January 2017 - mathfun@mailman.xmission.com

Re: [math-fun] approximating pi by 1
by Keith F. Lynch 27 Jan '17

27 Jan '17

James Propp <jamespropp(a)gmail.com> wrote: > (Of course when we're doing dimensional analysis we ignore > dimensionless factors like pi, but that's not what I mean; I'm > talking about replacing pi by 1 in calculations where getting an > answer that's off by just one order of magnitude is good enough.) Not Feynman, Fermi. See https://what-if.xkcd.com/84/

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[math-fun] Worst rational approximations to pi
by James Propp 27 Jan '17

27 Jan '17

Interestingly, it seems that nobody's studied the WORST rational approximations to pi! I'd better explain what I mean by this before anyone shouts "Hey, a *million* is a really bad rational approximation to pi!" Let nint(x) denote the nearest integer to x. For each denominator q, look at the discrepancy |q Pi - p| where p = nint(Pi q); call this disc(q). The best rational approximations to pi are those for which disc(q) is smaller than disc(1),...,disc(q-1), so it's natural to define the worst rational approximations to pi as those for which disc(q) is greater than disc(1),...,disc(q-1). If I haven't goofed, the denominators of the worst approximations to pi are *1,2,3,4,11,18,25,32,...* (a sequence that isn't in OEIS). Before I compute more terms, can anyone corroborate my numbers? It seems improbable to me that nobody's thought to look at this before. (If we play this same game with the golden ratio, we get *1,4,17,72,305,1292,...* which is in the OEIS (A001076 <http://oeis.org/A001076>).) Jim

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[math-fun] Tetromino puzzle from fivethirtyeight.com
by Tom Duff 27 Jan '17

27 Jan '17

The warmup to 538's weekly math puzzle asks how many ways Tetris pieces (tetrominos) can fill a 2x10 rectangle. (I'm assuming because of the context that reflections and rotations are considered different.) See https://fivethirtyeight.com/features/can-you-win-at-tetris-can-you-save-a-sp ecies/ The answer is 64 unless my program is busted. It's easy for my program to do other sized rectangles. For 2-by rectangles I get: size solutions 2x2 1 4x2 4 6x2 9 8x2 25 10x2 64 12x2 169 14x2 441 16x2 1156 18x2 3025 20x2 7921 The second column looks like the squares of the Fibonacci numbers. Is there an easy proof? -- Tom Duff. Uh, oh, we've got company.

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Re: [math-fun] Ford circles pi zoom
by Dan Asimov 27 Jan '17

27 Jan '17

What do you mean by the "Ford circle packing" of the upper half plane? —Dan ----- From: James Propp <jamespropp(a)gmail.com> Sent: Jan 27, 2017 10:08 AM Anyone care to make a short video (or GIF) that zooms in on the Ford circle packing of the upper half plane, converging on the point pi on the real axis, highlighting the circles we pass through (whose points of tangency to the real axis are the continued-fraction convergents to pi)? I searched YouTube but nothing like this seems to currently exist. -----

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[math-fun] They fit!
by Bill Gosper 27 Jan '17

27 Jan '17

ccc11.gif <http://www2.stetson.edu/~efriedma/cirincir/ccc11.gif> seems to assert that two unit disks precisely fit within a circle of nine. After much bovoparturition, Mathematica agrees! Can anyone provide some insight? --rwg

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[math-fun] Ford circles pi zoom
by James Propp 27 Jan '17

27 Jan '17

Anyone care to make a short video (or GIF) that zooms in on the Ford circle packing of the upper half plane, converging on the point pi on the real axis, highlighting the circles we pass through (whose points of tangency to the real axis are the continued-fraction convergents to pi)? I searched YouTube but nothing like this seems to currently exist. Jim Propp

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[math-fun] approximating pi by 1
by James Propp 26 Jan '17

