Oloid

Oloid structure

Oloid

The plane shape of a developed Oloid surface

Oloid-rainbow

An oloid is a geometric object that was discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by intersecting two congruent disks at right angles to each other, so that the distance between their centers is equal to their radius. In fact, since one third of each of the discs' perimeter lies inside the convex hull, from a geometric and kinematic point of view, only 240° circular arcs are needed to create an oloid.

The oloid is the only known object, other than some members of the sphericon family, that while rolling, develops its entire surface. 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 minimums and two maximums. 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. The contact line between the oloid and the rolling surface is of constant length, and is given by:

The surface area of an oloid is given by:

exactly the same as the surface area of a sphere with the same radius. No closed form is apparently known for its enclosed volume. A numerical calculation gives:

Schatz discovered in 1929 that the Platonic solids could be inverted, and one of the products of the inversion of the cube was the oloid. Based on two circles set perpendicular to each other, it rolls in a straight line such that its whole surface touches the plane on which it is rolled. The oloid visually resembles a Möbius strip. Schatz came to his geometric insights by studying the work of Rudolf Steiner, the founder of anthroposophy. Schatz obtained Swiss Patent no 500000 for his oloid mixer. Development has since continued, and the oloid is applied in ever more numerous ways.^{[citation needed]}

Another object is defined, when the distance of the intersecting disks is √2 times their radius. This is often called "Two circle roller". It is not solid as the oloid, it consists just of the two disks. It is interesting, because its center of gravity has a constant distance to floor, thus it rolls smoothly but straightforward, not as swinging as the oloid, with which it is sometimes confused.

References

• Hans Dirnböck, Hellmuth Stachel (1997). "The Development of the Oloid". Journal for Geometry and Graphics 1:2, pp. 105–118.
• View detail for Publication number: CH500000. Swissreg - Institute for Intellectual Property.

External links

• Rolling oloid, filmed at Technorama, Winterthur, Switzerland.
• Paper model oloid Make your own oloid

This geometry-related article is a stub. You can help Wikipedia by expanding it. v t e