , I show this value is actually *66.99999999999999999997299* --Ed Pegg Jr

2 2

[math-fun] The Orchard Problem
by Ed Pegg Jr 02 Feb '18

02 Feb '18

I wrote a blog on the Orchard Problem, with many new solutions. http://blog.wolfram.com/2018/02/02/cultivating-new-solutions-for-the-orchar… --Ed Pegg Jr

2 1

[math-fun] Polygon discriminant sequence
by Ed Pegg Jr 01 Feb '18

01 Feb '18

I've been looking at discriminants today. Specifically, sequence https://oeis.org/A193681 Discriminant of minimal polynomial of 2*cos(Pi/n) I've noticed that all these numbers are almost-powers of n. For n with the following values 1, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179 The power is exactly 1, 1, 2, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 9, 22, 19, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 52, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88 What about the rest? Here are values {n,p,a,b} so that discriminant of minimal polynomial of 2*cos(Pi/n) is (n^p a^b) discriminant( 2*cos(Pi/17) ) = 17^7 (Gauss's 17-gon) discriminant( 2*cos(Pi/168) ) = 168^48 189^(-8) discriminant( 2*cos(Pi/179) ) = 179^88 Is there any rhyme or reason to this sequence? --Ed Pegg Jr {{1,1,1,1}, {2,1,2,-1}, {3,1,3,-1}, {4,1,2,1}, {5,1,1,1}, {6,1,2,1}, {7,2,1,1}, {8,4,2,-1}, {9,2,1,1}, {10,3,2,1}, {11,4,1,1}, {12,4,3,-2}, {13,5,1,1}, {14,5,2,1}, {15,3,3,-1}, {16,8,2,-1}, {17,7,1,1}, {18,4,12,1}, {19,8,1,1}, {20,6,2,4}, {21,5,3,-2}, {22,9,2,1}, {23,10,1,1}, {24,8,3,-4}, {25,8,5,1}, {26,11,2,1}, {27,7,3,1}, {28,10,2,4}, {29,13,1,1}, {30,4,20,2}, {31,14,1,1}, {32,16,2,-1}, {33,9,3,-4}, {34,15,2,1}, {35,10,5,-1}, {36,9,2,6}, {37,17,1,1}, {38,17,2,1}, {39,11,3,-5}, {40,16,5,-4}, {41,19,1,1}, {42,12,189,-2}, {43,20,1,1}, {44,18,2,4}, {45,9,1,1}, {46,21,2,1}, {47,22,1,1}, {48,16,3,-8}, {49,19,1,1}, {50,17,40,1}, {51,15,3,-7}, {52,22,2,4}, {53,25,1,1}, {54,15,2,3}, {55,18,5,-3}, {56,24,7,-4}, {57,17,3,-8}, {58,27,2,1}, {59,28,1,1}, {60,16,45,-4}, {61,29,1,1}, {62,29,2,1}, {63,15,3,-3}, {64,32,2,-1}, {65,22,5,-4}, {66,20,2673,-2}, {67,32,1,1}, {68,30,2,4}, {69,21,3,-10}, {70,18,56,2}, {71,34,1,1}, {72,18,2,18}, {73,35,1,1}, {74,35,2,1}, {75,18,32805,-1}, {76,34,2,4}, {77,27,7,-2}, {78,24,9477,-2}, {79,38,1,1}, {80,32,5,-8}, {81,23,3,2}, {82,39,2,1}, {83,40,1,1}, {84,24,189,-4}, {85,30,5,-6}, {86,41,2,1}, {87,27,3,-13}, {88,36,2,12}, {89,43,1,1}, {90,18,2,6}, {91,33,7,-3}, {92,42,2,4}, {93,29,3,-14}, {94,45,2,1}, {95,34,5,-7}, {96,32,3,-16}, {97,47,1,1}, {98,38,112,1}, {99,27,3,-9}, {100,35,2,10}, {101,49,1,1}, {102,32,111537,-2}, {103,50,1,1}, {104,44,2,12}, {105,20,405,-2}, {106,51,2,1}, {107,52,1,1}, {108,30,2,12}, {109,53,1,1}, {110,30,42592,2}, {111,35,3,-17}, {112,48,7,-8}, {113,55,1,1}, {114,36,373977,-2}, {115,42,5,-9}, {116,54,2,4}, {117,33,3,-12}, {118,57,2,1}, {119,45,7,-5}, {120,32,45,-8}, {121,52,1,1}, {122,59,2,1}, {123,39,3,-19}, {124,58,2,4}, {125,46,5,-1}, {126,27,56,3}, {127,62,1,1}, {128,64,2,-1}, {129,41,3,-20}, {130,36,1352,4}, {131,64,1,1}, {132,40,2673,-4}, {133,51,7,-6}, {134,65,2,1}, {135,30,5,-3}, {136,60,2,12}, {137,67,1,1}, {138,44,4074381,-2}, {139,68,1,1}, {140,36,448,4}, {141,45,3,-22}, {142,69,2,1}, {143,55,11,-1}, {144,48,3,-24}, {145,54,5,-12}, {146,71,2,1}, {147,39,2711943423,-1}, {148,70,2,4}, {149,73,1,1}, {150,40,45,-10}, {151,74,1,1}, {152,68,2,12}, {153,45,3,-18}, {154,50,3872,2}, {155,58,5,-13}, {156,48,9477,-4}, {157,77,1,1}, {158,77,2,1}, {159,51,3,-25}, {160,64,5,-16}, {161,63,7,-8}, {162,47,384,1}, {163,80,1,1}, {164,78,2,4}, {165,36,820125,-2}, {166,81,2,1}, {167,82,1,1}, {168,48,189,-8}, {169,74,13,1}, {170,64,10625,-4}, {171,51,3,-21}, {172,82,2,4}, {173,85,1,1}, {174,56,138706101,-2}, {175,53,1715,-1}, {176,80,11,-8}, {177,57,3,-28}, {178,87,2,1}, {179,88,1,1}, {180,36,2,24}}

2 2

Re: [math-fun] Fourier Transform video
by Bill Gosper 01 Feb '18

01 Feb '18

On 2018-01-31 08:22, Henry Baker wrote: > I'm curious as to what others think about this Fourier Transform video > (1/26/2018). > > http://www.3blue1brown.com/ > > https://www.youtube.com/c/3blue1brown > I wish, very early on, when he's visualizing those simple chords, that he'd make a point of sliding around the relative phases, making visually dissimilar waveforms that sound the same. --rwg

1 0

Re: [math-fun] Fourier Transform video
by Dan Asimov 01 Feb '18

01 Feb '18

I just watched the first video, but compared to almost all similar videos (about the same length, teaching math) I found it to be SUPERB. It explains things extra clearly by explaining the idea of wrapping a periodic function around the circle, and accompanying each statement of the narrator with the appropriate pictures. Does anyone know who is responsible for this video? —Dan Henry Baker wrote: ----- I'm curious as to what others think about this Fourier Transform video (1/26/2018). http://www.3blue1brown.com/ https://www.youtube.com/c/3blue1brown -----

1 0