June 2017 - mathfun@mailman.xmission.com

[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) ?

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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

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[math-fun] crossword crystals
by Veit Elser 03 Jun '17

03 Jun '17

I’ve been playing with crossword puzzle variants. Here’s one that Dan A. will like. 1) The puzzle grid is an N by N torus. 2) In each row and column there should be exactly one black square. For example, here is a solved 7 by 7 puzzle (black -> blank): A D E R L E L E D S E A E R L E T T D W A N T E L A D D E R S E N D E R E A T E R H To avoid trivial solutions we should require that the horizontal words are not all the same — as in this one. Still, the solutions I’m finding all have lots of symmetry. Note that the vertical words in this solution match the horizontal words. Now here’s a challenge. Find a solution, with no repeated horizontal words, but where the black squares do not form a backward diagonal. -Veit

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[math-fun] The bathroom square dance
by Bill Gosper 02 Jun '17

02 Jun '17

I get _strong_ twisting from http://www.math.brown.edu/~rkenyon/gallery/squarespiral.gif (vs squarespiral2), and ascribed it to differential fatigue of my motion receptors. (Generalized moving bar illusion.) On 2017-06-01 09:21, Fred Lunnon wrote: > At second attempt, I did detect a slight twisting effect. > > I imagine this is the same mechanism that causes the room apparently > to rotate in the reverse direction after one suddenly ceases spinning about > a vertical axis (how do ballerinas cope, I wonder?); or the floor to roll and > pitch after one disembarks from a small boat. For many hrs after a local (1989, Richter 7) quake, the ground felt gelatinous. I assumed it was objectively measurable seismicity. Perhaps not. --rwg > > I doubt whether retinal cell configuration has any part to play in > such effects, > which are manifestations of a general mental process, occurring at many > levels, which seeks to filter out constant background from our perception, > leaving us conscious only of any unpredictable variation. > > I once encountered a spectacular example of this phenomenon, involving > hearing rather than sight. Standing beside the Victorian wall clock in the > hall of my home late one quiet night, I failed to hear its customary loud tick, > and concluded that it must have broken down. Visual inspection however > revealed that the pendulum continued to swing; and as I stared at it in some > perplexity, over several seconds its tick gradually faded in to become once > more clearly perceptible. > > Similarly as Sarah Silverman recently pointed out, it is impossible > to register > one's own body odour, whatever olfactory violence one may inadvertantly be > perpetrating upon one's neighbours. The distinctive feature of sight in this > context is that relative motion possesses sign, making its (sudden) absence > more directly contrasted with presence. > > Fred Lunnon > > > On 6/1/17, Bill Gosper <billgosper(a)gmail.com> wrote: >> What's supposed to happen after 20 sec? >> >> You're supposed to quickly redirect your stare to top left of your screen. >> >> >> (That's my sixties cred shot to pieces.) WFL >> >> >> Or just your retinal cone cells. If you do not see disturbing, startling >> twisting, >> >> consider an ophthalmological consult. --Bill >> >> >> >> On 5/31/17, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> 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 >>

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[math-fun] Edwards book on Integration
by Bill Gosper 02 Jun '17

02 Jun '17

Not(?) to be confused with https://archive.org/details/treatiseonintegr01edwauoft . --rwg On 2017-06-02 06:49, Hans Havermann wrote: > Guy Haworth: "His father was a Professor of Mathematics (hence the > 'Professor' above) and wrote a book on Integration solutions - my > school had a copy (with hard, orange covers)." > > The man was Reginald Walter Kenrick (RWK) Edwards (born 1862), > educated at Cambridge and taught at King's College, London. He > authored The Mermaid of Inish-Uig (1898), Dick Vaughan's First Term > (1901), A Radial Area-Scale (1904), and An Elementary Text-Book of > Trigonometry (1911). I can't find the Integration book to which you > refer but I'll note that Sir Alfred Lodge authored Integral Calculus > for Beginners (1905) about which is written (The Mathematical > Gazette): "The most novel feature of the book is perhaps the seventh > chapter dealing with approximate methods of integration. Here, after > the well-known rules of Simpson and Weddle, approximate formulae, > recently devised by Mr. R. W. K. Edwards and Professor Lodge himself, > are given, for dealing with the case in which the curvilinear boundary > of a required area cuts the axis at right angles; a case for which, as > is well known, rules of the Simpson type are not well fitted." >

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[math-fun] Bill & Rohan's G12 talk
by Bill Gosper 02 Jun '17

02 Jun '17

https://youtu.be/lXsVWwPa7bc --rwg http://www.gathering4gardner.org/g4g12-videos/

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[math-fun] Edwards book on Integration
by Guy Haworth 02 Jun '17

02 Jun '17

In the UK, 'Professor' Jimmy Edwards was a prominent comedian and actor through to 1970. His father was a Professor of Mathematics (hence the 'Professor' above) and wrote a book on Integration solutions - my school had a copy (with hard, orange covers). I can't find any trace of this book on the web but some may know of it. Clues welcome. Guy

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[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.

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Re: [math-fun] The bathroom square dance
by Fred Lunnon 01 Jun '17

01 Jun '17

At second attempt, I did detect a slight twisting effect. I imagine this is the same mechanism that causes the room apparently to rotate in the reverse direction after one suddenly ceases spinning about a vertical axis (how do ballerinas cope, I wonder?); or the floor to roll and pitch after one disembarks from a small boat. I doubt whether retinal cell configuration has any part to play in such effects, which are manifestations of a general mental process, occurring at many levels, which seeks to filter out constant background from our perception, leaving us conscious only of any unpredictable variation. I once encountered a spectacular example of this phenomenon, involving hearing rather than sight. Standing beside the Victorian wall clock in the hall of my home late one quiet night, I failed to hear its customary loud tick, and concluded that it must have broken down. Visual inspection however revealed that the pendulum continued to swing; and as I stared at it in some perplexity, over several seconds its tick gradually faded in to become once more clearly perceptible. Similarly as Sarah Silverman recently pointed out, it is impossible to register one's own body odour, whatever olfactory violence one may inadvertantly be perpetrating upon one's neighbours. The distinctive feature of sight in this context is that relative motion possesses sign, making its (sudden) absence more directly contrasted with presence. Fred Lunnon On 6/1/17, Bill Gosper <billgosper(a)gmail.com> wrote: > What's supposed to happen after 20 sec? > > You're supposed to quickly redirect your stare to top left of your screen. > > > (That's my sixties cred shot to pieces.) WFL > > > Or just your retinal cone cells. If you do not see disturbing, startling > twisting, > > consider an ophthalmological consult. --Bill > > > > On 5/31/17, Bill Gosper <billgosper(a)gmail.com> wrote: > > 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 >

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

31 May '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 >> >>

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