August 2012 - mathfun@mailman.xmission.com

[math-fun] That fractal pentagram again
by Bill Gosper 09 Apr '15

09 Apr '15

For some pictorial. mildly trick questions: http://gosper.org/foupent.pdf (1.6M) --rwg

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[math-fun] why can a line of slope 10^n determine n decimal digits of π?
by Gary Antonick 19 Jun '14

19 Jun '14

Just noticed this yesterday and can't figure out why it works. Draw a circle with radius 1 around 0,0. Draw a line with slope -10 that hits the circle at 1,0. This line will also hit the circle again and make a little arc. The length of the arc will be about 1/31 the perimeter of the circle. Now try again with a line of slope -100. The line will create an arc 1/314 of the circle. Starting to look a bit like pi. Am wondering why this is. In case I made a mistake here's what I'm trying to describe.. if the circle were a clock the first line would cut an arc from 15 minutes (3:00) to almost the 13 minute mark.

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[math-fun] sqrt(pi)
by Bill Gosper 24 Mar '13

24 Mar '13

Pairwise combining the terms of A&S 6.5.31 (upper Gamma CF), then adding last week's lower gamma CF: In[185]:= Equal[Gamma[ z], ((y^z*((1/(cfk[n*(z - n), -z + y + 2*n + 1, List[n, Infinity]] - z + y + 1)) + (1/(z + cfk[n*y, z - y + n, List[n, Infinity]] - y))))/(E^y))] z 1 Gamma[z] == (y (------------------------------------------- + -y + z + cfk[n y, n - y + z, {n, Infinity}] 1 y -----------------------------------------------------------)) / E 1 + y - z + cfk[n (-n + z), 1 + 2 n + y - z, {n, Infinity}] where cfk:=ContinuedFractionK and y>0 can be chosen to optimize convergence. Specializing, In[186]:= % /. z -> 1/2 /. y -> 1 Out[186]= 1 Sqrt[Pi] == (-------------------------------------- + 1 1 -(-) + cfk[n, -(-) + n, {n, Infinity}] 2 2 1 ------------------------------------------) / E 3 1 3 - + cfk[(- - n) n, - + 2 n, {n, Infinity}] 2 2 2 In[192]:= N[List @@ %186 /. cfk[num_, den_, _] :> ContinuedFractionK[num, den, {n, 9}]] Out[192]= {1.77245, 1.77244} This is the first non-asymptotic sum or CF I've seen for sqrt(pi). But the sum behind the lowergamma cf has been there all along, so I must not have been paying attention. --rwg

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Re: [math-fun] Golden Ratio discovered in uterus
by Bill Gosper 09 Sep '12

