Let $K$ be a solid, homogenous convex body in $\mathbb{R}^3$.
Place $K$ on an inclined plane, and let it roll down the plane,
under some reasonable assumptions of friction between $K$ and
the plane, and tilt angle of the plane, so that it rolls without slippage
under the influence of gravity.
Let the $y$-axis represent straight down the gradient of the plane,
and $x$ the distance of the contact point(s) to the left or right
of straight-down.
> **Q**. Does there exist some $K$ so that, from a particular starting position,
$K$ rolls such that the sideways displacement $x$ grows without bound
the further down $y$ it rolls?
For example, perhaps there is a $K$ so that this (entirely fanciful)
sideways "crab-walking" path is the trace of
the point of contact:
<br />
![RollingPathCrooked][1]
<br />
Or perhaps no such $K$ exists—It is mathematically impossible?
I neither have strong intuitions here, nor a clear idea on how to
settle the question one way or the other.
I have no particular candidate $K$, although this asymmetric [oloid-like object][2]
is what piqued my interest:
<br />
![Oloid12_36][3]
<br />
[1]: https://i.sstatic.net/M6gZ1.jpg
[2]: http://mathoverflow.net/q/141271/6094
[3]: https://i.sstatic.net/kYqKM.jpg