Geometric Model by Richard P. Baker, Wire Model of Clebsch's Diagonal Surface, Baker #365

This geometric model was constructed by Richard P. Baker, probably in the 1920s while 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.

The typed part of a paper label taped to this wire model reads: No. 365 (/) THE DOUBLE SIX FROM 367. Model 365 appears on page 10 of Baker’s 1931 catalog of models as “The double six from 367” under the heading Clebsch’ Diagonal Surface. Model 367 is the first model in this section and is listed as a plaster cast model title “Tetrahedral symmetry. 24 finite lines.” While model 367 is not among the models in the Smithsonian collections, the second model in this section, Model 366 (211257.055) “The lines of 367,” is.

The double six in the title of this model is a Schlaefli Double Sixes structure, named after 19th-century Swiss mathematician Ludwig Schlaefli. A double sixes structure consists of two sets of six lines that satisfy the following three properties: no lines in the same set intersect, each line is paired with a line of the other set which it does not intersect, and each line intersects the five lines in the other set with which it is not paired.

In this model, the twelve wire rods represent the twelve lines. It appears as if some of the rods have been repainted so it is no longer possible to distinguish six colors. However, it is likely that the paired rods were originally painted the same color. Labeling one of the sets of rods 1 through 6 and the other 1′ through 6′ as shown in one of the images, one can see that the yellow rod 6′ meets (left to right) rods 2, 4, 3, 1, and 5 but does not meet the yellow rod 6. Similarly the (clearly repainted) tan rod 4 meets (bottom to top) rods 5′, 6′, 1′, 3′, and 2′ but does not meet the purple rod 4′.

The surface on which this model is based, the Clebsch Diagonal Surface, is defined by the cubic equation x³+y³+z³+w³+v³=0 assuming x+y+z+w+v=0; it contains thirty six double-six structures.

Currently not on view