Re:Read the whole article? (Score 1)

Actually you got it backwards. Diameter approaching zero can ride on any road. The constraints are:

1) There is always a point on the wheel that is in contact with the ground

2) Any discontinuities in the road surface can be matched up to a discontinuity in the geometry of the wheel

3) Also from the article it seems that their definition of 'freely rolling' includes the axial of the wheel staying at the same elevation. This entails that on a net zero gradient surface that the sum of the vertical distance from the current contact point to the axial and height of the ground relative to the lowest point the ground has on the surface is a constant.

All of these constraints can be met with a wheel that is just a discontinuity(or a singularity).

The fun begins when one realizes that the discontinuity constraint implies that the line integral of the road surface from discontinuity to discontinuity imposes geometrical constraints on the possible shape of the wheel. You also have the issue of the shape being constrained by having to be able to fit into the 'holes' between dips. Not a trivial math problem to say the least.

For you physicists out there(or you math geeks that know some physics), I'm curious how you could apply the Euler-Lagrange methods of optimizing potential energies to the geometry of this problem. What I mean is this: to get the shape that a chain makes when suspended between two points we use Euler-Lagrange methods to optimize the energy of the system constrained by the fact that the chain has a fixed length, and the position of the mounting points. This give the caternary solution for the case of the two mount points being at the same elevation(ie at the same gravitational potential). How do we go from this to the geometry of the wheel? Or even more fun, given an arbitrary wheel or road, determine it's corresponding partner?