March 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Theorem-dependance: possibly trivial question
by Tom Duff 23 Mar '15

23 Mar '15

On Mon, 23 Mar 2015, Guy Haworth wrote: > > Let us say that, if Theorem 1 (T1) can be used to prove Theorem 2 (T2), then > T1 --> T2. > > Are there simple examples of 'equivalence', T1 --> T2 and T2 --> T1 (setting > up loop T1 --> T2 --> T1) that could be explained to school children? > > > Are there cases of longer loops T1 --> ... --> Tn --> T1 which cannot, or > cannot naturally, be shortened ? Schoolchildren-compatable preferred again > ! Not quite an answer to your question, but a large part of Marvin Minsky's "Computation: Finite and Infinite Machines" is devoted to proving that a bunch of models of computation (M[1], M[2], M[3], ... M[n]) have equivalent power by using M[i] to simulate M[i+1 mod n]. Until you get to the last link, where M[n] simulates M[1], all you know is that none of the models gives you more power than the previous ones. In the end you know that they're all equally powerful. I remember reading this when I was 17 and thinking that it was the best story I'd ever seen. -- Tom Duff. Lo! the raptured arithmetician!

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Re: [math-fun] Computational effort
by Henry Baker 23 Mar '15

23 Mar '15

Separating U235 from U238 is a pretty decent "proof of work" system. "Uraniumcoin" ? "Isotopecoin" ? "Hexofluorocoin" ? "Centricoin" ? These "coins" put the "rich" into "enriched"... https://en.wikipedia.org/wiki/Enriched_uranium Of course, you may only want to use Oralloy certificates in commerce, exchangeable* for the real thing. Who wants to carry around a lead wallet, with even heavier contents? * (Except by certain not-so-friendly countries.) At 07:16 AM 3/20/2015, James Propp wrote: >(b) the Manhattan Project,

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[math-fun] genus-1 polytopes in 4-space
by Tom Karzes 23 Mar '15

23 Mar '15

I was thinking about genus-1 polytopes in 4-space, and I was wondering if some of them would be considered "regular". For example, the cartesian product of two identical regular polygons is a two-dimensional surface (like the surface of a torus), which can be distorted to fit in 3-space but which is much more symmetrical in 4-space. As a specific example, here's a torus surface in 4-space whose cross sections are squares: max(|w|, |x|) = 1 max(|y|, |z|) = 1 Unless I'm mistaken, all of its faces are squares (16 of them, the cartesian products of the edges of the squares), and it is face-transitive, edge-transitive, and vertex-transitive. Does that make it regular? Tom

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[math-fun] genus-1 polytopes in 4-space
by Warren D Smith 22 Mar '15

22 Mar '15

> From: Tom Karzes <karzes(a)sonic.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] genus-1 polytopes in 4-space > Message-ID: <21773.64616.883302.948124(a)gargle.gargle.HOWL> > Content-Type: text/plain; charset=us-ascii > > I was thinking about genus-1 polytopes in 4-space, and I was > wondering if some of them would be considered "regular". For > example, the cartesian product of two identical regular polygons > is a two-dimensional surface (like the surface of a torus), > which can be distorted to fit in 3-space but which is much more > symmetrical in 4-space. > > As a specific example, here's a torus surface in 4-space whose > cross sections are squares: > > max(|w|, |x|) = 1 > max(|y|, |z|) = 1 > > Unless I'm mistaken, all of its faces are squares (16 of them, > the cartesian products of the edges of the squares), and it is > face-transitive, edge-transitive, and vertex-transitive. Does > that make it regular? > > Tom --By that defn, any cartesian product of any set of regular polytopes in any dimensions, would be "regular," right? I think you need to cut them down a bit, otherwise too much boring stuff is generated by your defn.

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[math-fun] Fish on a hyperboloid
by James Propp 21 Mar '15

21 Mar '15

Is it really possible that no one's rendered Escher's hyperbolic fish on the surface of a hyperboloid, where they would have more room to swim and feel altogether happier? I went to images.google.com but didn't find an image of such an object. I realize that a picture of such a surface would not do its symmetries justice. Maybe someone has created an animation that permits 3D rotation? That'd be almost as good as a tangible object. And if it permitted zooming in and zooming out, it'd be even better than the "real thing". Jim Propp P.S. Insert your own "snakes on a plane" joke here.