09 Sep '12

Here's a piecewise linear spiral approximation to a nautilus (shown me by Jack Holloway): http://www.amazon.com/The-Irrationals-Story-Numbers-Count/dp/0691143420/ref… --rwg Also, quarter-circles (in "four point ellipses") do interesting things<http://gosper.org/pumpstill.gif> . ----- Hi Jim, Yes, there is more to be said, but I had to use that curve (consisting of quarter circles) because the point I wanted to make is specifically about that curve. The curve is only a ruler-and-compass approximation to an exponential spiral and the differences may be important in some contexts. But it is a much larger error to claim that curve is a model for the nautilus, and that big error is continuously being copied and re-published. E.g., they both appear on the cover of Livio's popular book The Golden Ratio, and inside, his text about them confusingly does not clarify the difference. As to the curve not being a "true spiral" I don't know of a standard definition of "true spiral" that includes the many types of spiral that people often study, but doesn't include that assemblage of quarter circles. As far as I am concerned, it is a "true spiral" that happens not to be C2 continuous. Georgehttp://georgehart.com/ On 8/15/2012 11:02 PM, James Propp wrote:>>From 0:20 to 0:40 in his video, George shows a famous picture of the> "golden spiral". But as many of you probably know (no doubt including> George), this picture is bogus. Specifically, it conflates two different> curves: a curve made up of quarter-circular arcs (which isn't a true> spiral) and a curve made of true spiral arcs (which isn't tangent to the> rectangles).>> George doesn't discuss this in his video, since minor issues like this> would distract from his main point. But still, I wish the video hadn't> used the inaccurate picture.>> Yes, I'm being a bit picky. At times like this I am reminded of a motto I> saw on a tee-shirt once: "I'm not pompous. I'm pedantic. There's a> difference." :-)>> Jim>> On Tue, Aug 14, 2012 at 4:22 PM, George Hart <george(a)georgehart.com <http://gosper.org/webmail/src/compose.php?send_to=george%40georgehart.com>> wrote:>>> Hi Alex,>>>> Unfortunately, as a consequence of this being hosted by the Guardian, I>> expect all the folks who uncritically see the golden ratio everywhere will>> now be blathering about golden uteri. I've started making a series of math>> videos and here's one rebuttal to the golden ratio cult:>>>> http://www.youtube.com/watch?**v=_gxC8OjoQkQ<http://www.youtube.com/watch?v=_gxC8OjoQkQ>>>>> As to the image on that page, I hope you don't mind if I interpret your>> use of the middle finger as an editorial comment. I feel it's misleading>> to show precise metal calipers and print 1.618, as if finger measurements>> were precise enough to support four significant digits.>>>> George>> http://georgehart.com/>>>>>>>>>> On 8/14/2012 10:23 AM, Alex Bellos wrote:>>>>> I've just launched a maths blog for the Guardian>>>>>> Here's my first post: the research is serious, but any conclusions need>>> to be taken with a medium to large dosage of salt.>>>>>> http://www.guardian.co.uk/**science/alexs-adventures-in-**>>> numberland/2012/aug/14/golden-**ratio-uterus<http://www.guardian.co.uk/science/alexs-adventures-in-numberland/2012/aug/1…>>>>>>> alex>>> ______________________________**_________________

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[math-fun] Golden Ratio discovered in uterus
by Alex Bellos 01 Sep '12

01 Sep '12

I've just launched a maths blog for the Guardian Here's my first post: the research is serious, but any conclusions need to be taken with a medium to large dosage of salt. http://www.guardian.co.uk/science/alexs-adventures-in-numberland/2012/aug/1… alex

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[math-fun] max determinant circulant sign matrix of size 40, and summary
by Warren Smith 01 Sep '12

01 Sep '12

N=40 det=6.439842994660582748450 e31 circ=-+++---+-++---++-+-+---------++-++--+-++ think I'll quit finding them now. ---- In summary, all 2^N different NxN circulant sign matrices can have their determinants computed in O(N) arithmetic operations per matrix using the fourier trick combined with the gray code trick. If however we were to generate "binary necklaces" via a "3-gray code" for them, Mark Weston & Vincent Vajnovszki: Gray codes for necklaces and Lyndon words of arbitrary base, Pure Mathematics and Applications 17, 1-2 (2006) 175-182. http://homelinux.capitano.unisi.it/~puma/17_1_2/weston.pdf T.Ueda: Gray codes for necklaces, Discrete Maths. 219,1-3 (2000) 235-248. T.M.Y. Wang and C.D. Savage: A Gray code for necklaces of ﬁxed density, SIAM J. Discrete Maths 9,4 (1996) 654-673. then we could effectively compute the determinants of all NxN circulant sign matrices in O(1) arithmetic operations per matrix, which would have enabled me to reach, not N=40, but in fact about N=44, with the same amount of computing. Perhaps somebody will program that refinement.

