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?
Isn't one of the problems with security software and hardware to prevent the copying of currentcy, that it is assuming that the purpose is to conterfeit. Admitablely, a lot of people who copy bills are either counterfeiters, or are just curious, how good a copy the printer can make(and my decide later to use it to counterfeit because it looks good).
There are some useful reasons to copy currentcy. For example you may be a travel agency, and want to show your clients what the currency should look like so they don't get riped off. Or you are training clerks, for example in Canada, so they know what American bills should look like. Or how about the person who has came back from a vacation with several large denomination bills, and is really only interested in having the look of the bills not actually tying up his money in some foreign currency.
Do we not have the right to possess freely available images?(after all it's not the image your 'buying' when you exchange currancy but the associated value in the country your planning on going to). It reminds me of some open source companies who try to sell their documentation instead of the code. Here's the code and it's free, aren't we great!? Now pay us, so that we let you figure out how to use it. Still trying to make a buck, just moving the billable product around from the software to the instruction manual!
Slashdot Mirror
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?
Isn't one of the problems with security software and hardware to prevent the copying of currentcy, that it is assuming that the purpose is to conterfeit. Admitablely, a lot of people who copy bills are either counterfeiters, or are just curious, how good a copy the printer can make(and my decide later to use it to counterfeit because it looks good). There are some useful reasons to copy currentcy. For example you may be a travel agency, and want to show your clients what the currency should look like so they don't get riped off. Or you are training clerks, for example in Canada, so they know what American bills should look like. Or how about the person who has came back from a vacation with several large denomination bills, and is really only interested in having the look of the bills not actually tying up his money in some foreign currency. Do we not have the right to possess freely available images?(after all it's not the image your 'buying' when you exchange currancy but the associated value in the country your planning on going to). It reminds me of some open source companies who try to sell their documentation instead of the code. Here's the code and it's free, aren't we great!? Now pay us, so that we let you figure out how to use it. Still trying to make a buck, just moving the billable product around from the software to the instruction manual!