Model of a Cubic Surface by Richard P. Baker, Baker #430

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 one hundred of his models are in the museum collections.

A typed label attached to a side of this model reads No. 430 (/) CUBIC SURFACE xyz-y+z=0. An entry for this model is in Baker's 1931 catalog in the section on "Analytic Geometry (/) Cyclides." Cyclides are quartic, not cubic, surfaces (e.g. they are of degree 2, not degree 3), which makes this grouping puzzling.

Although the model appears to be divided into four regions, the cubic surface has three separate components. In the photographs NMAH-DOR2014-00228 and NMAH-DOR2014-00230 one can see that one of the three components is colored pink and appears to have all z values positive and that another which is diagonally opposite it appears to have all z values negative. As seen in the same photographs, the third component is comprised of the part of the surface that is colored yellow and the part of the surface that appears above the label.

We can determine which is the first octant, i.e., where all three coordinates are either 0 or are positive, by looking at the vertical diagonals, y= x and y=-x. By substituting y for -x in the original equation we find that vertical diagonal is not defined for x=1 and x=-1. As can be seen on the two photographs listed above, this implies that y=-x is the vertical diagonal that intersects the two small components and, therefore, that the portion of the surface for which x=y must be contained in the union of the first octant and the octant in which all the coordinates are either 0 or are negative.

On that vertical diagonal the equation of the surface becomes x²z – x + z=0 so z=x/(x² + 1). Since the denominator is always positive, z and x must have the same sign along the diagonal. It is clear from the first photograph listed above that the octant above the label satisfies the condition needed to be the first octant.

An entry describing this model appears in Baker’s 1931 catalog in an untitled subsection of the section on Analytic Geometry that follows the Cyclides subsection.

Reference:

Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 10.

Currently not on view