TITLE: Is there a sideways-walking rolling convex body?
QUESTION [6 upvotes]: 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:

 
 
 
 
 


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
is what piqued my interest:

REPLY [16 votes]: The answer is yes, and you can play with them at MoMath. Specifically, the Twist and Roll exhibit has several convex shapes, which will roll arbitrarily far sideways when started at an angle that is not directly downhill.  

Click the image to see a video of the exhibit, which shows the motion of several of the objects.