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[math-fun] T2
by Bill Gosper 21 Mar '15

21 Mar '15

Is it really possible that no one's rendered Escher's hyperbolic fish on the surface of a hyperboloid, where they would have more room to swim and feel altogether happier? I went to images.google.com but didn't find an image of such an object. I realize that a picture of such a surface would not do its symmetries justice. Maybe someone has created an animation that permits 3D rotation? That'd be almost as good as a tangible object. And if it permitted zooming in and zooming out, it'd be even better than the "real thing". Jim Propp P.S. Insert your own "snakes on a plane" joke here. For even greater realism, add some fish oil to the 3D printer resin. --rwg

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[math-fun] T2
by Bill Gosper 21 Mar '15

21 Mar '15

Presumably, an improvement on, e.g., http://formlabs.com/products/form-1-plus/ which is fairly old hat. It's not April 1st yet. Is it? http://www.theguardian.com/science/2015/mar/19/scientists-create-terminator… WFL For April 1st, how about In the rising tide of Rising Sun xenophobia, Japanese police are switching from pepper spray to wasabi spray, which incapacitates foreigners without bothering native Japanese. --rwg

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[math-fun] moonshine
by Fred Lunnon 21 Mar '15

21 Mar '15

Now that John McKay and Douglas Rogers have both commended https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-sha… I feel obliged to take a look. And the first thing I come across is << The 23 new moonshines ... >> There are 24 moonshines? Aw, come on! WFL

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[math-fun] T2
by Fred Lunnon 21 Mar '15

21 Mar '15

It's not April 1st yet. Is it? http://www.theguardian.com/science/2015/mar/19/scientists-create-terminator… WFL

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Re: [math-fun] Computational effort
by Bill Gosper 20 Mar '15

20 Mar '15

HGB> Some non floating point units of computational effort: Bitcoin POW (SHA-256 hashes/second) Other POW's https://en.wikipedia.org/wiki/Proof-of-work_system Password cracking Most of these problems are "embarrassingly parallel". Perhaps someone here has some ideas for a non-embarrassingly-parallel unit of computational effort. <hgb How about the Minsky Stock Index? Subj: What is the period of this integer sequence? On Sun, May 4, 2014 at 10:37 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > 0, 1, 7, -2, -8, 3, 14, -4, -16, 5, 21, -6, -24, 7, 28, -7, -25, 7, > 27, -7, -26, 7, 26, -6, -19, 5, 18, -4, -12, 3, 10, -2, -5, 1, 2, 0, > 2, -1, -6, 3, 16, -5, -22, 7, 30, -8, -30, 8, 30, -7, -23, 6, 22, -5, > -16, 4, 14, -3, -9, 2, 6, -1, -2, 1, 5, -1, -3, 1, 4, -1, ... > > a[0] = 0; a[1] = 1; > a[k_?OddQ] := a[k] = a[k - 2] - Floor[9*a[k - 1]/17]; > a[k_?EvenQ] := a[k] = a[k - 2] + Floor[15*a[k - 1]/2] > > This is the "Minsky Stock Index" that Corey and Julian ran fruitlessly > from a[-10^14] to a[10^14] a few years ago. "If you haven't looked at a > problem in the last few years, you haven't looked at it." --Ed Pegg, Jr. > There is a slight chance that the period is infinite. > (When it reached 18 trillion, we exclaimed AT&T!) > > The above recursive definition crashes Mathematica in under a million, > even with memoizing. Instead use the iteration > In[115]:= NestList[Function[xy, {#, xy[[2]] + Floor[15*#/2]} & > [xy[[1]] - Floor[9*xy[[2]]/17]]], {1, 0}, 35] > e.g., for {a[-1],a[0}, {a[1],a[2]},...{a[69],a[70]} > Out[115]= {{1, 0}, {1, 7}, ..., {-1, -4}} > > Background: > http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition > Unlike Collatz or twin prime searching, this is a very specific question > about > very specific quantities, rather than a question about the infinitude of > integers. > --rwg > So far, nobody(?) has figured out how to distribute this calculation. I conjecture finitude for all these monsters that don't blow right up with an obvious, provable pattern. The bigger they get, the proportionally smaller is the effect of the Floor steps, and thus the closer they get to pure ellipses. I.e. their growth rate slows down, and the density of points they visit will increase with increasing radius. Eventually they will run out of new points, and thus must return to the beginning, by reversibility. (Julian, Neil: Did we ever correlate monstrousness with poor rational approximabillity of the Floorless period?) At 07:16 AM 3/20/2015, James Propp wrote: I don't know what the relevant units of computation are --- floating-point operations are the ones I've heard about most, but what if you're not using floating point? --- but I'm wondering if anyone has tried to quantify "how much computing" took place in (a) code-breaking at Bletchley Park, (b) the Manhattan Project, and (c) the Apollo space program. It seems likely to me that my laptop, running some silly number-crunching problem in Mathematica and returning an answer after six hours that I could probably have figured out in my head if I'd thought a little bit more before jumping in to write some code, has them all beat. What do you think? (And while we're on the subject: What's the canonical example of wasted computational effort? Isn't there some well-known old story of someone who had an early computer calculate the determinant of a singular matrix?) Jim Propp Note that early Mersenne prime hacking was done in floating point. I.e., the big integer routines used floating point because that hardware was so much more optimized than the fixed. What could be more computationally wasteful than those ubiquitous, realistic shooter games? A few minutes on one of those GPUs should dwarf the Manhattan Project. (See Feynman's Surely You're Joking.) But the *cost* of this waste is vanishing by Moore's Law, which does not apply to all the young jellyware that these games tragically waste. --rwg

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