Intersecting cylinders

The following will discuss and illustrate various solids that result from the intersection of cylinders. By no means will all cases be covered but some Povray code will be provided (see later) that will allow for personal exploration.

Two perpendicular cylinders.

This is known as a Steinmetz solid, the surface area is 16r² and the volume is 16r³/3. This result was known as far back as 500BC by the Chinese astronomer and mathematician Tsu Ch'ung Chi who also calculated pi to 6 decimal places, namely 355/113. The PovRay scene file that generated the above is: scene.pov and scene.ini.

Cylinders through each vertex and the opposite face.
This is the same as cylinders through the opposite vertices of a cube.
Please note that in these and subsequent images the left and right images don't necessarily have the same viewing position, the exact views have been chosen based upon their highest illustrative value.

The volume is 12 (sqrt(8) - sqrt(6)) r³.

Cube (3 cylinders)

Cylinders through the center of each face.
This is the same as cylinders through the vertices of an octahedron.

The volume is (16 - sqrt(128)) r³.

Cube (6 cylinders)

Cylinders through the midpoint of each edge.

Volume is (16/3) (3 + sqrt(12) - sqrt(32)) r³

Dodecahedron (6 cylinders)

Cylinders through the midpoint of each face.

Icosahedron (6 cylinders)

Cylinders through the vertices.

Dodecahedron (10 cylinders)

Cylinders through the vertices.

Waterman Polyhedra

Cylinders through the polyhedra vertices.

p0005 (12 cylinders)

p0010 (12 cylinders)

Povray files

The following Povray model file will allow the reader to experiment with these volumes: cylinder.pov and cylinder.ini. It will be left as an exercise for the reader to work out how I achieved the face outlines with Povray :-).

As the number of cylinders increases the result is increasingly spherical, the following is the result from 100 randomly orientated (and randomly coloured) cylinders.

The "lines" that seem to run around the sphere (great circle lines) arise when two of the randomly placed cylinders are almost coincident.