January 2020 - mathfun@mailman.xmission.com

[math-fun] Triangle, 4-gones numbers, etc.
by Éric Angelini 04 Jan '20

04 Jan '20

Hello Math-Fun See those 3-digit (a, b and c) numbers as triangles with sides a, b and c: 111, 122,133,144,155,166,177,188,199, 212,221,222,223,244,... Is this (finite) seq in the OEIS? (I cannot compute its terms by hand -- all I know is that the biggest digit of n must be < the sum of the two others, like in 497). An equivalent seq could be computed for 4-gones numbers (starting with 1111,1112,1121,1211,... I guess), for 5-gones etc. Pardon me of this is old hat. Best, É.

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[math-fun] There are other kinds of corner reflector?
by Bill Gosper 03 Jan '20

03 Jan '20

E.g., square? big-red-buoy-removed-from-florida-beach-after-sitting-there-for-days <https://www.foxnews.com/us/big-red-buoy-removed-from-florida-beach-after-si…> (Kids, check out https://en.wikipedia.org/wiki/Corner_reflector . Three bounces and the laser beam returns to its source. But likewise, a bundle of *uncolilmated* beams! I've always wanted to put an anemometer made of corner reflectors on my car to confuse the heck out of police radar. Radar return ordinarily drops off with the *fourth power* of distance. But only 2nd power with a corner reflector. That's huge. Stealth in reverse. Radar transmitters are loud. In the 60s, I worked at a naval air station where a visiting photographer was severely burned when all the flashbulbs <https://upload.wikimedia.org/wikipedia/commons/f/fa/Brownie_Hawkeye_with_Fl…> went off in his pockets. (That sparkly stuff is "magnesium wool"!)) —rwg

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Re: [math-fun] There are other kinds of corner reflector?
by hbaker1 03 Jan '20

03 Jan '20

There are $100 baseball radar guns available for testing your device. Unfortunately, the test may be successful, costing you $100. You may want to "de-tune" the device and gradually tune it up so that you don't immediately burn it out. Alternatively, try it at a long distance -- e.g., half-mile (?), and then bring it closer. https://www.dickssportinggoods.com/products/baseball-radar-guns.jsp https://www.opticsplanet.com/radar-guns.html BTW, you've just designed a very cheap totally *passive* wireless anemometer, that you read by shining your radar gun at it. You should also be aware that with cheap 5GHz wifi and SDR radio components, you can build nifty things at home very inexpensively. https://hackaday.io/project/147-ism-s-band-lfmcw-radar-prototype/log/238-ea… https://hackaday.io/projects?tag=radar https://hackaday.com/2014/02/24/guest-post-try-radar-for-your-next-project/ https://luigifreitas.me/2018-11-23/software-defined-radar-cw-doppler-radar-… Also, due to the volume applications of radar today, you can buy quite inexpensive 60GHz components to play with. https://www.embedded.com/why-60ghz-mmwave-is-moving-into-the-mainstream/ -----Original Message----- >From: Bill Gosper <billgosper(a)gmail.com> >Sent: Jan 2, 2020 2:02 PM >To: math-fun(a)mailman.xmission.com >Subject: [math-fun] There are other kinds of corner reflector? > >E.g., square? >big-red-buoy-removed-from-florida-beach-after-sitting-there-for-days ><https://www.foxnews.com/us/big-red-buoy-removed-from-florida-beach-after-si…> >(Kids, check out https://en.wikipedia.org/wiki/Corner_reflector . Three bounces and the >laser beam returns to its source. But likewise, a bundle of *uncolilmated* beams! > >I've always wanted to put an anemometer made of >corner reflectors on my car to confuse the heck out of police radar. Radar return > ordinarily drops off with the *fourth power* of distance. But only 2nd power with a corner > reflector. That's huge. Stealth in reverse. Radar transmitters are loud. In the 60s, I >worked at a naval air station where a visiting photographer was severely burned when all the flashbulbs ><https://upload.wikimedia.org/wikipedia/commons/f/fa/Brownie_Hawkeye_with_Fl…> >went off in his pockets. (That sparkly stuff is "magnesium wool"!)) > —rwg

