Model of Sixteen Double Tangents to Two Cylinders by Richard P. Baker, Baker #358

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over 100 of his models are in the museum collections. This model, Baker number 358, appears on page 15 of Baker’s 1931 catalog, “Mathematical Models Made by R. P. Baker” under the heading “MECHANICS & THEORY OF SCREWS” and with the title “16 Double Tangents to two cylinders equally inclined to the axes.” However, only 14 of the original 16 metal rods remain.

The typed part of a paper label on the bottom of the wooden base reads: “No. 358 (/) 16 dble. Tangents to (/) 2 cylinders.” The right circular cylinders are made of wood. Each cylinder represents a screw whose threads lie along a spiral, not shown, on the surface of the cylinder. These two spirals have the same pitch and determine the angle made by the axes of the cylinders with all of the double tangents, which are represented on the model by rods.

There were originally four sets of four parallel rods. Looking from left to right at higher of the two cylinders in image NMAH-DOR2014-00034 the first four rods that are visible are one such set and in it one can see that there are two rods that meet the top cylinder on opposite sides. It can also be seen that the two rods on the nearer side of the upper cylinder meet the lower cylinder on opposite sides; the same is true of the rods on the other side of the upper cylinder. This pattern repeats with the other three sets of four parallel rods. Both missing rods come from the same set of parallel rods, with the placement of one of them visible at the right end of the upper cylinder.

While the cylinders have infinitely many double tangents one can use the equations defining the spirals to show that the collection of double tangents to the spirals consists of four sets of four parallel lines. For different pairs of screws with the same pitch there are at most four sets of at most four double tangents.

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