Uphill rollers: some double cones and a pseudosphere.

 The "uphill roller" is a physics demonstration first reported by the English Mathematician William Leybourn in 1964. In the original version, a double cone placed on two divergent inclined ramps appears to roll uphill, apparently violating the laws of physics.
The uphill roller is an example of the center of mass of an object descending under the influence of gravity.




Nikos Kassastogiannis, an inspired woodworker and an invaluable friend of mine made a nice wooden demonstration of the uphill roller on my request. You can watch the demonstration in the following video. Nikos  didn't only make the double cones I asked him to, but he also made a brilliant pseudosphere which you can watch in the video to oscillate before it reaches its equilibrium.

A pseudosphere is a surface with a constant negative Gaussian curvature. Revolving a tractrix of radius r about its asymptote generates this surface. Although the resulting surface has an infinite extension along its central axis, it has finite area ,exactly the same  as a sphere of radius r, and half its volume. The pseudosphere played an important mathematical role in the acceptance of non-euclidean geometry.

If this video whetted your appetite for more information, here they are some links to explore.
1.  A mechanical paradox or, a new and diverting experiment by William Leybourn

2.  "Pleasure with profit:: consisting of recreations of divers kinds, viz. numerical, geometrical, mechanical, statical, astronomical, horometrical, cryptographical, magnetical, automatical, chymical, and historical. Published to recreate ingenious spirits; and to induce them to make farther scrutiny intor these (and the like) sublime sciences. And to divert them from following such vices, to which (in this age) are so much inclin'd" by William Leybourn"

3. Defying gravity: the uphill roller.
4. Nikos Kassastogiannis, handmade wood