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Re: [math-fun] helical slide rule
by Bill Gosper 01 Sep '12

01 Sep '12

A few G4s ago, someone exchange gifted 3D-printed helic(oid)ally bisected screw-apart sintered plastic cubes. It would be interesting to determine the dimensions, for a given pitch, of a helic(oid)ally bisected cylinder that maximize torsional stiffness. Probably, the fatter the better. Alan Adler tells me stiffness scales with the cube of thickness, so gluing two planks together makes them eight times stiffer than letting them slip at the interface. (Extreme case: a ream of paper.) I think I once mentioned here that horizontal tree limbs often fail gradually by splitting internally along a horizontal plane through the axis. This ought to reduce their breaking strength fourfold and their stiffness eightfold. Perhaps sagging or too-slowly-swaying trees could be flagged in this age of near universal video surveillance. I don't understand the need for a transparent slide rule component, except for maybe a sleevelike cursor. I wonder if Shapeways can work tungsten, which is surprisingly cheap. I was shocked that the Mars rover dumped 330lb of tungsten ballast before landing. Remember the incredulity when Sputnik-1 claimed to weigh 184lb? Depleted uranium is pretty stiff and abundant, but its mechanical properties are complicated. (John McCarthy once considered trying to corner the DU market in anticipation of the the eventual acceptance of breeding, and found that the supply was inexhaustible.) McCarthy also collected stories of untimely inventions, e.g. the non-guttering candle, but I'm not sure he digitized them. --rwg ------------- > On 8/31/2012 10:53 AM, Allan Wechsler wrote:> Don't cut the pieces out of a hollow cylinder. Cut them out of a> solid cylinder. I bet George Hart can just print out the pieces for> you... mrob>You still get two helices, with no solid core because the "C scale" and "D scale" pieces meet each other at the axis. I think it would still be springy enough to cause excessive error, unless it were solid diamond or tungsten (-:, and you still have to solve the axis alignment problem (making sure the two helices remain co-axial) which is much more easily solved by the transparent concentric hollow cylinder solution that I mentioned at the end. On 8/31/12, George Hart <george(a)georgehart.com <http://gosper.org/webmail/src/compose.php?send_to=george%40georgehart.com>> wrote:> [...] A cursorless helical slide rule that doesn't "spring"> could be designed using a two-color process to make the inner cylinder> solid with the scale of a contrasting color, and the outer (hollow)> cylinder using a clear material for its background color. (The scale> markings could be above the line for one and below the line for the> other.) However, the technology is not yet as accurate and inexpensive> as one would like for two-color 3D-printing processes. Wow, 3-D printing the whole thing, that's an approach I wasn't willing to consider. I'd be asking how we make the end product snug enough to keep the two helices co-axial. Clearly you'd print them as two pieces, smooth off the mating surfaces, then screw then together -- but how do you smooth the inner surface of the transparent outer piece, whilst keeping it transparent? I guess I'm getting rather far afield of math-fun here, unless we start citing relevant formulas. -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 -mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com

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[math-fun] maxdet circulants, Clark's issues
by Warren Smith 31 Aug '12

31 Aug '12

>WE CLARK: Starting at N = 29 using the values of -1 and + 1 given by Warren and using Maple with Digits = 1000, I get different values when I round his determinants to the nearest integer. --good. My values were inexact. > For example, for N = 29 I get Maple's value = -668472148627227148288 Warren's value = 668472148627227148000 It seems that in Warren's numbers there are not enough significant digits above N = 28. --true. >Also in the case of N = 29 I get a different sign for the determinant. --above 28 I printed out |det| not det. --Note: It is conceivable, but highly unlikely, that my maxdet circulants are not really maxdet, because there is some other one with a determinant like 1.00000000001X higher. I doubt that pathology has happened but it might be a concern if this table were extended further to say N=50 or 60, and I cannot promise with 100% certainty that it did not happen already. Of course it would be possible to enhance my program to overcome these quibbles, but I did not. --If Clark has computed all the exact integer values via MAPLE, please email them to me warren.wds AT gmail.com

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[math-fun] maxdet, Clark issues
by Warren Smith 31 Aug '12

31 Aug '12

Also keep in mind that any NxN sign matrix has determinant divisible by 2^(N-1) so you should NOT be rounding to nearest integer, but to nearest multiple of 2^(N-1).

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[math-fun] zooming in on the reals
by James Propp 31 Aug '12

31 Aug '12

Has anyone created interactive software that lets one zoom in on the real line? (One possible design would be an inverse symbolic calculator connected to a Google Maps interface.) Jim Propp

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