June 2015 - mathfun@mailman.xmission.com

[math-fun] decorated flexagon on the hong kong $10 note?
by Thane Plambeck 23 Jun '15

23 Jun '15

I just noticed that there seems to be a decorated flexagon on the hong kong $10 note. Here's a combined photo of the front and the back. They line up exactly when you hold the bill up to the light https://www.flickr.com/photos/thane/shares/zu9962 -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] A001110 again
by Bill Gosper 23 Jun '15

23 Jun '15

Closed form (as square = triangular): ( (sqrt(2)+1)^(2*n)/(4*sqrt(2)) - (1-sqrt(2))^(2*n)/(4*sqrt(2)) )^2 = (1/2) * ( ( (sqrt(2)+1)^n / 2 - (sqrt(2)-1)^n / 2 )^2 + 1 )*( (sqrt(2)+1)^n / 2 - (sqrt(2)-1)^n / 2 )^2. - Bill Gosper <http://oeis.org/wiki/User:Bill_Gosper>, Jul 25 2008 And for the Mathematica section(?): ChebyshevU[-1 + n, 3]^2 == Binomial[ChebyshevT[n/2, 3]^2, 2]; % /. n -> Range@9 Out[20]= True

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[math-fun] counting topologies
by Warren D Smith 23 Jun '15

23 Jun '15

The Superstring Theorists have the idea (which I am highly skeptical of, but never mind) that string theory is the "final theory" of physics and will predict the values of all fundamental physical constants, uniquely. However, the uniqueness parade got rained on when the notion of the "landscape" came along. That is, in string theory, the universe is not 3+1 dimensional like you thought, but actually has 6 (or 7 for "M theory") extra dimensions, all "microscopic" and having the topology of a "Calabi Yau manifold." The question then is WHICH Calabi Yau manifold? Also, there are various integer valued "fluxes" or "winding numbers" trapped in this topology. So, to specify the Final Laws Of The Universe, you do not actually get unique laws -- you get a large set of possible laws depending on which CY manifold and which integer vectors you choose. Oops. But, supposedly, once you'd specified that stuff, from then on everything was going to be unique and all physical constants were going to get predicted exactly from the power of pure thought (if only anybody was smart enough to do the computation), etc. So: How many Calabi-Yau manifolds and acceptable integer vectors are there? One paper guesstimated the number as 10^500. This suggests that quite possibly, specifying that stuff will require more bits than it takes to specify all known physical constants, which would be kind of a philosophical failure. But anyhow, nobody has even proven the CY-manifold count is even BOUNDED. It might be "infinite." And apparently, it is now conjectured that the CY-manifolds come in a finite number of families, each containing a countably infinite number of topologically inequivalent members. Which leads me to my QUESTION -- can we even prove it is at most COUNTABLY infinite? That, at least, might be easy. Maybe even trivial. Or already known. The inequivalent 2-dimensional-manifold compact topologies were classified by Mobius and plainly are countably infinite in number. For 3D, the Thurston Geometrization Conjecture proven by G.Perelman classifies all possible compact topologies, and I believe makes it clear that they too are countable. More generally, in any fixed dimension, if we regard a topology as defined by a simplicial complex having that same topology ("piecewise linear" manifolds) -- well, it seems clear the number of PL topologies is countable. Proof: specify the number V of D-simplices you want. Then their connection pattern is defined by a (D+1)-regular graph with V vertices and each edge labeled with up to D! permutations. All that information happens in only a countable set of possible ways. In fact we could easily program a Turing machine to print them all out (runs forever, and probably generates a superset, but does print them all) which proves countability kind of "constructively." QED I suspect/hope that "wild" topologies (not describable in piecewise linear manner) are forbidden in stuperstring theory (and physics generally) in which case this would prove countability. Do the string theorists claim that, once you specify the topology of the CY, then its geometry (metric) is forced ("rigidity")? Anyhow the hope I here am trying to hope is: that at least, if superstringers are right, then the laws of physics can be specified using a FINITE number of bits, because of countability. If it had been uncountable, then that would not be obvious. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] corollary sniping
by Warren D Smith 23 Jun '15

23 Jun '15

APG denigrates "corollary sniping." I express disagreement. My view is, solve problems, whichever way you can. Sometimes, you solve it in a way which makes it look easy, including sometimes by using somebody else's theorem. Other times, you solve it in a way which makes it look hard, and makes you look clever and/or more original. Sorry, I do not regard the latter as necessarily better. Especially to the degree APG seems to imply, which seems to be that it's practically unacceptable to use somebody else's theorem if the way you used it, is a lot easier than his proof of it. I say: That's just nuts. However, yes: in the particular case APG had in mind about Heawood unit distance graph, I would indeed contend this was a C.Snipe - or should have been.

