Oloid

An oloid is a three-dimensional curved geometric object that was discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the other circle. The distance between the circle centers equals the radius of the circles. One third of each circle's perimeter lies inside the convex hull, so the same shape may be also formed as the convex hull of the two remaining circular arcs each spanning an angle of 4π/3.

The surface area of an oloid is given by:

exactly the same as the surface area of a sphere with the same radius. In closed form, the enclosed volume is

where K and E denote the complete elliptic integrals of the first and second kind respectively. A numerical calculation gives:

The surface of the oloid is a developable surface, meaning that patches of the surface can be flattened into a plane. While rolling, it develops its entire surface: every point of the surface of the oloid touches the plane on which it is rolling, at some point during the rolling movement. Unlike most axial symmetric objects (cylinder, sphere etc.), while rolling on a flat surface, its center of mass performs a meander motion rather than a linear one. In each rolling cycle, the distance between the oloid's center of mass and the rolling surface has two minima and two maxima. The difference between the maximum and the minimum height is given by:

Where r is the oloid's circular arcs radius. Since this difference is fairly small, the oloid's rolling motion is relatively smooth.

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