Sorry about the missing Subject line(s). On Tue, Dec 31, 2013 at 4:23 AM, Bill Gosper <billgosper@gmail.com> wrote:

DanA>

A few years ago I tried to solve for the tightest (smallest constant slope) helical rope about the z-axis in R^3.

I.e., the lowest-slope helix about the z-axis such that if each point P of the helix has a flat, closed, perpendicular disk of radius = 1 centered at P that touches the z-axis, the interiors of all such disks are disjoint.

This results in a transcendental equation that can only be solved numerically, but it was a fun endeavor and is in some sense the simplest "tightest-knot" type question I can think of.

--Dan It might be easier to think in terms of a moving sphere.

Was your equation amenable to Lambert-W, or equivalent to Kepler's? (http://www.tweedledum.com/rwg/pizza.html)

Did you grab 20 digits and ask http://isc.carma.newcastle.edu.au ? In any event, I'd count this as resoundingly solved. All we need is *any* system of equations admitting routine numerical methods.

--rwg

The pizza page ends with a sequence of polynomials in (Lambert) w, 1, 2 + 3 w, 9 + 26 w + 20 w^2, 64 + 273 w + 404 w^2 + 210 w^3, 625 + 3524 w + 7672 w^2 + 7692 w^3 + 3024 w^4,... whose "Rodrigues-like" formula can be considerably simplified:

(D[t^(2*k)/w[t]^(2*k), {t, k}]*w[t]^k*(1 + w[t])^(-1 + 2*k))/(t^k*(2*k)) (none of which is a multiple of 1+w; they all seem irreducible). The high order coefficients are (2 n - 3)!/(n - 1)! (A006963). The constant terms are n^(n-1) (A000169). So why not edit the page instead of muttering here? Because the underlying notebook reads into NeilB's Student Edition but not my Mma 9.0 ! --rwg