May 2017 - mathfun@mailman.xmission.com

[math-fun] Big Splat theory
by James Propp 04 Sep '17

04 Sep '17

Has anyone ever proposed in the Annals of Improbable Research or a similar venue that, in addition to undergoing periodic reversals of its magnetic field, Earth is also subject to occasional short-lasting reversals of its gravitational field? Such a phenomenon would handily explain selective extinction of megafauna: a mouse could survive an upward fall for five seconds followed by a downward fall for five seconds followed by an extreme deceleration, but a mastodon would not fare as well. :-) Jim Propp

7 9

Re: [math-fun] math-fun Digest, Vol 171, Issue 35 re Bathroom square dance tilings
by Stuart Anderson 07 Jun '17

07 Jun '17

> But all that's completely irrelevant to my purpose, which is simply to > enquire if these tilings have an established name --- preferably one rather > more euphonious than my current nomenclature: The bathroom floor tiling! > > Fred Lunnon Wikipedia has a good article on these tilings; Wikipedia.org/wiki/Pythagorean_tiling There are 4 names given: Pythagorean tiling Two squares tessellation Hopscotch pattern Pinwheel pattern (not to be confused with pinwheel tiling) The only other thing about these tilings I am aware of that is not mentioned in the article is that these tilings can tile a torus. See Geoffrey Morley's article www.squaring.net/sq/st/st.html Stuart Anderson

4 4

[math-fun] Number theory question
by David Wilson 04 Jun '17

04 Jun '17

Let !n = 0! + 1! + ... + n! = A003422(n) For integer k, is it true that !a == k (mod a) and !b == k (mod b) together imply !c == k (mod c) where c = lcm(a, b) ?

7 8

Re: [math-fun] How does a dull spoon manage to cut a stringy pieceof meat
by Bill Gosper 03 Jun '17

03 Jun '17

Actually, close questioning by Mike Beeler jogged me to remember what is probably a big clue: Somewhat prior to the creeping, I noticed on that shoe the grommetoidoid (occulted, opposite the circled one) in gosper.org/shoe.png can now freely pivot. After the next creep cycle, I'll try gluing it. What the heck are the correct names for grommetoid and (the simply connected) grommetoidoid? On 2017-04-27 13:33, Hilarie Orman wrote: >> From: Bill Gosper <billgosper(a)gmail.com> >> On Sun, Jan 15, 2017 at 4:57 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> > while sliding back an forth against the inside of a slippery bowl? --rwg >> > > >> No one took this question seriously, but I claim the phenomenon needs >> explaining. >> It definitely requires some friction with the bowl: Just now I had it work >> improbably >> well on a piece of meat, yet fail on a tenderer but slipperier partially >> stewed tomato >> in the same bowl. > > I did express concern about your diet. > > Tomato skins are tough. The gold standard for a perfect knife blade > is one that with a light touch will cut a tomato skin. > > I've not found any standard measurements for muscle fiber strength. > The usual measure of tenderness relates to compression. Presumably > you are twisting and tearing the fibers with rotational force. > >> OK, on to the next quotidian mystery: For years I've worn the same pair of >> https://ll-us-i5.wal.co/asr/fd205dcf-4974-49eb-b2d2-6166402bc2b3_1.ecc95c27… >> (only with five pairs of grommetoids (discontinued?) >> instead of four). Possibly coincident >> with replacing the original laces with Kiwis >> several weeks ago, a strange phenomenon has emerged: Despite two different >> lacing patterns, the right lace >> creeps rightward an inch or two per week, until it is so lopsided as to >> drag on the ground. This time, I'm >> going to mirror-image the disparity. This happens automatically if you simply juxtapose the aglets before redistributing all the slack to the shorter lace. Evening them is harder. You only want to distribute half the slack. >> Prior to this, both shoes were coming >> untied, until I modified the knot. Circle the thumbnail twice. Pull hard. Works great. (Is this Propp's "mathematician's knot"?) >> The left shoe doesn't creep. Nor has any other shoe in my experience. The >> creep is sufficiently insidious >> that I can't tell if it's while I'm walking or tying. Probably tying. My foot is so narrow that the knot is immobilized between the top two grommetoidoids. --rwg >> --rwg Some Web news service recently ran filler claiming that CIA agents signaled each other with binary patterns in their shoelaces. The practice was discontinued after feminist protests alerted the KGB. > > Does the left lace creep forward when you are in Australia? > > Hilarie

