I foresee a whole branch of ipod-spell-correction–inspired mathematics. Cynical permutations, cynical field extensions,...
Indeed, I rechecked on my iphone, and it jadedly auto-corrects cubical --> cynical.
But in 3-d it's trickier. The events that we're recording in order involve our arbitrary line passing through unit squares parallel to one of the coordinate planes. The "wiggling the lines" argument, though, only seems to guarantee that your boundary cases touch a parallel-to-the-coordinate-axis line, not that you can get a boundary case to touch a vertex.
Anyone with clarity on the 3-d picture, please enlighten us by (doing a better job than I just did of) explaining it.
Here's something about the 3-d picture: As has been noted, the interleaving of a triple of arithmetic progressions <--> the sequence of intersections of a line L in R^3 with the faces of the cubical grid. Let T be standard 3-torus, R^3 mod the integral lattice. Let's consider the special case that L has rational slope. In this case the sequence will be periodic, where each period has a fixed numbers of each type: A a's, B b's, and C c's, assumed to have gcd =1. How many different periodic sequences can occur? Consider the projection of the 3-torus a 2-torus T/L obtained as the quotient by L. In this 2-torus are 3 curves, the images of gridlines parallel to the a, b and c axes (which are each single circles in T). These curves divide T/L into a collection of 2-cells, where for any line parallel to L that maps to a point in the cell, the word (in a, b and c) is constant up to cyclic permutation, but at any edge, it changes by switching the order of some pair of adjacent letters. How many cells are there? Let's count the vertices. It's standard and not hard to see that when you have simple curves on a standard 2-torus in the classes (s,t) and (u,v), the number of intersections is the determinant of the matrix. s u t v. One way to think of it: the torus R^2 / < (s,t), (u,v) > is a covering space of the standard torus R^2 / <(1,0), (0,1)> with degree given by the determinant. Similarly, in our current situation: if you think of the 2-torus in the (a,b) plane, you can see directly that it's a covering space of T/L, with degree C. Therefore the image of the a-curve intersects the image of the b-curve C times (but only one of these is the image of a lattice point). By symmetry, there are a total of A+B+C - 2*#(triple points) vertices in the subdivision of T/L. There is at least one triple point: the image of (0,0,0). I'm feeling too lazy to figure out about others at the moment, but I suspect they could occur if to pairs among (a,b,c) have non-trivial common divisors but not the whole triple. For the moment, perturb each triple point to give 3 separate intersections: this gives us (A+B+C) vertices and adds one additional (spurious) 2-cell. Each vertex is an endpoint of 4 edges, each edge has 2 endpoints, so there are 2(A+B+C) edges, so since the Euler characteristic is 0, there are A+B+C 2-cells. In the actual subdivision where triple points haven't been perturbed, there are (A+B+C) - #triple points cells. It follows that the number of cyclically inequivalent sequences with A a's, B b's and C c's grows linearly with A+B+C ------ the triple point issue only effects at worst the linear rate of growth. So, for periodic sequences with fixed period P, we have O( P^2) choices of A, B and C adding to P, for each choice there are O(P) up to cyclic permutation each having O(P) cyclic permutations, resulting in a total of O(P^4). ---- Maybe with a little number theory one could get the exact formula for the number of periodic triple-interleavings of period P. It appears it should boil down to counting issues about greatest common divisors. *** I think you can proceed from this picture to analyze something about the general (non-periodic) case, but I don't have time to think on it more right now. --Bill T
--Michael
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun