February 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Tricylinder puzzle
by Bill Gosper 12 Feb '15

12 Feb '15

Dan Asimov>Perhaps I should define my terms. The solid angle at a point on a convex surface S is the limit of the solid angle subtended by the surface in an epsilon-neighborhood N(eps) of the point, as epsilon -> 0. And that is the area on the unit sphere of the set of all perpendicular unit vectors to the supporting hyperplanes* to S on N(eps) (when the vectors are translated to the origin). --Dan ________________________________________________________ * < http://en.wikipedia.org/wiki/Supporting_hyperplane > On Feb 2, 2015, at 10:27 AM, Daniel Asimov <asimov(a)msri.org> wrote: Consider the tricylinder T -- the intersection of three cylinders each of unit radius in 3-space with perpendicular axes. Let S denote its surface bd(T). PUZZLE: Find the solid angle in closed form at a point of S common to the 3 cylinders. --Dan The tricylinder, a.k.a. triroller, a.k.a. Steinmetz solid, is the union of four copies of the convex hull of the three ellpses {{Cos[t], Sin[t], Sin[t]}, {Sin[u], Sin[u], Cos[u]}, {Sin[v], Cos[v], Sin[v]}} which, when t=u=v=-π/4±π/2 coincide at the two points ±(1,1,1)/√2 , where their tangent vectors are proportional to (1,-1,-1), (-1,-1,1), (-1,1,-1) and (-1,1,1),(1,1,-1),(1,-1,1). The angle between any pair of these (at the same vertex) is acos(-1/3), the tetrahedral bond angle, gving a solid angle of π. So four Steinmetzes should fit snugly around a single point. --rwg

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Re: [math-fun] Short programs for e, etc. (was Re: Factorial base)
by Steve Witham 12 Feb '15

12 Feb '15

rcs(a)xmission.com, he say, > The program, in assembly language, easily fit on one page. A gem. What's fun has something to do with the right level of challenge. For me this has been one of the most fun items on the list in a while. Anyway, in Python... > import math > print repr(math.e), "= math.e" > > from sys import stdout > > stdout.write("2.") > # 32! is about 10**35. d[0] and d[1] are unused. > d = [1] * 33 > for i in range(35): > c = 0 > for j in range(len(d) - 1, 1, -1): # len(d)-1 down to 2. > c, d[j] = divmod(d[j] * 10 + c, j) > stdout.write(str(c)) # (Prints a digit without adding a space.) > print It's cache-friendly as-is. It's a pipeline so it's nicely parallelizable. I think I could do sin(1) and sin(3) but not sin(3 + sin(3)) this way. You can speed it up by a factor of four by converting to base 10**4. ...for that matter, why not generate all the digits in one pass? --Steve > Stanley Rabinowitz, Stan Wagon, > A Spigot Algorithm for the Digits of $\pi$, > The American Mathematical Monthly, vol.102, no.3, pp.195-203, > (March-1995). > > http://www.mathpropress.com/stan/bibliography/ > Unbounded Spigot Algorithms for the Digits of Pi > > http://www.cs.ox.ac.uk/jeremy.gibbons/publications/spigot.pdf --Steve

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[math-fun] Short programs for e, etc. (was Re: Factorial base)
by Keith F. Lynch 12 Feb '15

12 Feb '15

rcs(a)xmission.com wrote: > The only theorem I know of is that a rational number always has a > finite representation in factorial base; therefore e is irrational. How does that follow? The existence of an infinite representation for e in factorial base doesn't prove there isn't also a finite representation. > Back in the day, I was looking through the file of paper tapes in > the PDP1 room (this served as the software library), and I came > across one labeled simply "e". The handwriting might have been > either Eric Jensen or Bill Ackerman. It was only a couple of folds, > a few hundred characters. A few years ago I wrote the following obfuscated c program, which prints a thousand digits of e: void main(){int j,k,l[1002];for(j=1776;j;j--){for(k=0;k<1001;k++){l[k+1]+= 10*(l[k]%j);l[k]/=j;}l[0]++;}for(j=0;j<1001;j++){printf("%d",l[j]);if(!j) printf("%c",46);}printf("\n");} I'm especially proud of operating on an uninitialized array, just to drive readers insane. "It *can't* work, but it does." Can anyone write an equally short c program to print the first thousand decimal digits of any other interesting irrational number, such as pi, phi, zeta(3), cos(x)=x, ln(2), gamma, or sqrt(2)? (An irrational number contrived to have its first thousand digits have some simple pattern (e.g. all 0) is not interesting.) (Okay, nobody knows if gamma is irrational, but it's close enough for me.)

