November 2018 - mathfun@mailman.xmission.com

Re: [math-fun] Finding a polynomial from an integer sequence
by Steve Witham 09 Nov '18

09 Nov '18

Here's my favorite way to fit a sequence to a polynomial, in working Python. (As I mentioned before, this is O(2^len(ys)), but can be cured of that.) def fit(ys, x): """ The x at the first of the ys is 0. """ d = len(ys) - 1 # (max x) - (min x) if d == 0: return ys[0] else: left = fit(ys[:-1], x) # Up to the 2nd-last y. right = fit(ys[1:], x - 1) # Start with the 2nd y. return ((d - x) * left + x * right) / d If x is a number, the function just processes numbers, as in any sensible language: ys = [0, 4, 16] fit(ys, -1 ) => 4 fit(ys, 0 ) => 0 fit(ys, 0.5 ) => 1.0 fit(ys, 1 ) => 4 fit(ys, 1.5 ) => 9.0 fit(ys, 2 ) => 16 fit(ys, 2.5 ) => 25.0 But if the language happens to let you define objects that act like polynomials, then the same code processes polynomials: X = UniPoly([0, 1]) P = fit(ys, X) => UniPoly([0, 0, 4]) P( -1 ) => 4 P( 0 ) => 0 P( 0.5 ) => 1 P( 1 ) => 4 P( 1.5 ) => 9 P( 2 ) => 16 P( 2.5 ) => 25 --Steve

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[math-fun] Draft of November 2018 blog post
by James Propp 08 Nov '18

08 Nov '18

I've posted a draft of my next essay at http://mathenchant.org/041b.pdf ; as usual, I'll be grateful for thoughtful criticism, both positive and negative. I'll ponder any comments you care to send and will acknowledge them at the end of the essay if they've helped me make the piece better. My main fear is that the piece won't really succeed in communicating its main point to people who don't already understand the main point. (Example: I can imagine a reader who feels that "All men are mortal, Socrates is a man, hence Socrates is mortal" is valid whereas "All men are immortal, Socrates is a man, hence Socrates is immortal" is nonsense, and who stops reading at the point in the essay where I seem to be giving them equal respect.) Are there other pieces of writing you can think of that do a better job of saying what I'm trying to say here? Thanks, Jim Propp P.S. I already know about the "Lobachevkian" typo; I'm sure there are others I've missed.

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Re: [math-fun] conic sections
by Dan Asimov 08 Nov '18

08 Nov '18

I would stay away from calculus, but there are so many cool things you can do without it. Of course, a pencil in a loop of string around 2 nails in a board will draw an ellipse. But also there is a beautifully simple proof that a plane through one nappe of a cone cuts an ellipse. Very surprising at first, because you expect the fat part of the cone to result in a different shape — but the two ends of the ellipse are the same! And it's very cool that a billiard ball shot from one focus bounces straight to the other one. (Hmm, what kind of billiard table would the space between the arms of a hyperbola be?) —Dan ----- I’m going to talk to my daughter’s 8th grade class about conic sections. I want to focus on foci (ha), and how curves with beautiful geometric descriptions also have nice algebraic descriptions in Cartesian geometry. But I found it surprisingly tricky to work out examples. Consider an ellipse with foci at (-1,0) and (+1,0), and define the set of points where the sum of its distances from these two is 4. Using Pythagoras’ theorem produces an equation with a bunch of square roots. Squaring both sides eventually turns this into 3x^2 + 4y^2 = 12 but this takes a bunch of steps of algebra, and mysterious cancellations of 4th-order terms. Similarly, it takes a fair amount of work to get from the hyperbola with foci at (+2,+2) and (-2,-2), where the difference in distances is 4, to the simple equation xy = 2. Am I doing something wrong? Is there an easier way to get from foci and distances to these simple quadratic equations - without recourse to canonical forms, linear transformations, polar coordinates etc.? Of course, I then want to talk about light waves bouncing from one focus to another… I’m not sure how to justify this without a little calculus. -----