2 2

[math-fun] Calculus (?) problem ?
by Henry Baker 01 Jun '17

01 Jun '17

[A modern Zeno problem.] Achilles runs a total distance D=2^k meters in a total time T seconds. For example, Achilles might run 2048 meters in 491.52 seconds (= 8:11.52). Achilles wasn't terribly fast by today's world record standards. :-) Achilles's average *pace* (=1/speed) for the first half (=D/2) is 5 seconds/500m (= 0.01 seconds/meter) *faster* than his average pace for the second half. Achilles's average pace for the first quarter (=D/4) is 0.01 seconds/meter faster than his average pace for the second quarter. and so forth. Achilles's motion is a smooth analytic function of time and/or distance, excepting perhaps the origin. What is the functional form of his motion -- e.g., distance as a function of time, time as a function of distance, speed as a function of time, speed as a function of distance, pace as a function of time, pace as a function of distance -- any one of these forms should be useful. If I haven't made a mistake, Achilles's absolute *average pace* over the entire run shouldn't matter; clearly if the average pace for the first half differs by 5 seconds/500m from the average pace for the second half, then we could simply set the average pace for the entire run to zero (i.e., Newton relativity), so he may end up going backwards part of the time (i.e., relative to a tortoise moving at constant pace = Achilles's average pace). But this still leaves the problem of the functional form of his speed.

1 1

[math-fun] The bathroom square dance?
by Bill Gosper 01 Jun '17

01 Jun '17

On 2017-05-29 05:48, James Propp wrote: > Can a square-tiling of the plane, like a square-tiling of a square, be > converted into an electrical network (a la Brooks, Smith, Stone, and > Tutte)? That might give a picture that would be suggestive of a different > name than "bathroom floor tiling". > > Incidentally, I've always thought that the tiling of the plane by squares > and octagons was the "bathroom floor tiling": The tiles of the MIT AI Lab bathroom were just squares, randomly dark and light. I remember because, as I looked at them, they'd spontaneously run a couple of Life generations, so long had I spent watching the CRT. Strangely, the dark ones were 1 and the light ones 0. Maybe the dark ones were sparser? --rwg > > https://www.houzz.com/photos/48092404/Victorian-Octagon-With-White-Dot-Porc… > > Jim Propp > > On Sun, May 28, 2017 at 7:24 PM, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > >> As on page 494 of the handy 900-page >> http://paulbourke.net/dataformats/postscript/psref.pdf >> for instance. >> >> I cannot honestly pretend that "PostScript type 10 halftone dictionary" >> constitutes a great improvement on "bathroom floor tiling". >> >> Where are the ancient Greeks, the Moors (Moroccan, not Yorkshire), >> the Alhambra when we need them? >> >> WFL >> >> >> On 5/28/17, Mike Speciner <ms(a)alum.mit.edu> wrote: >> > This reminds me of PostScript's type 10 halftone dictionaries. In >> > ancient history, when I was implementing halftones in firmware, I >> > realized that any rational-tangent halftone could be represented by a >> > single orthogonal rectangle of values that repeated horizontally but was >> > offset vertically. Later, PostScript 3 appeared with type 10 halftone >> > dictionaries, defined by two orthogonal squares tiling the plane. It >> > took me a little while to realize that these approaches are equivalent. >> > (Halftones are normally defined by a single square of values with some >> > arbitrary angle to the coordinate axes. When the angle has a rational >> > tangent, we can get either of the above techniques.) >> > >> > I guess I didn't answer your question. >> > >> > --ms >> > >> > On 28-May-17 11:13, Fred Lunnon wrote: >> >> A well-known tiling of the plane employs offset square tiles of two >> >> sizes. >> >> See the example below using tiles of 2 and 3 units, somewhat perturbed >> >> on this occasion by proportional spacing rather than incompetent home >> >> improvement skills. >> >> >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> >> >> I can't resist remarking that such diagrams provide a particularly >> >> elegant >> >> demonstration of Pythagoras' theorem, apparent when the period along >> >> a horizontal axis is calculated. >> >> >> >> A particularly humiliating programming exercise involves generating a >> >> matrix in which unoccupied cells above are represented by 0 and >> >> asterisked cells by +1 or -1 , chosen to satisfy the local >> constraints >> >> on >> >> an integer number wall (frame theorems). >> >> >> >> But all that's completely irrelevant to my purpose, which is simply >> to >> >> enquire if these tilings have an established name --- preferably one >> >> rather >> >> more euphonious than my current nomenclature: The bathroom floor tiling! >> >> >> >> Fred Lunnon >> >>

