September 2012 - mathfun@mailman.xmission.com

[math-fun] Constructing a hyperbolic geodesic
by Mike Stay 04 Sep '12

04 Sep '12

Given the unit circle U, a line L that intersects that circle, and a point P within the circle on the line, how do I construct the one circle C that is both tangent to L at P and intersects U at right angles? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

2 2

[math-fun] n-dimensional volume
by Erich Friedman 03 Sep '12

03 Sep '12

greetings. does anyone know the n-dimensional volume V(k,n) that satisfies: 0 <= x_i <= 1 (for 1 <= i <= n) and \sum_{i=1}^n x_i <= k (where k is some integer 0 <= k <= n)? this is just the portion of the unit hypercube that is underneath the hyperplane x_1+...+x_n <= k. or, if you're a probabilist like me, V(k,n) is the chance that the sum of k iid uniform [0,1] random variables is larger than the sum of (n-k) other iid uniform [0,1] random variables. surely these are important enough to have been determined before, though i could not find these in the OEIS. easy multiple integrals and symmetry show: V(0,n) = 0 V(n,n) = 1 V(1,n) = 1/n! V(n-1,n) = 1 - 1/n! V(n/2,n) = 1/2 V(n-k,n) = 1 - V(k,n) is there a general formula for V(k,n)? i couldn't even find a good way to calculate V(2,5).... erich friedman

4 4

[math-fun] GMail is embarrassing me
by Bill Gosper 02 Sep '12

02 Sep '12

I said [I have no idea how or why GMail sent two copies of this.] Due to marginal connectivity, > ½ the time I click Send, GMail says "An error occurred and your message was not sent". This sometimes gives me a chance to add a couple of afterthoughts before retrying. But GMail turns out to be lying--the message *was* sent, at least to some recipients, so it looks like I'm bothering you all with resends of trivial afterthoughts. --rwg

6 8

[math-fun] How do you print a number? DUHHHHH!!!!!!!
by Warren Smith 02 Sep '12

02 Sep '12

Gay's algorithm to print a number is available as C code here: http://netlib2.cs.utk.edu/fp/dtoa.c code is over 70 pages long!! Here are some thoughts about how to be a little more sane: 1. Start with a IEEE standard 64-bit real X. Assume X>0 wlog. 2. From the base-2 exponent field, we can deduce (up to +-1 error) the decimal exponent field via 1 table lookup (precomputed table). 3. The exact thresholds (in the form of 64-bit IEEE reals) for where the base-10 exponents increment, can be precomputed and stored in another table. Checking if > or X<=threshold, we then deduce the exact decimal exponent field in just 1 more instruction. 4. That tells us a power of 10 to multiply X by, to convert it to a real Y, 1<=Y<10. (This power again can be in precomputed table.) If our goal is to print that real with N decimal places, we want to multiply by a further 10^N too, converting Y to an (N+1)-digit integer (plus fractional part, which we eliminate via rounding). 5. Do that multiplication (call this the "critical multiplication") and rounding to get Y. 6. Now print Y as an integer in decimal -- I assume we already know how to print integers! 7. Done. The "critical multiplication" to produce Y could be done in exact arithmetic, or just extended enough beyond the usual 53-bit IEEE mantissa's precision. Intel processors have internal 80-bit reals, right? Would that not be enough? And we can also work in 128-bit integer arithmetic from then on if necessary. So I don't see why this has to be so freaking hard.

2 1

Re: [math-fun] GMail is embarrassing me
by Bill Gosper 01 Sep '12

01 Sep '12

MKleber> There's no theoretical way to prevent this, right? That is, any system in which connectivity might be lost must in some case either claim a message was sent when it wasn't, or that it wasn't when it was? --Michael On Sep 1, 2012 3:53 PM, "Bill Gosper" <billgosper(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote: > I said> [I have no idea how or why GMail sent two copies of this.]>> Due to marginal connectivity, > ½ the time I click Send, GMail says> "An error occurred and your message was not sent". This sometimes> gives me a chance to add a couple of afterthoughts before retrying.> But GMail turns out to be lying--the message *was* sent, at least to> some recipients, so it looks like I'm bothering you all with resends of> trivial afterthoughts.> --rwg> _______________________________________________ I think you're right, it's like the Synchronization Problem--a single memory can't serve two synchronous computers in bounded time, and instant service four-way red lights must sometimes malfunction (Hang, or split-pulse double green). A flipped coin might land on edge. But this often?? It happened again when I sent the foregoing, only this time, somebody other than math-fun got the extra copy. --rwg PS, Neil says There seems to be an inverse trend associated with the price of (certain) text books and their length. Stewart's Calculus (Stewart being the guy who built a mansion off the profits), at 1250 pages, is $80. The Heart of Mathematics (Burger & Starbird) , 640 pages, is $114. Linear Algebra (David C. Lay), 429 pages, was bought for $160 at the Barnes and Noble SJSU Bookstore.

1 0

[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

8 11

[math-fun] How do you print a number? DUHHHHH!!!!!!!
by Warren Smith 01 Sep '12

01 Sep '12

You have a 64-bit IEEE floating point real. You wish to print it in decimal in the format 1.2345e17. Note: Any such real always has a finite-length printout. But here is the thing: I want that printout to be accurate. If you, during the conversion to base 10, make any roundoff error at all, then you lose. If you are required to make the printout fit inside N characters, then roundoff is ok, but only the smallest possible roundoff error consistent with fitting the output into N characters. (W.Edwin Clark and I are wondering whether the C printf function actually does the right thing, or whether for like 30 freaking years they've been using a bogus print function without telling anybody.) A google search found the following paper about their former paper with same title: Guy L. Steele Jr. & Jon L White: How to Print Floating-Point Numbers Accurately http://grouper.ieee.org/groups/754/email/pdfq3pavhBfih.pdf where it said the former paper was "20 years in the making." If it took them 20 years to figure it out, and they published in >=1990, then I claim that the authors of the C language could not know about this paper, hence quite likely they screwed it up. Further, this paper contains various depressing words, indicating the considerable likelihood that a programming language near you has a print statement which does not work.

3 3

[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.

4 4

[math-fun] Make your own rhombic polyhedra
by Adam P. Goucher 01 Sep '12

01 Sep '12

Tim and Dave (and everyone else), You might want to make some of these projects with your children. With just one component, you can make polyhedra, Wieringa roofs (raised Penrose tilings) and quasicrystals amongst other things: http://cp4space.wordpress.com/2012/09/01/activities-with-golden-rhombi/ Everything you create in this way is compatible with everything else. Rhombic triacontahedra and hexecontahedra fit snugly inside the 'valleys' of a Wieringa roof. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/

1 0

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

3 2