Nick Baxter points out that in answering
what is the smallest constant c so that the graph of the function f(x) = x^3 - c x contains the vertices of an equilateral triangle?
I somehow edited sin((%pi/9)) into sin(pi)/9. Please imagine c = sqrt(8 - 8 sqrt(3) sin(pi/9)) ~ 1.80578 . Claim: The midpoint of one triangle side is the origin. Otherwise, centering some side and rotating its endpoints back onto the curve will intersect the other two sides with the curve, indicating that c is non-minimal. So let y(x) := x^3 - c x, and the vertices be z := x + i y(x), -z, and z sqrt(-3) = w + i y(w). Eliminate w from the last eqn, leaving a polynomial p(x,c) = 0 biquartic in x and cubic in c. For minimality, dc/dx = 0. This is equivalent to resultant(p,dp/dx,x) = 0, which gives a bicubic in c. Experimentally, c=1 gives no real roots x. c=2 gives two pairs. The critical c should be where these merge into double roots, i.e., when p and dp/dx have a common factor. Which is what the resultant finds. ------- A useful formula I can't find in MathWorld is for the solid angle Omega of a trihedral vertex with wedge (or sphere arc) angles a, b, c: 2 (cos(c) + cos(b) + cos(a) + 1) Omega = acos(-------------------------------------- - 1) (cos(a) + 1) (cos(b) + 1) (cos(c) + 1) With this you can readily solve for the edge or inradius of regular and nearly regular n-sided solids by setting Omega = 4 pi/n. Sucker bet: Which is greater, the edge of a regular icosahedron or its circumradius? Answer: e/r = sqrt(2-2/sqrt(5)) . I had to stare at a picture to believe this. --rwg