July 2018 - mathfun@mailman.xmission.com

Re: [math-fun] n-dimensional geometry puzzle
by Dan Asimov 07 Jul '18

07 Jul '18

Here's my solution to the "n-dimensional geometry puzzle" below: The center of the smallest sphere containing (1,0,...,0), ..., (0,...,0,1) in R^n must be at their centroid, which is C = (1/n, ..., 1/n). The distance from the point C to any of the basis vectors is sqrt(1-1/n). The distance from C to the origin is sqrt(1/n). Since sqrt(1-1/n) >= sqrt(1/n) for all n >= 2, (exercise), sqrt(1-1/n) solves the problem. As far as I know, only Tom Karzes sent in a solution (almost instantaneously). —Dan ----- Puzzle: ------- In n-dimensional space R^n, find the radius R = R(n) of the smallest sphere containing (whether inside or on the surface) the standard basis vectors {e_k} = {(1,0,...,0), ..., (0,...,0,1)} and the origin 0 = (0,...,0). I.e., R(n) = inf {r > 0 | for some c in R^n ||p - c|| <= r for p = 0 and all p = e_k} Apologies if this has been asked already; I don't recall. -----

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[math-fun] Today's XKCD comic is about OEIS
by Keith F. Lynch 07 Jul '18

07 Jul '18

Today's XKCD comic is about OEIS. See xkcd.com/2016

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Re: [math-fun] Political spectral analysis
by Dan Asimov 06 Jul '18

06 Jul '18

Perceptrons were the 1957 invention of Frank Rosenblatt (not Minsky). —Dan ----- Check out Minsky's (1950's?) work on "Perceptrons". He was trying to show how weak they were at certain decision tasks. -----

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Re: [math-fun] dual content determinant
by Dan Asimov 06 Jul '18

06 Jul '18

It is true that the unit normals u_i to the hyperplanes F_i satisfy u_i = (F_i1, ..., F_in) and so the determinant | F_ij | is the volume of the n-parallelepiped spanned by the u_i: | F_ij | = volume({Sum t_i u_i in R^n | 0 <= t_i <= 1}). —Dan ----- Let F_0, ... F_n denote hyperplanes in (Euclidean) n-space, where each F_i satisfies an equation F_i1 x_1 + ... + F_in x_n = -F_i0 , normalised so that (F_i1)^2 + ... + (F_in)^2 = 1 . The square matrix determinant | F_ij | has the property that it vanishes when the hyperplanes have some common point. More interestingly, its sign may be used in conjunction with a fixed frame F_1, ..., F_n --- or equivalently, with their intersection point P_0 --- to define the (chromatic) orientation of a variable plane F_0 in a scene as observed from viewpoint P_0 . With F_i0 and F_0j omitted, the remaining order n minor is (related to?) what Coxeter somewhere calls an "n-dimensional cosine", and thence to Gaussian curvature. But does the entire determinant have any established name; does its value represent some natural geometric property of a simplex? -----

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[math-fun] dual content determinant
by Fred Lunnon 06 Jul '18

06 Jul '18

Let F_0, ... F_n denote hyperplanes in (Euclidean) n-space, where each F_i satisfies an equation F_i1 x_1 + ... + F_in x_n = -F_i0 , normalised so that (F_i1)^2 + ... + (F_in)^2 = 1 . The square matrix determinant | F_ij | has the property that it vanishes when the hyperplanes have some common point. More interestingly, its sign may be used in conjunction with a fixed frame F_1, ..., F_n --- or equivalently, with their intersection point P_0 --- to define the (chromatic) orientation of a variable plane F_0 in a scene as observed from viewpoint P_0 . With F_i0 and F_0j omitted, the remaining order n minor is (related to?) what Coxeter somewhere calls an "n-dimensional cosine", and thence to Gaussian curvature. But does the entire determinant have any established name; does its value represent some natural geometric property of a simplex? Fred Lunnon

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[math-fun] Political spectral analysis
by Keith F. Lynch 04 Jul '18

