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Oloid

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 Title: Oloid Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

Oloid

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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.

Surface area and volume

The surface area of an oloid is given by:

$\!A = 4\pi r^2$

exactly the same as the surface area of a sphere with the same radius. In closed form, the enclosed volume is [1]

$\left(2/3\right) \left(2 EllipticE\left[3/4\right] + EllipticK\left[3/4\right]\right)r^\left\{3\right\}$.

A numerical calculation gives:

$\!V \approx 3.0524184684r^\left\{3\right\}$

Kinetics

The oloid is one of the only known objects, along with 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:

$\Delta h=r\left(\frac\left\{\sqrt\left\{2\right\}\right\}\left\{2\right\}-\left\{3\right\}\frac\left\{\sqrt\left\{3\right\}\right\}\left\{8\right\}\right)\approx 0.0576r$

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:

$\!l = \sqrt\left\{3\right\} r$

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.

History

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.

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. [2]

References

• Hans Dirnböck, Hellmuth Stachel (1997). "The Development of the Oloid". Journal for Geometry and Graphics 1:2, pp. 105–118.

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