26 Jan '17

Can anyone help me remember a quote (maybe from Feynman) about approximating pi by 1? I googled "approximate pi by 1", "replace pi by 1", and a few other similar phrases, but didn't find what I'm looking for. (Of course when we're doing dimensional analysis we ignore dimensionless factors like pi, but that's not what I mean; I'm talking about replacing pi by 1 in calculations where getting an answer that's off by just one order of magnitude is good enough.) Thanks, Jim Propp

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[math-fun] Julian has constructed a spacefill dense with sextuple points
by Bill Gosper 26 Jan '17

26 Jan '17

using a 25-way conditional. ClearAll[piecewiserecursivefractal]; piecewiserecursivefractal[x_, f_, which_, iters_, fns_] := piecewiserecursivefractal[x, g_, which, iters, fns] = ((piecewiserecursivefractal[x, h_, which, iters, fns] := Block[{y}, y /. Solve[f[y] == h[y], y]]); Union @@ ((fns[[#]] /@piecewiserecursivefractal[iters[[#]][x], Composition[f, fns[[#]]], which, iters, fns]) & /@which[x])); (I cannot praise this function enough.) In[363]:= trifil25[t_] := piecewiserecursivefractal[t, Identity, Piecewise[{{{1}, #1 == 0}, {{Ceiling[25*#1]}, Inequality[0, Less, #1, LessEqual, 1]}}, {}] & , {25*#1 + 0 & , -25*#1 + 2 & , 25*#1 - 2 & , 25*#1 - 3 & , 25*#1 - 4 & , 25*#1 - 5 & , 25*#1 - 6 & , 25*#1 - 7 & , 25*#1 - 8 & , -25*#1 + 10 & , -25*#1 + 11 & , 25*#1 - 11 & , -25*#1 + 13 & , -25*#1 + 14 & , 25*#1 - 14 & , -25*#1 + 16 & , 25*#1 - 16 & , -25*#1 + 18 & , 25*#1 - 18 & , 25*#1 - 19 & , -25*#1 + 21 & , -25*#1 + 22 & , 25*#1 - 22 & , -25*#1 + 24 & , 25*#1 - 24 & }, {#1/5 & , (1/10)*(3 + I*Sqrt[3] + (-1 - I*Sqrt[3])* #1) & , (1/10)*(3 + I*Sqrt[3] + #1 - I*Sqrt[3]*#1) & , (2 + #1)/5 & , (1/10)*(6 + #1 + I*Sqrt[3]*#1) & , (1/10)*(7 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (1/5)*(3 + I*Sqrt[3] - #1) & , (1/10)*(4 + 2*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , (1/10)*(3 + I*Sqrt[3] - 2*#1) & , (-(1/10))*I*(-I + Sqrt[3])*(-2 + #1) & , (1/5)*(2 + I*Sqrt[3] - #1) & , (1/5)*(2 + I*Sqrt[3] + #1) & , (1/10)*(5 + 3*I*Sqrt[3] + #1 - I*Sqrt[3]*#1) & , (1/10)*(4 + 2*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , (1/10)*(4 + 2*I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (-(1/10))*I*(-I + Sqrt[3])*(-4 + #1) & , (1/5)*(2 + 2*I*Sqrt[3] + #1) & , (1/10)*(5 + 3*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , (1/10)*(5 + 3*I*Sqrt[3] + 2*#1) & , (1/10)*(7 + 3*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , (1/5)*(4 + I*Sqrt[3] - #1) & , (1/10)*(9 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (1/10)*(9 + I*Sqrt[3] - 2*#1) & , (1/10)*(8 + I*(I + Sqrt[3])*#1) & , (4 + #1)/5 & }][[1]] E.g., In[361]:= trifil25[2/25 + 2/75] // ComplexExpand Out[361]= {1/2 + (I Sqrt[3])/10} In[367]:= Graphics[{Hue@#, Line[ReIm@{trifil25@#, trifil25[# + 5^-5]}]} & /@ Range[0, 1 - 5^-5, 5^-5]] gosper.org/trifil25.png (Those little white hexagons are sextuple points.) The inverse fractal, constructed more or less automatically: In[380]:= untrifil25[t_] := piecewiserecursivefractal[t, Identity, If[0 <= Re[#1] <= 1 && 0 <= Im[#1] <= Sqrt[3]/2, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, {}] &, {ComplexExpand[5 #1] &, ComplexExpand[1/4 (6 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-4 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[-2 + 5 #1] &, ComplexExpand[1/4 (-6 + 6 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (4 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[3 + I Sqrt[3] - 5 #1] &, ComplexExpand[1/4 (10 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (3 + I Sqrt[3] - 10 #1)] &, ComplexExpand[1/4 (8 - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[2 + I Sqrt[3] - 5 #1] &, ComplexExpand[-2 - I Sqrt[3] + 5 #1] &, ComplexExpand[1/4 (4 - 8 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-10 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-2 + 6 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (16 - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[-2 - 2 I Sqrt[3] + 5 #1] &, ComplexExpand[1/4 (-14 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (-3 I Sqrt[3] + 5 (-1 + 2 #1))] &, ComplexExpand[1/4 (16 - 4 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[4 + I Sqrt[3] - 5 #1] &, ComplexExpand[1/4 (6 + 10 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (9 + I Sqrt[3] - 10 #1)] &, ComplexExpand[1/4 (8 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[-4 + 5 #1] &}, {#1/25 &, (2 - #1)/25 &, (2 + #1)/ 25 &, (3 + #1)/25 &, (4 + #1)/25 &, (5 + #1)/25 &, (6 + #1)/ 25 &, (7 + #1)/25 &, (8 + #1)/25 &, (10 - #1)/25 &, (11 - #1)/ 25 &, (11 + #1)/25 &, (13 - #1)/25 &, (14 - #1)/25 &, (14 + #1)/ 25 &, (16 - #1)/25 &, (16 + #1)/25 &, (18 - #1)/25 &, (18 + #1)/ 25 &, (19 + #1)/25 &, (21 - #1)/25 &, (22 - #1)/25 &, (22 + #1)/ 25 &, (24 - #1)/25 &, (24 + #1)/25 &}] Here it finds the preimages of the simplest sextuple point: In[381]:= untrifil25[1/2 + I Sqrt[3]/10] Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} --rwg

