# Stability of Polygons Inscribed in an Ellipse

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This Demonstration concerns polygons with sides inscribed in an ellipse with semimajor axis 1 and semiminor axis . If there exists a perpendicular line from a side that intersects the center of gravity, then the side is stable. The stable sides are shown in green.

[more]

Every convex polygon can be defined by a function in a polar coordinate system with origin at the center of gravity of an object with cross section . On horizontal surfaces, all objects start rolling in a way that sends the center of gravity lower such that decreases at the point of contact with the underlying surface. Equilibria occur if at this point. A balance point is stable at the minima of , where .

The number of vertices because in these cases, the center of gravity of the polygon and the ellipse are equal.

[less]

Contributed by: Tímea Deichler (June 2015)
Suggested by: Gabor Domokos
Open content licensed under CC BY-NC-SA

## Details

References

[1] G. Domokos, Z. Lángi, and T. Szabó, "On the Equilibria of Finely Discretized Curves and Surfaces," Monatshefte für Mathematik, 168(3–4), 2012 pp. 321–345.

[2] "The Gömböc." (Jun 25, 2015) www.gomboc.eu.

## Permanent Citation

Tímea Deichler

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send