December 2018 - mathfun@mailman.xmission.com

[math-fun] Can anyone source this mean joke?
by James Propp 14 Dec '18

14 Dec '18

“I had a weak doctoral student once. I didn’t mind telling him what problem to work on. I didn’t mind solving it for him. I didn’t even mind writing up the proof. But I really resented having to EXPLAIN it to him.” (heard at UC Berkeley in the 1980s?) Jim Propp

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Re: [math-fun] spacefilling surface
by Dan Asimov 13 Dec '18

13 Dec '18

Still, I wonder if there exist space-filling surfaces that are limits of surfaces S_n in R^3 as n —> oo, such that ... if for some eps > 0, the epsilon neighborhood of S_n (meaning {x in R^3 | ||x-p|| < eps for some p in S_n}) is denoted by N_eps(S_n), then *hypothetically*: N_eps(S_n) contains N_(1/eps)(0) which we require must hold for all n, for some sequence of eps = eps_n > 0 that decreases to 0 as n —> oo. And of course we require that for some sequence h_n of embeddings h_n : R^2 —> R^3 of the plane into R^3 with h_n(R^2) = S_n, we have that the limit lim_{n—>oo} h_n exists and is continuous. It's space-filling by the hypothetical construction. —Dan Andy Latto wrote ----- Aren't there even simpler solutions? If f:R -> R^3 is space-filling, define g:R^2 -> R^3 by g(x,y) = f(x). >On Thu, Dec 13, 2018 at 4:34 PM Bill Gosper <billgosper(a)gmail.com> wrote: >> >> Rohan requested a continuous map from a 2D surface onto a non-porous patch >> of >> positive 3D volume.

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[math-fun] spacefilling surface
by Bill Gosper 13 Dec '18

13 Dec '18

Rohan requested a continuous map from a 2D surface onto a non-porous patch of positive 3D volume. There's one under our noses, but maybe not where you think. We certainly have continuous maps from 1D onto 3D, so it is tempting to compose such a volume-filling function with some sort of (necessarily multivalued) 1D inverse of a planefilling function. But such inverses are wildly discontinuous, no matter which values you choose as the definition. Although 1/2+i/2 is "merely" triple point, the Hilbert "curve" is dense with quadruple points. The preimage of a sufficiently small neighborhood of such a point will comprise at least four disjoint pieces. Could the 1D to 3D map possibly smooth everything out? Maybe, but there's a much cleaner solution. SPOILER: gosper.org/3dsno.png gives a continuous mapping of an equilateral triangle onto a standard 3 simplex. —rwg

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[math-fun] a Christmas carpet
by Fred Lunnon 13 Dec '18

13 Dec '18

Glad tilings of great joy: a Christmas carpet. Herewith the only greeting card I am ever likely to get around to sending, folks. Make the most of it --- but don't spy this at phone! In connection with Penrose tilings, `empire' denotes the subset of tile placements forced by some given subset of compatible placements; `country' denotes any connected region in such an empire. Modulo geometric congruence and de/inflation, there are just 7 distinct classes of country, identifiable by their simple polygonal boundary shapes. The graphic illustrates the empire of a `free wheel', or cartwheel for which a location is specified, but not an orientation. It somehow appeals to this supercilious misanthropist as vaguely festive, even to the extent of omitting any coffin for Tiny Tim: https://www.dropbox.com/s/ixfdnjax12kn7id/glad_tiling.png Amongst others, the recent paper by Fang Fang, Dugan Hammock, Klee Irwin, "Methods for Calculating Empires in Quasicrystals" (2017) https://www.mdpi.com/2073-4352/7/10/304/htm provides more concrete illustrations of 3 of the 7 home-country empires, populated with "Penrose rhomb" slings-and-arrows tiles, rather than visually preferable kites-and-darts. Figs. 9a, 9b, 9f illustrate instances of single worms (sandwiches) with increasing sizes; 9c, 8b, 9e cartwheels (coffins); 9d, 8a pentagonal stars. By the way, I am in the process of providing more material on this topic, available to anybody sufficiently reckless to enquire further. A Dickens of a lot more ... Bah! Humbug! WFL

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[math-fun] Curious parametrized family of loci
by Dan Asimov 12 Dec '18

12 Dec '18

Let z be a complex variable, t real. Then the locus L(t) given by L(t) = {z in C | |z-1|^(1/|z+1|) = t} goes through an interesting bifurcation as t passes through 1.260522 . I doubt I would have predicted it. —Dan

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Re: [math-fun] Egyptian Fractions (was Re: Messages in pi (cont'd))
by Dan Asimov 12 Dec '18

12 Dec '18

There is the greedy-harmonic-series-with-signs method: Let C be the irrational real number to approximate; say C > 0. Then let p1 be the least integer for which 1 + 1/2 + 1/3 + ... + 1/p1 > C. Now let n1 be the least integer > p1 for which 1 + 1/2 + 1/3 + ... + 1/p1 - (1/(p1+1) + 1/(p1+2) + ... + 1/n1) < C Continue this way, alternately adding and subtracting groups of consecutive terms, resulting in a signed harmonic series that converges to C. —Dan Keith Lynch wrote: ----- ..... What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original. -----

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[math-fun] How AB-normal can numbers get?
by Henry Baker 12 Dec '18

12 Dec '18

This talk of pi being 'normal' got me to thinking: "How bad can an ab-normal number get?" I'm not talking about numbers like 1.000000...; I'm talking about numbers which "look" random to the unaided eye, but which are quite abnormal -- e.g., they might be quite compressible by something as simple as zip or gzip. In particular, a gzip'd number might be 1/100'th the size of the un-gzip'd number. Are there easy constructions for such numbers -- hopefully constructions which don't involve individual bits or Turing Machines ? Perhaps using only "calculator" functions ?

