February 2019 - mathfun@mailman.xmission.com

[math-fun] What is the ratio, Kenneth?
by Dan Asimov 28 Feb '19

28 Feb '19

Let C(n) denote the 2^n vertices of the n-cube [-1, 1]^n in R^n: C(n) = {-1, 1}^n Let S(n) denote the n+1 vertices of the n-simplex in R^(n+1): S(n) = {e_j | 1 <= j <= n} where { 0, i ≠ j <e_j, e_k> = delta(i, j) = { { 1, i = j (so all interpoint distances = sqrt(2)). Question: --------- For n >= 2, what is the smallest number f(n) of isometric copies of S(n) needed to cover C(n) ??? E.g., f(2) = 2; f(3) = 2, f(4) = ??? Question: --------- What is a nice function asymptotic to f(n) as n —> oo ??? —Dan

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[math-fun] Integration
by Bernie Cosell 28 Feb '19

28 Feb '19

Today's XKCD: <https://xkcd.com/2117/> /b\ Bernie Cosell bernie(a)fantasyfarm.com -- Too many people; too few sheep --

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[math-fun] Hey Kids: Did Norman Johnson (and PolyhedronData) forget AugmentedHeptagonalPrism?
by Bill Gosper 27 Feb '19

27 Feb '19

A nice little three-liner: http://gosper.org/johnson7.pdf . Is it convex? This pdf is a spoiler for a programming lab (https://lab.open.wolframcloud.com/app/) notebook because I have yet to figure out how to unrestrict public access. But wouldn't it be a great way for math-funsters to share typesetting, images, animations, and interactions? > If not convex, exactly what is that nearly straight angle? > Note that evil (and misnamed) FindGeometricTransform coerces to > floating point! > > [Mathematica developers: Needs Exact->True option. MachinePrecision > isn't quite sufficient > for RootApproximant to reconstruct the algebraics here. A consolation would be WorkingPrecision.] > ] > I think at some point I exhaustively searched PolyhedronData for Johnson glue jobs and found even narrower convexity failures. Johnson was working with cardboard? —rwg I need to tweak gosper.org/johnsonposter.pdf to disambiguate some hidden undersides.

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Re: [math-fun] surdburger
by Bill Gosper 26 Feb '19

26 Feb '19

> > From: Fred Lunnon <fred.lunnon(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > > Er --- and your point is? WFL > > 1. We are no longer stuck with Maple's numerics. 2. Seven terms, 600 digits of (1/4)!. —rwg > > On 2/25/19, Bill Gosper <billgosper(a)gmail.com> wrote: > > Fred (and everyone), try entering these three statements (successively) > > into Wolfram's > > Programming Lab: > > > > (1/4)! - 2^(-9/8) \[Sqrt]5 (1 + \[Sqrt]\[Sqrt]5) GoldenRatio^(3/2) > > \[Pi]^(3/4) Sum[(-1)^n/E^(30 \[Pi] (n + 1/6)^2), {n, -3, 3}] > > > > N[%, 33] > > > > $MaxExtraPrecision = 9999; N[%%, 33] > > > > Whether or not Maple is actually crumbling, I'd say as of now Wolfram > > pretty much owns numerics. > > (But I wonder when he'll give 0 for the infinite sum!-) > > > > -------- Original Message -------- > > Subject: Re: [math-fun] surdburger > > Date: 2019-02-23 18:50 > > From: Fred Lunnon <fred.lunnon(a)gmail.com> > > To: math-fun <math-fun(a)mailman.xmission.com> > > Reply-To: math-fun <math-fun(a)mailman.xmission.com> > > > > ############################################## > > > > # Gosper surdburgers: http://gosper.org/surdburger.pdf > > Digits := 99; > > evalf( (1 + 3^(2/3))^2 - ((1 + 3^(1/3))*(1 + 2*3^(1/3))) ); > > if (1 + 3^(2/3))^2 = ((1 + 3^(1/3))*(1 + 2*3^(1/3))) > > then true else false fi; > > Digits := 10; > > > > ############################################## > > > > Digits := 99 > > -0.1e-97 > > false > > Digits := 10 > > > > ############################################## > > > > I could understand Maple surrendering and returning > > a failure message --- indeed, it might be argued that the > > arguments of the (complex) roots are indeterminate --- > > > > That is a terrible argument. √4 = ±2 is a thought crime. > > —rwg > > > > however _immediately_ returning a wrong answer appears > > to qualify for a monumental raspberry. > > > > And while I'm having my customary Maple moan, why the devil > > must logical expressions be buried inside if-expressions to persuade > > the beast to evaluate them at all, correctly or otherwise? > > Which expressions incidentally may appear nowhere else but > > at top level, or as the result of a function --- oh why, why, why?! > > > > These things are sent to try us ... Amen. WFL > > _______________________________________________ > > math-fun mailing list >

