November 2007 - mathfun@mailman.xmission.com

Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by James Propp 26 Nov '07

26 Nov '07

>I've often wondered if it is somehow a mistake for >mathematics to distinguish between i and -i. > >There is absolutely no mathematical way to distinguish >between the two. Perhaps they should only be referred >to as a pair, and never one at a time? > >Opinions? My first opinion is that this is a great question! :-) My second opinion is that topos theory gives a way to make Dan's insight precise. Galois theory and the theory of Riemann surfaces do this too, but in a more limited, less mind-blowing way. With topos theory we can distinguish between two kinds of two-element sets: those for which it's possible to distinguish between the elements, and those for which it's not possible. The relevant geometrical model is a double cover of the circle: locally, each fiber consists of two points, but if you go around the loop you find that the two points have traded places. This is just the standard Riemann surfaces picture of the two-valuedness of the square root function. But with topos theory, this kind of ambiguity gets elevated to the level of ontology, which is the sort of thing I think Dan had in mind. FYI, what little I know of topos theory I picked up "on the street" as it were, so others who know more should do me the favor of alleviating my ignorance. Jim

2 1

[math-fun] Five geometry problems
by Dan Asimov 25 Nov '07

25 Nov '07

Let V be the set of vertices of one of the Platonic solids in 3-space. QUESTION: For which 3D rotations of V does the orthogonal projection P of V to the xy-plane realize the maximum possible minimum distance among points of P(V) ? E.g., for the tetrahedron this might occur only when the vertices project to the corners of a square. (Only essentially distinct solutions are of interest: rotations for which the projections of V are not isometric.) For definiteness we could assume that V lies on the unit sphere. --Dan P.S. For a warmup, the same question could be asked about orthogonal projections to the x-axis. Also, a related question is to minimize the maximum possible distance: find the projection of V with the smallest diameter.

1 0

Re: [math-fun] A Property of 17
by Dan Asimov 25 Nov '07

25 Nov '07

Robert Baillie wrote: << Choose numbers a, b, c, ... in the interval (0, 1) so that a and b are in different halves of the interval; a, b, c are in different thirds; a, b, c, d are in different quarters, etc. Not more than 17 such numbers can be chosen. First, can 17 such numbers be chosen? Second, why can not more than 17 such numbers be chosen? >> 17 is in fact the maximum; this result is from a 1970 paper of Ron Graham and Elwyn Berklekamp: "Irregularities in the Distributions of Finite Sequences." J. Number Th. 2, 152-161, 1970. See also < http://mathworld.wolfram.com/18-PointProblem.html >. --Dan

2 1

Re: [math-fun] A Property of 17
by N. J. A. Sloane 24 Nov '07

24 Nov '07

Reminds me of the other property of 17: call a string of consecutive numbers (like 6,7,8,9) "stapled" if for any number in the string there is a mate in the string with which it has gcd > 1. Here 7 has no mate so 6,7,8,9 is not stapled. A stapled string of length n exists iff n >= 17. The string of length 17 starting at 2184 is the first one that exists. See A090318 for more information. Neil

1 0

[math-fun] A Property of 17
by Robert Baillie 24 Nov '07

24 Nov '07

For a long time, I've been stumped by the following property of 17. This is from David Wells' book, "Dictionary of Curious and Interesting Numbers": Choose numbers a, b, c, ... in the interval (0, 1) so that a and b are in different halves of the interval; a, b, c are in different thirds; a, b, c, d are in different quarters, etc. Not more than 17 such numbers can be chosen. First, can 17 such numbers be chosen? Second, why can not more than 17 such numbers be chosen? Bob Baillie

1 0

Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by Dan Asimov 24 Nov '07

24 Nov '07

Fred wrote: << Which reminds me --- quite irrelevantly --- of a claim I fear I may have made previously in these august pages: that a most effective means of causing a polite gathering to descend into an incoherent brawl is to remark that "a mirror exchanges left with right" and then to enquire innocently "so, why doesn't it also exchange up with down?" >> No problem -- you just rotate the mirror 90 degrees, and voilà! --Dan

1 0

[math-fun] The Axiom of Choice for roots of z^2 + 1
by Dan Asimov 24 Nov '07

24 Nov '07

I've often wondered if it is somehow a mistake for mathematics to distinguish between i and -i. There is absolutely no mathematical way to distinguish between the two. Perhaps they should only be referred to as a pair, and never one at a time? Opinions? --Dan P.S. In a faintly related vein, if we define i^z := exp(pi*z*i/2), then iterating this function on the starting value of z = i approaches a limit L of approx. 0.6528812343931018 + i 0.3675743023883531 (or so says my C program). Hmmm, Mathematica seems to give a rather different answer. (Can someone please recompute; my program iterated i^z on the starting value z = i forty times before the new value was within 10^(-9) of the last one, but maybe my computation was done in by roundoff error.) In any case: 1. Is it possible that this limit L of towers of exponentiated i's can be identified as some familiar number? 2. In any case, what can be said about its number-theoretic properties? (In a sense L^(1/L) = i, so apparently Gelfond-Schneider implies L can't be an algebraic irrational.) --Dan

6 5

Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by Dan Asimov 24 Nov '07

24 Nov '07

Kerry wrote: << [I wrote:] << There is absolutely no mathematical way to distinguish between [the two complex square roots of -1]. Perhaps they should only be referred to as a pair, and never one at a time? >> Ok, I'll show my ignorance here. How can we distinguish between 1 and -1 and not be able to distinguish between i and -i? Isn't the simple fact that they're distinct numbers (or distinct points in the complex plane) enough? What am I missing? >> Well, 1 and -1 have different algebraic properties. E.g., 1 is the unique complex number whose product with any other number z is z. -1 is the unique complex number that's unequal to 1 but whose square is 1. On the other hand, there is no distinguishing property that holds for one of i,-i but not the other. This follows from the fact that there exists a field isomorphism f:C -> C interchanging i and -i. (The simplest example is complex conjugation: f(x+iy) = x-iy.) --Dan

3 2

Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by N. J. A. Sloane 23 Nov '07

23 Nov '07

> P.S. In a faintly related vein, if we define i^z := exp(pi*z*i/2), > then iterating this function on the starting value of z = i > approaches a limit L of approx. 0.6528812343931018 + i > 0.3675743023883531 (or so says my C program). and for further information see entries A077589 and A077590 in the OEIS! Best regards Neil

1 0

Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
by Dan Asimov 23 Nov '07

23 Nov '07

David Cantrell wrote: << [I wrote:] > In any case: > > 1. Is it possible that this limit L of towers of exponentiated i's > can be identified as some familiar number? Isn't it just 2 i/pi W(-i pi/2) where W denotes the principal branch of the Lambert W function? >> I suppose, but I don't believe in the Lambert W function (:-)>. (What I mean is, it adds no new information, as far as I'm concerned.) --Dan

1 0