2 1

[math-fun] The bathroom square dance
by Bill Gosper 31 May '17

31 May '17

On 2017-05-29 10:55, Stuart Anderson wrote: > To make the tiles really dance ... > Danny Caligari > ; > https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-… > Bad URL there. Use http://www.math.brown.edu/~rkenyon/gallery/squarespiral2.gif While you're there, stare at http://www.math.brown.edu/~rkenyon/gallery/squarespiral.gif for ~20 secs, then the top corner of your screen. --rwg

2 1

[math-fun] Mechanical computer for kids: Turing Tumble
by Henry Baker 31 May '17

31 May '17

FYI -- https://www.kickstarter.com/projects/871405126/turing-tumble-gaming-on-a-me… Turing Tumble: Gaming on a Mechanical Computer Build mechanical computers powered by marbles to solve logic puzzles. Escape planet Eniac and discover how computers work. Turing Tumble is a new type of game where players (ages 8+) build mechanical computers powered by marbles to solve logic puzzles. It's fun, addicting, easy-to-learn, and while you're playing, you discover how computers work. I'm all about teaching kids to code. When I was a professor at the University of Minnesota, I saw how valuable it is for all students to be coders. I have three young kids and I've tried all sorts of games to build their interest in coding. The problem is that they all treat computers like abstract, black boxes. They overlook the fundamental, most amazing concept: how simple switches, connected together in clever ways, can do incredibly smart things. Kids learn best when they use their senses to explore concepts. Turing Tumble is the only game that lets kids see and feel how computers work. The logic isnÂt hidden inside a computer chip Â it's all right there in front of them. It builds logic and critical thinking skills, fundamental coding concepts, and grounds their understanding of computers. And adults, this game is not just for kids! It's a game unlike any you've ever played. It takes creative thinking to come up with a solution to each puzzle - you can't just stumble across the solution by trial and error - and it's extremely gratifying to build it and watch it work. Even if you're an expert programmer, I guarantee you'll find the latter puzzles a challenge. ... Also, this page has some animated gifs showing some of the mechanical logic at work: https://boingboing.net/2017/05/30/cool-marble-powered-mechanical.html

1 0

[math-fun] The bathroom square dance
by Stuart Anderson 30 May '17

30 May '17

To make the tiles really dance ... Danny Caligari ; https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-…

5 6

[math-fun] The bathroom square dance?
by Fred Lunnon 29 May '17

29 May '17

A well-known tiling of the plane employs offset square tiles of two sizes. See the example below using tiles of 2 and 3 units, somewhat perturbed on this occasion by proportional spacing rather than incompetent home improvement skills. [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] I can't resist remarking that such diagrams provide a particularly elegant demonstration of Pythagoras' theorem, apparent when the period along a horizontal axis is calculated. A particularly humiliating programming exercise involves generating a matrix in which unoccupied cells above are represented by 0 and asterisked cells by +1 or -1 , chosen to satisfy the local constraints on an integer number wall (frame theorems). But all that's completely irrelevant to my purpose, which is simply to enquire if these tilings have an established name --- preferably one rather more euphonious than my current nomenclature: The bathroom floor tiling! Fred Lunnon

4 5