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[math-fun] Intersection of random cosets of Z[exp(2πi/3)]: When is this set empty?
by Dan Asimov 13 May '19

13 May '19

Let E denote the triangular lattice Z[w] of Eisenstein integers, where w = -1/2 + i sqrt(3/4) is a primitive 1^(1/3) in C. For j = 1, 2, ...,n let A_j = K_j + L_j w ≠ 0 and B_j = M_j + N_j w be arbitrary elements of E with all A_j nonzero. Multiplication by A_j carries E into a subset of itself by uniform scaling, and translation by B_j carries E onto itself isometrically. Thus for each j, 1 <= j <= n, the set X_j = A_j E + B_j = {A_j u + B_j | u in E} is a subset of E isometric to a uniformly scaled copy of itself. Question: --------- Find an algorithm in terms of the A_j and B_j that determines whether or not the sets X_1, ..., X_n have nonempty intersection. —Dan

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Re: [math-fun] The Doormat Singers at MIT
by Bill Gosper 12 May '19

12 May '19

I believe that was written by PDP-1 Teco creator and WTBS DJ Dan Murphy, for performance in the annual All Tech Sing, or whatever they called it. (Wherein the "co-eds" substituted "Kenneth R. Wadleigh" for "Officer Krupke".) Big Julie was President Julius Adams Stratton, after a TV skit about Julius Caesar. (Stratton was the guy for whom Bill Ackermann rented a tux, just so he could say, "It certainly is a pleasure!".) —rwg I should have added in my Stirling post that the errors are monotone decreasing (= "antitone"). On Sun, May 12, 2019 at 2:02 PM rwg <rwg(a)ma.sdf.org> wrote: > > > -------- Original Message -------- > Subject: Re: [math-fun] The Doormat Singers at MIT > Date: 2019-05-12 13:40 > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Reply-To: Dan Asimov <dasimov(a)earthlink.net>, math-fun > <math-fun(a)mailman.xmission.com> > > Hans! Thanks for a wonderful blast from the past — about 54 years ago. > > The mp3 was better than any lyrics! > > —Dan > > > > DA: "Can anyone recall the lyrics, or know where they can be found, to > > a song from '64-'65 that had words like: As I walked out, one May > > morning, along the shady route, ..." > > > > Look for "The Institute Screw" here: > > > > http://www.dpbsmith.com/doormat/ > > > > No lyrics but they have an mp3. > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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Re: [math-fun] The Doormat Singers at MIT
by Dan Asimov 12 May '19

12 May '19

Hans! Thanks for a wonderful blast from the past — about 54 years ago. The mp3 was better than any lyrics! —Dan >DA: "Can anyone recall the lyrics, or know where they can be found, to a song from '64-'65 that had words like: As I walked out, one May morning, along the shady route, ..." > >Look for "The Institute Screw" here: > >http://www.dpbsmith.com/doormat/ > >No lyrics but they have an mp3.

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Re: [math-fun] Stirlingoid figure of merit
by Dan Asimov 12 May '19

12 May '19

Can anyone recall the lyrics, or know where they can be found, to a song from '64-'65 that had words like: ----- As I walked out, one May morning, along the shady route, There I met Big Julie, the keeper of the 'Tute. I said to him, "Big Julie, how are your students all?" He said to me every one of them is screwed right to the wall! ----- I had no luck trying to google this. —Dan

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Re: [math-fun] Draft of my May 17, 2019 essay
by Dan Asimov 11 May '19

11 May '19

I regret that I cannot concur with my esteemed colleague on this matter: I don't think it's an great idea to begin a title with the word "But". Intoxication by alcohol alone is not necessarily sufficient to achieve Buddha-wisdom! —Dan Adam Goucher wrote: ----- The latter, definitely: it flows better to have the quote as the title, and the description as the subtitle. (As I happened to drink wine this evening, to a non-trivial extent, I claim to be in a position to judge* matters pertaining to both mathematicians and intoxication.) * no pun intended. -----