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Re: [math-fun] Egyptian Fractions (was Re: Messages in pi (cont'd))
by Dan Asimov 12 Dec '18

12 Dec '18

Yes, any divergent series whose terms approach 0. Say we're choosing the sign of each term. Then all possible sign choices amount to the countable product K (cartesian power) of {-,+}, which has the size of 2^aleph_0 = the continuum. The product K naturally has the topology of the Cantor set. So for each point of the Cantor set K — i.e., for each choice of signs S : Z+ —> {-,+} — this S gives rise to the real number Sum_{n in Z+} S(n)/n. So assuming the probability of convergence is 1 (which it is), we get a map K —> R from the Cantor set onto the real numbers. This should by rights be continuous, so it's a funny map — a continuous surjection from the totally disconnected Cantor set onto the real numbers. But basic theorems in topology rule this out (the image must be compact). Therefore, mathematics is inconsistent. Another related thing is to look at the distribution of values obtained when choosing signs randomly. Or for more fun, use points on the unit circle S^1 in the complex plane, chosen independently at random from S^1, as the coefficient for each term of the harmonic series. What does the distribution of sums in the plane look like? —Dan Cris Moore wrote: ----- This applies to any divergent series, right? And the fun thing is that any C has a series with any initial subsequence. I suppose this means any C has countably many sequences, although it’s not clear that taking a finite initial subsequence and then turning greedy covers all of them… > On Dec 10, 2018, at 3:43 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > > There is the greedy-harmonic-series-with-signs method: ... -----

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[math-fun] High school physics problem
by Henry Baker 10 Dec '18

10 Dec '18

Suppose we have a large container not 100% filled with ball bearings (all of exactly the same material & hence density) but with a wide & random variety of radii. The largest ball bearings are << the smallest dimension of the container, and the smallest ball bearings can get into every corner of the container. Initially, the bearings of all sizes are equally distributed w.r.t. location within the filled portion of the container. We're on Earth with its standard gravity g and standard atmospheric pressure & temperature (if that matters). Now shake the container vigorously for quite a while (but not so vigorously enough that the balls become dented or heated!). What is the resulting distribution of sizes after this shaking? A. The same as before. B. The larger balls become more likely in the upper layers. C. The larger balls become more likely in the lower layers. I don't have a proof, but I have an intuition about which answer is correct. Has this problem been studied before?

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Re: [math-fun] The moving now (was: Messages in pi)
by Dan Asimov 10 Dec '18

10 Dec '18

Thoughts about what Mike wrote below: 1. I don't know about this arrow of time. I haven't kept up to date, but isn't there a theorem that if you reverse charge, parity, and time, any movie of something physically possible would still be possible if run backward? (And if charge and parity are not involved, then you can just run the same movie backward without change.) Of course, in our neighborhood of spacetime, we are living in a part of the universe where stars radiate energy in the positive direction of time (t+). Given that fact, things like eggs breaking in t+ but seemingly never reassembling follow from the existence of evolution. But in principle if we could imagine where the recent past of [all spacetime that could affect our present] is being copied — in a CPT-reversed way, at least T — presumably the same past would unfold just like our own future. But Mike himself pointed out that "entropy increases" isn't by itself a fully adequate explanation. 2. Mike asks "More carefully, how could you distinguish the experience of living the events out of order from living them in order, or from living them simultaneously?" My first answer is, It sure *feels* as if the recent past is something that I lived very recently. And the recenter the past, the recenter it feels. It's enough to make one want to locate one's focus in a new place, to recenter oneself. *Maybe* it's just a coincidence that all these moments fit together into a continuous narrative. It feels much more likely that something crucial about how the moments came into being is real-numbery, i.e., somehow 1-dimensional. I think the real numbers R can be characterized as the only connected topological space such that the removal of any point leaves just two connected components. But there may be another condition like, y'know, Hausdorffness or local compactness or homogeneity, that I'd rather not mention, necessary for this uniqueness. But: Would we indeed feel as we in fact do, if all phases of our lives were being lived, kind of, "at the same time" ? Who is feeling all those other moments that we lived or will live? (Not 100% serious, but still.) —Dan "All you have is the state of your brain at that given point in time." I think. —Dan ----- On Wed, Nov 28, 2018 at 1:47 PM Dan Asimov <dasimov(a)earthlink.net> wrote: > > That's probably my preferred view, but there are problems reconciling it with > everyday experience. It would mean that all ages t of (say) me are living their > lives equally, in some sense of equally. > > But how does that explain the experience we have of living our lives in the order > of the real numbers, and the fact that it keeps getting later? A too-cute answer is, "Because the arrow of time points in the direction of increasing entropy." More carefully, how could you distinguish the experience of living the events out of order from living them in order, or from living them simultaneously? All you have is the state of your brain at that given point in time. -----

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