04 Jul '18

I hope I can be forgiven a brief digression into politics on this, the US's 242nd Independence Day. In response to the pending resignation of one of the nine current Supreme Court justices, charts showing all nine, in order of their alleged position on the supposed political spectrum, have been in the news lately. I've long thought that the "left" and "right" are more like random and shifting assemblages of ideas than any sort of coherent, persistent, or self-consistent political, legislative, legal, economic, or judicial philosophies. I usually can't tell who is what. And people who know me strongly disagree with each other on where on the political spectrum I am (except that nobody thinks I'm moderate). Can we use math to determine whether the idea of a single one-dimensional political spectrum has any utility in explaining the behavior of Supreme Court justices, senators, U.S. states, etc.? Can we arrange the justices (or senators, etc.) in a line such that for every yes-no decision they made, there's a point which everyone who voted yes is on one side of and everyone who voted no is on the other side of? The point can be in a different place for each such decision, and yes and no can swap sides. But, for instance, if justices A, B, and C are in that order, A and C should never agree with each other except when B agrees with both of them. The answer is almost certainly no. But how close can we get? Can we get it to work for 99% of their decisions? For 90%? What's an efficient algorithm for finding out? With just nine, it's practical, with a modern computer, to try all 181440 (9!/2) possible orders. But with 50 states, 100 senators, or 535 members of congress, that would obviously be intractable. Some claim that the two extremes of the spectrum are identical, or close to it. We could test that by making the spectrum a circle, and seeing if there's always a line that will divide everyone who voted one way from everyone who voted the other. (A different line each time, but with the justices (or senators, etc.) always in the same order around the circle.) Or we could try a two-dimensional spectrum. Place the justices on a plane such that for every decision a line exists that divides the yes votes from the no votes. Again, the spectrum may wrap around in one (a cylinder) or both (a torus) dimensions. Similarly with higher dimensions. Of course with N voters, if you choose N-1 dimensions you can always find an N-2-plane that divides the yes votes from the no votes, even if they're completely random. Just as any N points can be perfectly, but uselessly, fit by some order N-1 polynomial. But I'm way over my pay grade, even for two dimensions. Just how many "orderings" of N voters in a plane are there, for the purposes of finding whether such a line exists? Has anyone already done such work, especially with real-world data? Also, where can I find complete machine-readable real-world voting records for justices, senators, states (in presidential elections), etc.? Thanks.

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[math-fun] n-dimensional geometry puzzle
by Dan Asimov 04 Jul '18

04 Jul '18

Puzzle: ------- In n-dimensional space R^n, find the radius R = R(n) of the smallest sphere containing (whether inside or on the surface) the standard basis vectors {e_k} = {(1,0,...,0), ..., (0,...,0,1)} and the origin 0 = (0,...,0). I.e., R(n) = inf {r > 0 | for some c in R^n ||p - c|| <= r for p = 0 and all p = e_k} Apologies if this has been asked already; I don't recall. —Dan

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[math-fun] Draft of June 2018 blog post
by James Propp 02 Jul '18

02 Jul '18

I've written two thousand-word essays (shorter than my usual essays) and posted them in draft form here: https://mathenchant.wordpress.com?p=2397&shareadraft=5b2e33269254c One essay is on topology; the other is on set theory. I've tried to make them accessible to my core audience of gifted-high-school-students and professionals-who-loved-math-in-high-school-and-college-but-went-on-to-do-other-things-with-their-lives, but perhaps I aimed too high in the essay on Skolem's paradox. I'd welcome comments, but as usual, if you do comment, I hope you'll say who you are since WordPress won't tell me (and I may have follow-up questions for you). Jim Propp

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[math-fun] Dan Romik's ambidextrous moving sofa constant
by Bill Gosper 01 Jul '18

01 Jul '18

(I'm sure he knows) can be written Out[436]= 1 + Gudermannian[ArcSinh[1]/3] + 4 Sinh[ArcSinh[1]/3]^2 In[437]:= N[%, 22] Out[437]= 1.644955218425440851669 https://www.math.ucdavis.edu/~romik/movingsofa/ billgosper$ ries/ries -l7 1.64495521842544085 Segmentation fault: 11 (Can we ever have enough hardware?) --rwg

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