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[math-fun] Fill in the blank.
by Bill Gosper 26 Feb '19

26 Feb '19

pentiprisms <http://gosper.org/pentiprisms%3F.png> —rwg (Note Julian's tympanohedron in the upper left.)

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Re: [math-fun] surdburger
by Bill Gosper 26 Feb '19

26 Feb '19

Fred (and everyone), try entering these three statements (successively) into Wolfram's Programming Lab: (1/4)! - 2^(-9/8) \[Sqrt]5 (1 + \[Sqrt]\[Sqrt]5) GoldenRatio^(3/2) \[Pi]^(3/4) Sum[(-1)^n/E^(30 \[Pi] (n + 1/6)^2), {n, -3, 3}] N[%, 33] $MaxExtraPrecision = 9999; N[%%, 33] Whether or not Maple is actually crumbling, I'd say as of now Wolfram pretty much owns numerics. (But I wonder when he'll give 0 for the infinite sum!-) -------- Original Message -------- Subject: Re: [math-fun] surdburger Date: 2019-02-23 18:50 From: Fred Lunnon <fred.lunnon(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> ############################################## # Gosper surdburgers: http://gosper.org/surdburger.pdf Digits := 99; evalf( (1 + 3^(2/3))^2 - ((1 + 3^(1/3))*(1 + 2*3^(1/3))) ); if (1 + 3^(2/3))^2 = ((1 + 3^(1/3))*(1 + 2*3^(1/3))) then true else false fi; Digits := 10; ############################################## Digits := 99 -0.1e-97 false Digits := 10 ############################################## I could understand Maple surrendering and returning a failure message --- indeed, it might be argued that the arguments of the (complex) roots are indeterminate --- That is a terrible argument. √4 = ±2 is a thought crime. —rwg however _immediately_ returning a wrong answer appears to qualify for a monumental raspberry. And while I'm having my customary Maple moan, why the devil must logical expressions be buried inside if-expressions to persuade the beast to evaluate them at all, correctly or otherwise? Which expressions incidentally may appear nowhere else but at top level, or as the result of a function --- oh why, why, why?! These things are sent to try us ... Amen. WFL

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[math-fun] For a mild surprise, try
by Bill Gosper 25 Feb '19

25 Feb '19

https://lab.open.wolframcloud.com/objects/wpl/GetStarted.nb Overpaste the 2+2 with (-1 + 2^(1/3))^(1/6) Sqrt[1 + 2^(1/3)] Shift Enter (shift+return) On the next line type FullSimplify@% (shift+return) Optionally, type TraditionalForm[%%==%] (shift+return) —rwg

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[math-fun] Recommend Geoffrey Campbell as member
by David Wilson 24 Feb '19

24 Feb '19

I would like to recommend Geoffrey Campbell to the math-fun mailing list. His Facebook profile is here: https://www.facebook.com/geoffrey.campbell.180 He says he has attempted to subscribe via https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun but "It kept declining my email address and proposed password - the entry requirement." At any rate, Geoffrey regularly posts some interesting math puzzlers on LinkedIn Number Theory group, which is how I encountered him.

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Re: [math-fun] Fwd: Near misses
by Dan Asimov 24 Feb '19

24 Feb '19

Dang, I was about to bring up Christine Jorgensen, but now I won't even mention him or her. —Dan Bill Gosper escribe: ----- >Dang. I thought this would be about transgenders. ... -----

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[math-fun] minor denesting puzzle
by Bill Gosper 24 Feb '19

24 Feb '19

n>a>b are positive integers, a and b coprime, a^(1/n) irrational, Can √(a^(1/n)+b^(1/n)) denest? (:= linear finite combination of real algebraic monomials.) —Bill (Apologies if this is trivial because I forgot to add a condition.)

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