World Library  
Flag as Inappropriate
Email this Article

Oloid

Article Id: WHEBN0012599909
Reproduction Date:

Title: Oloid  
Author: World Heritage Encyclopedia
Language: English
Subject: Convex hull, Sphericon
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Oloid

File:Oloid structure.pdf

File:Oloid development.pdf


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]

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

A numerical calculation gives:

\!V \approx 3.0524184684r^{3}

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(\frac{\sqrt{2}}{2}-{3}\frac{\sqrt{3}}{8})\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{3} 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.

External links

  • Rolling oloid, filmed at Technorama, Winterthur, Switzerland.
  • Paper model oloid Make your own oloid
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.