Shape Arithmetic

David Koski was reminding me tonight of the happy mnemonics twixt ancient Greek results and our Synergetics concentric hierarchy.

A cone inscribed in a cylinder of same base and height has 1/3rd its volume.

A cone having half the cylinder's height, its tip at the equator line, has 1/6th its volume.

Lets define two cylinders, one twice the height of the other, called "oil barrel" and "tuna can" respectively.

Two tuna cans stacked up = one oil barrel.

Then we have the following volumes table:

Oil Barrel:  6
Tuna Can: 3

Cone in Oil Barrel: 2
Cone in Tuna Can: 1

Sphere in Oil Barrel: 4
Two Tuna Can Cones: 2

Oil Barrel - Two Tuna Can Cones (hourglass looking) = 6 - 2 = 4 = Sphere in Oil Barrel. [link]

Also:  Cone in Oil Barrel + Sphere in Oil Barrel = Oil Barrel (2 + 4 = 6). [link]

We could make the following identifications and size accordingly:

Oil Barrel = Rhombic Dodecahedron (6)
Tuna Cone = Tetrahedron (1)
Sphere = Octahedron (4)

We could pour sand from objects on the left into corresponding objects on the right.