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[math-fun] Does zero divide zero ...
by Guy Haworth 12 Feb '15

12 Feb '15

This is a deeper question than it appears ... and there are of course some well-informed books on the history of zero. A colleague of mine created an algebra J = {(a, b) | a, b in the Reals R} ... with (a, b) being analogous to a/b in R. Thus, addition, inverses, subtraction and multiplication can be defined naturally ... and division by (a, b) is defined as multiplication by (b, a) without any exceptions. There is also an equivalence relationship: (ca, cb) is equivalent to (a, b). Thus, he has elements in the algebra like (a, 0), (-a, 0) and (0, 0) with a > 0. He gives these the names 'infinity', '-infinity' and 'nullity'. Anything equivalent to nullity is also 'nullity'. Nullity is what I (and maybe others) would call an 'annihilator' ... if it is combined with anything, the result is 'nullity'. Let us add 0/0 as a notional 'number' to the set of numbers a/b in the Reals R. Let this algebra be R'. Thus, there is a 1-1 mapping from R into J. Any expression not featuring 0/0 in R' maps in an obvious way to J ... and expressions featuring 0/0 in R' map to nullity in J. So J is equivalent to a number system with 0/0 being treated as a number rather than an exception. However, it does not follow that '2', i.e. 2/1, equals '1', i.e. 1/1, in this algebra. "All expressions have a value" might be a neat mantra ... but we haven't really created - IMHO - a number system which is different from what we have now. The history of new numbers - minus 1, zero, famously sqrt(2) and presumably the unfortunately named complex numbers ... is that they are not universally received with peace of mind. They had a troubled birth. I have not been able to persuade my colleague 'J' that he has not created anything new. One cannot simply ignore the fact that a number is zero if it is effectively being divided into something. Many mathematical methods have 'if zero' get-out clauses. The standard formula for the roots of a quadratic does not work if the coefficient of x^2 is zero. Maybe there is a neat proof that this concept of 'nullity' does not bring anything to the party: I sure wish there was ... and then I could save my colleague's time writing conference papers and being convinced that he is onto something. I believe his interest in 0/0 stemmed from an interest in Project Geometry, and then in Interval Arithmetic ... where he was trying to tackle the problem of Instability in computations. Guy

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Re: [math-fun] Factorial base
by Bill Ackerman 12 Feb '15

12 Feb '15

> ---------- Forwarded message ---------- > From: <rcs(a)xmission.com> > Date: Wed, Feb 11, 2015 at 2:57 PM > Subject: Re: [math-fun] Factorial base > To: math-fun(a)mailman.xmission.com > Cc: rcs(a)xmission.com > > > The only theorem I know of is that a rational number always has a > finite representation in factorial base; therefore e is irrational. > > Back in the day, I was looking through the file of paper tapes in > the PDP1 room (this served as the software library), and I came > across one labeled simply "e". The handwriting might have been either > Eric Jensen or Bill Ackerman. It was only a couple of folds, a few > hundred characters. Curious, I printed it out, and then ran the > assembler on it. Indeed, it promptly printed out a few thousand > digits of e. I examined the code more closely, and discovered it > created a factorial base representation of (the fractional part of) e, > one digit per machine word, and then did a simple decimal conversion, > multiplying the factorial base representation by decimal 10, and > printing the integer "carry" that falls off the front end; rinse, repeat. > The program, in assembly language, easily fit on one page. A gem. > > Rich > The program was most likely written by the late Richard P. Howell, one of the noteworthy PDP-1 folks in the late 60's. I seem to recall he had some extremely clever program for calculating e. I don't remember ever examining it. Bill

