Wolfram Demonstrations Project

12,000+ Open Interactive Demonstrations
Powered by Notebook Technology »

Equal Cores and Shells in Circles and Spheres

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

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

The area of the red circular ring between the radii and is given by . This is equal to the area of the blue disk of radius given by if the three radii satisfy . Visually, the equality of the areas of the shell and disk is often not very obvious, which might loosely be classed as an optical illusion.

[more]

The 3D analog compares the volume of a red spherical shell and a blue central sphere. Recall that the volume of a sphere of radius is given by . The relation between radii is now given by . The spheres are shown in transparent hemispherical cross section.

[less]

Contributed by: S. M. Blinder (July 2013)
Open content licensed under CC BY-NC-SA

Snapshots

Details

Integer triplets , , such that such as , , , and so on are the well-known Pythagorean triplets. By the Fermat–Wiles theorem (formally known as Fermat's last theorem), there are no analogous integer triplets for cubes.

Snapshot 1: equality of the areas of the shell and disk is not visually obvious

Snapshot 2: these radii belong to the simplest Pythagorean triplet , scaled by factor of 4

Snapshot 3: when , the circle is bisected azimuthally

Embed Code

Take advantage of the Wolfram Notebook Emebedder for the recommended user experience.

Related Demonstrations More by Author
Browse all topics
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