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[math-fun] Earliest life
by Warren D Smith 23 Jun '15

23 Jun '15

http://www.jbc.org/content/early/2015/06/18/jbc.M115.642876 is a new paper which claims to find some clues about the nature of the earliest life. More precisely, "the earliest life that has left traces of itself in present-day life, that anybody has noticed so far."

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Re: [math-fun] Fano Plane puzzle
by Warren D Smith 23 Jun '15

23 Jun '15

It is trivial to see that any graph can be embedded as a unit-distance graph, if the space-dimension is high enough. The fact that the Heawood and Doyle graphs have unit distance embeddings in the plane is pretty remarkable, I guess. But the NATURAL QUESTION now is: for your favorite family of graphs, what is the minimum space dimension D you need, to unit-distance-embed it? This is an interesting (new?) integer-valued graph invariant. The question is especially interesting for "distance regular" graphs. For them, there exist natural embeddings that arise as follows. If the graph has graphical diameter d, then the NxN symmetric adjacency matrix of the graph has d+1 distinct real eigenvalues. The N eigenvectors all wlog lie on a unit sphere surface in N dimensions. For any two vertices at graphical distance J from you, their corresponding eigenvectors should lie at geometrical distance f(J) from you. In particular, we get a constant-edge-length embedding of the graph drawn on a sphere. Now for the vertices at graphical distance J from somebody, they lie on a lower-dimensional sphere, the dimension being the dimension of the appropriate eigenspace. Now let us consider "projective planes" encoded via their point-line NxN bipartite incidence structures. Always N=k^2+k+1 and N is prime power in all known examples. ** The Heawood graph is the case N=7 and shows D(7)=2. ** The K33 graph is the case N=3 and since the K23 subgraph is *not* unit-distance graph in the plane, this is not either, but K23 can be embedded in 3-space with unit distances (triangular bipyramid) and Kjj can be embedded in 4-space for any value of j: place the j red points on a circle of radius 1/sqrt(2) in the xy plane, and the j blue points on another such circle in the zt plane, both centered at (0,0,0,0). I can prove K33 is not embeddable in 3-space: consider unit spheres centered at the 3 red points; their intersection is necessarily at most 2 points if the 3 red points are distinct. QED. (If coincident points were allowed, then any bipartite graph would be embeddable on the line.) So D(3)=4. ** The next cases would be N=13 and N=21. The projective planes viewed as bipartite graphs with N+N vertices, N=k^2+k+1, always have graphical diameter=4. The eigenvalues of the NxN red-blue adjacency matrix? The squared matrix equals J+k*I, where I=identity matrix and J=all ones matrix, which makes spectrum determination for the squared matrix easy: its eigenvalues are (k+1) with multiplicity 1, and sqrt(k) with multiplicity N-1. The eigenvalues of the 2Nx2N full adjacency matrix then should be +-(k+1) with multiplicity 1 each, and +-sqrt(k) with multiplicity N-1 each. This eigen-information does not seem to lead to any interesting low dimensional embeddings, sorry.

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[math-fun] PhudgIng phi
by James Propp 22 Jun '15

22 Jun '15

What is the small but non-zero angle of tangency between the golden spiral and the associated nest of golden rectangles? And does this angle occur elsewhere in geometry? (Lots of phi-fans pretend the angle is zero; hence the title of this post.) See https://books.google.com/books?id=Gw-lqvE6fNgC&pg=PT102&lpg=PT102&dq=golden… Jim Propp

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[math-fun] Pisot & Salem numbers
by Warren D Smith 22 Jun '15