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[math-fun] When in doubt, use 0, 1, or take the square root.
by Ed Pegg Jr 11 May '19

11 May '19

Square or take the square root. In my Substitution Tilings program at http://demonstrations.wolfram.com/SubstitutionTilings/ , I have a routine called SqrtSpace. Pictures at https://community.wolfram.com/groups/-/m/t/1679969 In the Psi Quad tiling, https://community.wolfram.com//c/portal/getImageAttachment?filename=PsiQuad… , the algebraic number field is Psi, ψ, the https://en.wikipedia.org/wiki/Supergolden_ratio . The coordinates in the tiling are NOT in the algebraic number field of ψ, since sextics are involved. However, with a sign-preserving squaring or square root operation, all coordinates work perfectly within the algebraic number field. I expand on this more at https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-su… Squaring/sqrt makes coordinates elegant for extremal orchard solutions. https://blog.wolfram.com/2018/02/02/cultivating-new-solutions-for-the-orcha… Squaring makes all vertices elegant for degenerate power simplex coordinates. http://demonstrations.wolfram.com/DegeneratePowerSimplices/ "If it doesn't work, try squaring your values." -- I've heard this since college, but I didn't expect it to work so well in algebraic number fields. Is there a name for this? Ed Pegg Jr

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[math-fun] Draft of my May 17, 2019 essay
by James Propp 11 May '19

11 May '19

I've written another essay (a little shorter than usual), which I'll either call A Mathematician in the Jury Box, or, “But how should we define ‘intoxicated’?” or “But how should we define ‘intoxicated’?”, or, A Mathematician in the Jury Box (I can't decide which is better). See http://mathenchant.org/048-draft2.pdf Comments are welcome! Jim

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[math-fun] Impossible triangle raytracer
by Neil Bickford 10 May '19

10 May '19

Here's a neat visual trick: This shader program (by yx) renders a fully illuminated Reutersvard*/Penrose impossible triangle: https://www.shadertoy.com/view/ttXGWr All of the GLSL code used to generate the image on the left (minus Shadertoy's framework) is listed and editable on the right, and you can edit the code on the right to see what it does. Note how the geometry is locally consistent, and how the shadows between cubes, other cubes, and the ground all match up without any hard seams. Here's some context behind what makes this especially interesting. This program simulates global illumination using Monte Carlo path tracing: For each on- screen pixel, we trace a ray from a virtual camera into the scene, find the first place where the ray intersects the scene geometry, choose a new direction for the ray, trace the ray until it intersects the scene geometry again, and repeat until the ray reaches a light source (in this case, the sky) or reaches some maximum number of bounces (in this case, 4). (There's a lot I'm leaving out here, but this is sort of the basis behind it.) For geometry in R^3, this makes sense; finding ray-object intersections can sometimes be difficult, but are usually feasible. But here's the impossibility: how can you find the intersection of a ray and an object that can't exist? Left-click and drag over the image on the left (ideally while the shader is playing) to see a hint of how the shader works. *With a diaresis over the a: Reutersvärd --Neil Bickford (encoded partial solution (swaps a<->z): FromCharacterCode[219 - ToCharacterCode["zeviztv{tvlnvgib{levi{grnv"]].)

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[math-fun] Lots of Substitution Tilings
by Ed Pegg Jr 10 May '19

10 May '19

I've been parameterizing substitution tilings by their algebraic spaces in my program at http://demonstrations.wolfram.com/SubstitutionTilings/ I've put a bunch of pictures at https://community.wolfram.com/groups/-/m/t/1679969 Additions, Comments, Corrections are all welcome. Ed Pegg Jr

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Re: [math-fun] 170
by Dan Asimov 10 May '19

10 May '19

There are also a few fun facts here: https://en.wikipedia.org/wiki/170_(number)#In_mathematics , like that 170 is the smallest number whose phi(170) (= 64) and whose sum of divisors, sigma(170) (= 324), are perfect squares. —Dan

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