Equal Cores and Shells in Circles and Spheres
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]
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