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Re: [math-fun] Computer Science alternative to "Pi Day" ?
by Keith F. Lynch 08 Nov '18

08 Nov '18

"Adam P. Goucher" <apgoucher(a)gmx.com> wrote: > Henry Baker <hbaker1(a)pipeline.com> wrote: >> So what would be the single number that most computer science >> people would recognize and acknowledge as being representative >> of computer science? > Chaitin's constant represents computer science in a deep way: > it embodies the eventual behaviour of every computer program. > The only problem is that its numeric value depends on your choice > of prefix-free encoding of computer programs. A bigger problem is that it is uncomputable. So even if we settled on an encoding, we wouldn't know which day to celebrate. We could celebrate June 21, the date the first stored program computer ran (in 1948). (Wow, I've been using computers for 2/3 of the time there have been computers.) Or October 30, the date the first message was sent on the ARPAnet (in 1969). Or June 23, Turing's birthday. Or June 14, Alonzo Church's birthday. Or December 26, Charles Babbage's birthday. Or November 2, George Boole's birthday. Other important constants that vary with the computer, but that at least are computable, are the largest integer that can be represented, the largest real number that can be represented, and the smallest real number greater than zero that can be represented. Or negative zero, which in some computer architectures, unlike in standard math, is distinct from positive zero. After all, the reciprocal of positive zero is positive infinity, but the reciprocal of negative zero is negative infinity. You can't get more different than that. :-) We would then celebrate computer science on the zeroth day of the zeroth month, because computer people like to count starting with zero. Or maybe on the negative zeroth day of the negative zeroth month. Or we could celebrate on the anniversary of the computer's epoch. That's January 1 (1970) for Unix, November 17 (1858) for VMS. Unix time is counted in seconds from the epoch. Instead of celebrating annually, we could celebrate when it hits some multiple of a large power of ten. Or better yet, when it hits some multiple of a large power of two. VMS time is the same, except that it isn't counted in seconds, but in microfortnights. But I think the best choice is NaN ("Not a Number"). It's unique to computer science, as it often appears in computers, but never appears on the real number line or in the complex number plane. We would celebrate on NaD ("Not a Date").

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Re: [math-fun] Upending the sphere — the discrete version: Puzzle
by Dan Asimov 08 Nov '18

08 Nov '18

Closely related questions are the same but for rolling a polyhedron on another one with the same faces: upending an octahedron or an icosahedron, rolling on a tetrahedron, or on an octahedron, or on an icosahedron. —Dan >I am pretty sure the cuboctohedron can be done, though. > >On Wed, Nov 7, 2018 at 7:24 PM Allan Wechsler <acwacw(a)gmail.com> wrote: > >> The rhombic dodecahedron also can't be done, but for entirely different >> reasons. I think. >> >> On Wed, Nov 7, 2018 at 7:17 PM Mike Stay <metaweta(a)gmail.com> wrote: >> >>> On Wed, Nov 7, 2018 at 5:10 PM Allan Wechsler <acwacw(a)gmail.com> wrote: >>> > >>> > Wait, I don't have an answer for the octohedron, and suspect it can't be >>> > done. >>> >>> I think you're right, since you can color both the octahedron and the >>> triangular tiling with two colors such that any move also changes the >>> color of the tangent faces. >>> >>> -- >>> Mike Stay - metaweta(a)gmail.com >>> http://math.ucr.edu/~mike >>> https://reperiendi.wordpress.com >>> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun(a)mailman.xmission.com >>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Upending the sphere — the discrete version: Puzzle
by Dan Asimov 08 Nov '18