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[math-fun] (Corollary of) Kepler's 3rd Law:
by Bill Gosper 03 Jan '20

03 Jan '20

The orbital period of a planet depends only on its major axis. gosper.org/4planets.gif . Does anybody feel like proving that with this phasing, these planets are collinear only twice per period? —rwg

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[math-fun] Mathpuzzle update
by ed pegg 02 Jan '20

02 Jan '20

I've started making updates to http://www.mathpuzzle.com/ again. --Ed Pegg Jr

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[math-fun] When decades start
by James Propp 02 Jan '20

02 Jan '20

I’ve heard that, “officially”, the 21st century began on January 1, 2001, not January 1, 2000. Do those officials (whoever they are) also contend that the Aughts (or whatever they’re called) didn’t start until 2001, so that likewise the Teens didn’t start until 2011 and the Twenties won’t start until a year from now? Or do they say that the last year of each century is also the first year of a new decade? Sorry if I’m resurrecting a thread from ten or twenty years ago... Jim Propp

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[math-fun] Prefix for one-third?
by James Propp 01 Jan '20

01 Jan '20

Is there such a prefix? I’ve done about ten minutes of web-searching, but can’t find out if there’s a prefix for one-third, though it seems that trito- and triento- (from Greek and Latin respectively) would be natural choices. Jim Propp

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Re: [math-fun] Boustrophedon primes
by Walter Trump 01 Jan '20

01 Jan '20

There exists another "Boustrophedon"-sequence in OEIS. I like the name "Boustrophedon primes" and appreciate all the work that was done in a short time for A330339. What I do not like is that Boustrophedon primes can only occur on the left ends of the lines and never on the right ends (except of the prime 2). Therefore I thought about a slightly different definition and found another "Boustrophedon"-sequence which already exists in OEIS. In the following graphic all primes are represented by colored dots and all other natural numbers by small black dots. https://www.trump.de/definition-A330339-A282178.gif Neil already added in the description of A282178 a link to A330339. A big b-file with the first 846 elements already exists. A282178 was created by Samuel B. Reid in 2017. A282178 starts with number 1 and has a higher density than A330339. Please have a look at 3 pictures demonstrating A282178. !!! Zoom in to read all primes !!! https://www.trump.de/A282178-01.gif https://www.trump.de/A282178-02.gif Especially I like the definition using a perfect zigzag path of the natural numbers with primes in the vertices of the path: https://www.trump.de/A282178-zigzag.gif Walter

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[math-fun] A First Look at "Snow Crystals"
by Brad Klee 01 Jan '20