22 Jun '15

For the benefit of anybody who did not know, 1. The "Pisot numbers" X are the real roots X>1 of monic integer polynomials, such that every other root r of that poly obeys |r|<1. Equivalently, the Pisot numbers are the non-integer "algebraic integers" X whose powers X^n, where n=0,1,2,3..., lie exponentially close to ordinary integers. Example: the golden ratio g=1.61803... obeying g^2=g+1 and the "plastic constant" 1.3247179572 4474602596 0908854478 0973407344 0405690173 3364534015 0503028278 5124554759 4054699347 9817872803... obeying p^3=p+1. The plastic constant is the minimum Pisot number (proven by CL Siegel in 1944). 2. The "Salem numbers" X are the real roots X>1 of monic integer polynomials, such that every other root r of that poly obeys |r|<=1, with at least one case of equality. Examples: Salem number minimal polynomial 1.72208380573905 1-z-z^2-z^3+z^4 1.40126836793985 1-z^2-z^3-z^4+z^6 1.28063815626776 1-z^3-z^4-z^5+z^8 1.17628081825992 1+z-z^3-z^4-z^5-z^6-z^7+z^9+z^10 (Lehmer) 1.21639166113827 1-z^4-z^5-z^6+z^10 1.23039143440722 1-z^3-z^5-z^7+z^10 1.20002652398739 1-z^3-z^4+z^7-z^10-z^11+z^14 Lehmer's Salem number conjecturally is minimum. This has been proven in the subcases where the polynomial is demanded to have odd coefficients or when the polynomial degree is <=54. Salem numbers S have the property that L*S^n is close to an integer for every n>=0. More precisely: for any epsilon>0, there exists an L>0, and indeed an everywhere-dense set of such L>0, such that L*S^n is within epsilon of an integer for every n>=0; and that sentence is true about Salem numbers but untrue for non-Salem non-integer reals>1. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] The "God problem" -- designing the universe as a linear programming problem
by Warren D Smith 22 Jun '15

22 Jun '15

> From: meekerdb <meekerdb(a)verizon.net> >> If you now want to assess "how likely" our >> life-friendly universe was, or how likely other >> observed properties of universe are, so you can >> compute how much of a "miracle" it all is, >> then you can do so by computing >> polytope volume ratios. One volume being, >> say, the parameter-vectors yielding a life-friendly universe, > > But what do you do for the volume in the denominator? What range of values > is > possible?...natural?...canonical? --Well, actually, the volume just mentioned could BE the denominator. What the numerator & denominator are, depend on what "Goal(s)" for "God" you are trying to think about. In particular, if you want to remove "observer existence bias" (OEB) then you want to only consider the polytope of universes in which human-like beings could exist, then to assess some goal God might have, you'd use that goal intersect humans-exist as the numerator, and humans-exist as the denominator, in your volume-ratio computation. It's hard to assess human-like-life's existence as a possible god-goal in itself, because it is completely contaminated by OEB. However, you might be able to assess that within a wider "ecosystem" of possible goals. For example, humans exist in situation A rather than humans exist in situation B -- that's free of observer existence bias, and if A seems a priori improbable compared to B, then that seems to tell us something both statistically significant, and free of OEB. I have some specific things in mind that Meeker knows about, but I won't discuss them at present. Some of them did look statistically significant to me based on a preliminary examination. For now, if anybody wants to help, you could (a) suggest possible additional physics formulas to add to the list (b) suggest possible god-goals that you'd like assessed. (c) whatever else. And maybe eventually this will get clearer. I think this area has in the past been a mess. Can it be cleaned up? To some extent.

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[math-fun] a bit of precision fraud
by Bill Gosper 22 Jun '15

22 Jun '15

too late for April 1st: Out[666]= EllipticTheta[2, 0, Catalan] EllipticTheta[3, 0, Catalan] == -π/Log[Catalan] In[667]:= N[%] Out[667]= True In[668]:= N[List @@ %%, 69] Out[668]= {35.7908268633193865771253088986699000335476024418014552592829359941761, 35.7908268633193865771253088986699000335476024418014552592829359941761} In[676]:= $MaxExtraPrecision = 999; N[Subtract @@ %666, 22] Out[676]= -3.101101704291470266693*10^-96 (Based on the last formula in gosper.org/pathi.pdf) --rwg

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