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[math-fun] Knuth update
by Henry Baker 22 Jan '17

22 Jan '17

FYI -- from DEK: https://cs.stanford.edu/~uno/news.html Volume 4B will begin with a special section called 'Mathematical Preliminaries Redux', which extends the 'Mathematical Preliminaries' of Section 1.2 in Volume 1 to things that I didn't know about in the 1960s. Most of this new material deals with probabilities and expectations of random events; there's also an introduction to the theory of martingales. You can have a sneak preview by looking at the current draft of pre-fascicle 5a (52 pages), last updated ***18 January 2017.*** As usual, rewards will be given to whoever is first to find and report errors or to make valuable suggestions. I'm particularly interested in receiving feedback about the exercises (of which there are 125) and their answers (of which there are 125). https://cs.stanford.edu/~uno/fasc5a.ps.gz

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[math-fun] Posts going astray with dud return address !!
by Fred Lunnon 22 Jan '17

22 Jan '17

On 1/18/17, James Propp <jamespropp(a)gmail.com> wrote: > I suspect this didn't get posted (since I believe that Jeff is not a member > of math-fun and that non-members of math-fun can't post), so I'm forwarding > it. > > Jim > ---------- Forwarded message ---------- > From: Jeffrey Shallit <shallit(a)uwaterloo.ca> > Date: Wed, Jan 18, 2017 at 10:25 AM > Subject: Re: Fwd: [math-fun] Thue-Morse and cubefree firs seq > <snip> This was already posted on math-fun, and JS is replying to other math-fun posts. However, the return address on his appears to be his own, instead of math-fun; hence our replies to his posts are NOT arriving at the remainder of the list. Until this gets fixed, check and insert math-fun address into your own posts! WFL

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