Is an oloid a solid of constant width??

Question

An oloid is defined to be the convex hull of two linked congruent circles in perpendicular planes each of which passes through the center of the other. The Wikipedia page has more info. I came across a couple of places where this is characterised as a solid of constant width. See here and here. I am not sure it is though. Can anyone clarify??

Answer 1

(1) When we roll the body on the plane, then the mid point between centers of two circles (of radius 1) has a const height from the plane : Consider sphere $S$ of radius $\frac{3}{4}$ and a loop $c: [0,l]\rightarrow S$. Then there is $v(t)\in T_{c(t)}S,\ |v(t)=1|$ s.t. $$ \bigg\{ c(t) + mv(t)\bigg| t\in [0,l),\ -a(t)\leq m\leq b(t) \bigg\}$$ is the oloid. Here oloid contains the sphere $S$.

Let me explain why $c$ is in the sphere, intuitively not rigorously : Consider a smooth loop $\alpha :[0,2\pi] \rightarrow \mathbb{R}^3$ containing sets $\{(\cos \ t,\sin\ t,0)| 0<\varepsilon <t<2\pi-\varepsilon \}$ and $(1-\varepsilon,0,0)$. Note that this is not in a sphere. When $M =_{let} \{ \alpha (s)+ l(0,0,1) | 0\leq s\leq 2\pi,\ -1\leq l\leq 1\}$ and we roll $M$, then it will stop. But oloid comes out from rolling a linked two circles.

(2) Further, intersection between the body and the plane is always exactly a line segment of const length, i.e. $a(t)+b(t)$ is constant.