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[math-fun] inner core of earth
by Warren D Smith 11 Feb '15

11 Feb '15

http://news.illinois.edu/news/15/0209innercore_XiaodongSong.html http://www.nature.com/ngeo/journal/vaop/ncurrent/full/ngeo2354.html Allegedly the earth has an inner core inside its core. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] wolframtaps
by Bill Gosper 09 Feb '15

09 Feb '15

On Mon, Feb 9, 2015 at 4:57 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > On Fri, Oct 17, 2014 at 11:07 AM, Bill Gosper <billgosper(a)gmail.com> > wrote: > >> NeilB tweeted three superfluous characters: >> >> (*primes*)d[n_,t_,k_]:=If[k<1,Boole[n<1],If[t>n,0,d[n-t,0,k-1]+d[n,1/2+t+Sqrt[2t+1/4],k]]];2Select[Range[99],2#+2==d[#,0,4]&]+1 >> Can you find them? —rwg >> > SPOILER: Zack Chroman found Range@99. I replaced Sqrt[] with √(). > Zack: "That is so cheating." > So I finally broke down with my all-time maiden tweet > https://pbs.twimg.com/media/B8T3djoIYAEO6Y-.png > which crashed in flames when wolframtap botched the π character! > Retrying, https://twitter.com/wolframtap/status/559917026748407808/photo/1 > This hasn't yet shown up at http://wolframtap.tumblr.com/archive/2015/1 > but neither has anything subsequent (27 Jan). NeilB's primes generator > never showed up, presumably for lack of eye candy. > The candy collector must now judge my latest: > pic.twitter.com/MVKeGDqiBG <http://t.co/MVKeGDqiBG> . Who is this idiot > tweeting a trivial heptagram? > E.g. ListLinePlot[{Im[#], Re[#]} & /@ ((-1)^Range[0, 6, 6/7])] . But this Lerchy heptagram is actually a continuous graph, not just seven vertices. Proof: pic.twitter.com/AqowSLllaP --rwg > Wolfram is right--nobody wants to look at code when there are pictures. > Let's > display the code too small to read. > --rwg > For an incredibly cheap thrill, Colorize[JuliaSetPlot[I]] . >

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[math-fun] In https://twitter.com/wolframtap/status/522518755168243712 , profligate pleonast
by Bill Gosper 09 Feb '15

09 Feb '15

NeilB tweeted three superfluous characters: (*primes*)d[n_,t_,k_]:=If[k<1,Boole[n<1],If[t>n,0,d[n-t,0,k-1]+d[n,1/2+t+Sqrt[2t+1/4],k]]];2Select[Range[99],2#+2==d[#,0,4]&]+1 Can you find them? —rwg

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[math-fun] mathematics in art
by James Buddenhagen 07 Feb '15

07 Feb '15

My wife called this to my attention, and it occurred to me some in this group might be interested in both mathematics and art, so I thought I'd pass it along. I hope it is not too far from "math-fun" to mention here. https://t.co/pTMmJmmxEq

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[math-fun] Unique factorization (connection to music?) question
by Warren D Smith 05 Feb '15

05 Feb '15

>Victor Miller victorsmiller at gmail.com >According to SAGE the field Q(2^{1/12}) has class number 1, and so does >have unique factorization. In addition Z[2^{1/12}] is the maximal order in >this field. --so: for which N does Q(2^{1/N}) have class number 1? http://oeis.org/A005848 = {1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84} claims to answer the same question for Q(Nth root of unity) duplicating the Main Theorem of J. Myron Masley and Hugh L. Montgomery: Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976) 248-256 which shows that all such N obey N<=84 and gives the full list. Amazingly enough, all the historically-used music-scale sizes {4,5,6,7,8,12} happen to be on this list (asterisk: M&M explain that if k is odd and on their list, then 2k automatically is also, so therefore they omit mentioning all 2k with k odd). Which actually is not so amazing since ALL numbers 1-12 are on the M&M list.

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