01 Jan '20

1. The substantial work of Kenneth Libbrecht "Snow Crystals" [1] was released in October 2019, and recently popularized by Quanta [2]. Once again, the Quanta article buys in to egocentric optimism, and fails to produce a critical book review (my opinion: what they should be doing here). Yet this article does discuss relatively original and timely research. Also, awesome video header [3]! 2. The book is very long at 500+ pages, but if nothing else at least read the first chapter for history from Kepler to Ukichiro Nakaya. Nakaya, a contemporary of Tomonaga, was the first scientist to chart a temperature-dependent phase diagram for snow crystals, see figures 1.24, 1.25, 1.12 (copied on quanta as "A Wintry Mix"). 3. Figures 3.3 and 3.4 explain discrepancy in adsorption; however, they also contradict the phase diagram figure 1.12. The author even admits that this formation paradox remains incompletely resolved, so still an open question of science. Chapter 3-4 contains in-depth science worth more serious critical reading. 3b. Quanta figure "Growing Snowflakes" is extremely confusing, and excessively over-simplifed. The use of "face" and "edge" is flat out _Wrong_. This is not a good summary of chapters 3-4. 4. Chapter 5 "A Progression of Snow Crystal Models" is an exciting prospect; though, somewhat of a disappointment in execution, especially considering diligent history of Chapter 1. 4b. The "Packard Snowflake" *is not* the earliest relevant C.A. model of dendrite growth. Unfortunately Packard 1986 [4] does not cite Ulam 1962 [5]; though, OEIS does, see [6-8]. Ulam mentions that Holladay already knew of 2^n extinctions as early as 1960 [8]! 4c. A151723 includes a picture [9] that shows the structure of a sectored plate, and features the same 2^n extinctions. This picture is drawn over the half-hexagon tiling. From a mathematical perspective, half-hex is a better analog to 2^n C.A. than the Koch Snowflake. 4d. A similar comparison between Chair tiling and another growth C.A. is available at [10]. After this, it is possible to show that Code 686 C.A. in fact *builds* the chair tiling[11]. It follows in some sense that 686 ~ trilobite and crab. (Did C.G.S. know this?) 4e. We do not know if a variation of the C.A. from A151723 can be used to build the Socolar-Taylor tiling, or any of its variants [12]. This is a worthwhile open question. 5. Even before Holladay & Ulam, L. Pauling began to develop the "ice-type model" in 1935 [13]. Why isn't this cited? 5b. Pursuant to 4b. to 4e. a perspective combining matching-rule tiling theory and cellular automata *is not being presented* in Libbrecht's "Progression of snow crystal models". Does the author have a reason for such an exclusion? 5c. Hypothetical: In a matching-rule tiling theory for describing snow crystals, obviously the binding energies would change with thermodynamic conditions, but also, the matching-rules could change over temperature, pressure, vapour saturation, etc. 6. Chapters 6-9 are out of my price range, but describe incredible experiments. Thankfully lots of pictures and videos are available for purpose of data analysis. 7. Chapter 10, WOO-HOO! Libbrecht speaks: "Keeping an eye out for interesting crystals is a fine pursuit whenever you happen to be outside during a light snowfall. You could be riding the chair lift at your local ski area, taking a stroll through the winter woods, or just waiting in your car somewhere. If the snow is falling all around you, why not have a look from time to time to see what you can find?" 7b. Couldn't agree more, great advice Kenneth! Here are some specimens I collected at about 9800 feet elevation in the Rocky Mountains on Christmas morning: https://0x0.st/zDFW.jpg 7c. The photographs were rushed, but do show plates, sectored plates, stellar dendrites, and simple stars. From the existence of these structures, I think we can infer temperature of formation in the magic range around -15c, and decently high vapour saturation. 7d. Similar structures persisted at least 1000 feet above the height of base camp; though, I did not collect photo evidence. 8. HATE WARNING: The website "snowflakevictory.com" is not actually a website about physical snowflakes. It is a political site for the impeached Donald Trump, and uses the word "snowflake" as derogatory slang [14]. 8b. I supposed that domain name registration is solely a question of money ($$$) at this point. However, if decisions were being made by more fair criteria, I would think that snowflakevictory.com would simply redirect to Kenneth Libbrecht's awesome website: http://www.snowcrystals.com/ Any thoughts on 4e., 5c., or snowflake calculations in general? Anyone else getting decent snowflake pictures this season? Happy new-flakes, --Brad [1] https://arxiv.org/abs/1910.06389 [2] https://www.quantamagazine.org/toward-a-grand-unified-theory-of-snowflakes-… [3] https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/12/SnowStructure_2880x16… [4] http://www.scipress.org/e-library/sof/pdf/0095.PDF [5] https://oeis.org/A002858/a002858.pdf [6] https://oeis.org/A002858 [7] https://oeis.org/A151723 [8] https://oeis.org/A322662 [9] https://oeis.org/A151723/a151723.png [10] https://oeis.org/A147562/a147562_1.png [11] https://demonstrations.wolfram.com/Code686BuildsTheChairTiling/ [12] https://demonstrations.wolfram.com/LimitPeriodicTilings/ [13] https://en.wikipedia.org/wiki/Ice-type_model [14] https://en.wikipedia.org/wiki/Snowflake_(slang)

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