08 Nov '18

Label each unit grid square Q_(K,L) = {(x,y) in R^2 | |x-K| <= 1/2 and |y-L| <= 1/2} in the plane by the pair of integers (K, L) that are the coordinates of its center. Think of the plane R^2 as R^2 x {0} in R^3, an infinite tabletop. At time t = 0, place a unit cube C on Q(0,0). Assume the cube C rests on Q(K,L) at time t = N. A legal move, taken once per unit time step, is to rotate C by an angle of 90º about one of its 4 edges that are touching the tabletop at time t = N, so that it ends up at time t = N+1 resting on one of Q(K+1,L), Q(K-1,L), Q(K,L+1), or Q(K,L-1), with the side down at t = N+1 being one of the 4 sides of C that were vertical at t = N. Puzzle: ------- a) What is the smallest time t at which C can end up resting on Q(0,0) with its initially down side now facing up? Now what if we had used instead b) an octahedron or c) an icosahedron, on the triangular grid, with the corresponding question? (I.e., on each move the solid is rotated by π - (dihedral), so that it ends up on an adjacent triangle, resting on one of its faces adjacent to the one it had just been resting on.) d) The same question for a dodecahedron! (It doesn't matter that there is no tiling of the plane by regular pentagons; the question makes sense anyway.) —Dan

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[math-fun] square root approximation?
by Stuart Anderson 07 Nov '18

07 Nov '18

Is this a known method? Not particularly accurate though https://mystudentvoices.com/how-an-old-mans-ravings-awakened-the-mathematic…

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Re: [math-fun] Lower bound in superpermutation problem proven by anonymous anime fan
by Dan Asimov 06 Nov '18

06 Nov '18

I wonder about the two related problems a), b) where we want the smallest N such that: a) each permutation of n letters must occur as a counterclockwise substring of some circular ordering on N letters; b) each permutation of n letters must occur as a substring of some circular ordering on N letters. (Each substring can occur either counterclockwise or clockwise.) I can't tell whether the N of case a) is likely to be bigger or smaller than the N for the original problem (a linear ordering containing each permutation of n letters). —Dan Warut Roonguthai wrote: ----- Mystery Math Whiz and Novelist Advance Permutation Problem https://www.quantamagazine.org/sci-fi-writer-greg-egan-and-anonymous-math-w… On Thu, Oct 25, 2018 at 10:24 PM James Propp <jamespropp(a)gmail.com> wrote: > Maybe the anonymous prover was Satoshi Nakamoto? > > Jim Propp > > On Wednesday, October 24, 2018, Allan Wechsler <acwacw(a)gmail.com> wrote: > > > At http://mathsci.wikia.com/wiki/The_Haruhi_Problem, we have an > intriguing > > situation. > > > > The problem is, what is the shortest string that contains every > permutation > > on N letters as a substring. > > > > The problem has been around for a while, but somebody posted it on > Wikia, a > > wiki dedicated to fans of various TV programs, mostly anime. > > > > Then an anonymous commenter posted a proof of a lower-bound that had been > > conjectured earlier, that the shortest string must have length at least > > n! + (n-1)! + (n-2)! + n - 3, for n > 2. The style is a little bit > > amateurish but the proof seems correct. Robin Houston, a mathematician > who > > has worked on the problem, has vetted it. > > > > Nobody knows who the prover is. -----

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[math-fun] Lower bound in superpermutation problem proven by anonymous anime fan
by Allan Wechsler 06 Nov '18

06 Nov '18

At http://mathsci.wikia.com/wiki/The_Haruhi_Problem, we have an intriguing situation. The problem is, what is the shortest string that contains every permutation on N letters as a substring. The problem has been around for a while, but somebody posted it on Wikia, a wiki dedicated to fans of various TV programs, mostly anime. Then an anonymous commenter posted a proof of a lower-bound that had been conjectured earlier, that the shortest string must have length at least n! + (n-1)! + (n-2)! + n - 3, for n > 2. The style is a little bit amateurish but the proof seems correct. Robin Houston, a mathematician who has worked on the problem, has vetted it. Nobody knows who the prover is.

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Re: [math-fun] Computer Science alternative to "Pi Day" ?
by Dan Asimov 06 Nov '18

06 Nov '18

I assume this should be Gibbs's constant: << Gibb's constant ~ 1.8519 (Inevitable overshoot for digital/square waves) —Dan

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