By George Hart for the Museum of Mathematics

A lathe is used to turn wood into baseball bats, spindles, and other shapes with rotational symmetry. It can also be applied to making many types of mathematical models. Bob Rollings made this construction from spindles that form the edges of an icosahedron inside of a dodecahedron. Custom spindles are required as there are two sizes, with their lengths in the golden ratio.

Another example of Rolling’s lathe work has a spindle icosahedron assembled around a solid great stellated dodecahedron. Although the great stellated dodecahedron has flat faces and is nothing like a baseball bat, it was completely fabricated by turning on a lathe. Sets of five triangular faces are coplanar and so are turned in one operation. (A plane, though flat, has rotational symmetry, so can be cut on a lathe.) Twelve such planes are turned, on twelve different centers, with the piece held in a spherical chuck.

Even fancier lathe work is needed to release a shape from inside a turned cage. Custom cutting tools are used to create an annular hole and separate the core from the cage, carefully leaving a protruding spindle attached to the core in each hole.

Here, Rollings has created a large icosidodecahedron and the five Platonic solids: icosahedron, dodecahedron, octahedron, cube, and tetrahedron, each with flat exterior faces and a movable, spindled core, all done on a lathe.
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8 thoughts on “Math Monday: Mathematical lathe work”

1. gyziger says:

I can’t quite envision the flat faces being made on a lathe. Could it be explained a little further?

1. migpics says:

I don’t know if this helps but you have to envision coming in from the the head of the lathe rather than the side.
We used to make planes on aluminum stock by cutting straight across the top. If you look close enough you can see the arcs at the base of the triangles.

2. George Hart says:

You position the piece in the spherical chuck so that the plane you want to cut is perpendicular to the axis of rotation. Then you move the cutting tool along a line in that plane.

2. Gareth Branwyn says:

Thanks for the question, gyziger. I see if I can get Geo Hart over here to explain.