National
          Science Foundation
National Curve Bank

A New Geometric Shape?
A Relative of the Sphericon-Oloid Family

The Parabolicon

Deposit #167

David Hirsch






sphericon2
                                            animation

Sphericon Animation


animation

Parabolicon Animation


David Hirsch named his new solid the PARABOLICON. It is a new member to the line segment bodies (solids whose surfaces are created by self-intersecting ruled surfaces that create closed 3D space). Like the sphericon and the oloid, co-members of this special family, its whole surface is made of a single smooth developable surface. Like the sphericon, and unlike the oloid, while rolling on a flat surface, its center of mass maintains constant altitude, so its motion is smooth. Unlike the sphericon and the oloid, the distance between any two parallel opposite planes that are tangent to its surface is constant, so that during all rolling duration its height doesn't change. It has 6 parabolic edges and 8 vertices, 6 of which, define a regular hexagon, which divides the PARABOLICON into two congruent parts. The PARABOLICON circumscribes a regular octahedron whose 6 vertices lie on the 6 PARABOLICON's edges.

David Hirsch also writes that the PARABOLICON is only one of an infinite number of shapes belonging to this new family of bodies. For example, a new shape made of six cone segments can be constructed to have a middle cross-section of a regular hexagon; thus, an appropriate name might be the HEXACON, a special case of the PARABOLICON. Other future possibilities include the Octacon, Decacon, etc.



Copyright Notice:  The animations and all images within are under copyright by David Hirsch and may not be copied, electronically or otherise, without his express permission.

David Hirsch,  playidea@gmail.com



References
Websites: https://en.wikipedia.org/wiki/Sphericon >.  Please note the patent statements.


Printed
References
    Ian Stewart,  Mathematical Recreations: Introducing the Sphericon, a new solid shape.
    Scientific American, vol 281 (4), October, 1999, pp. 116-117.


David Hirsch   signature  2018