Model of a Cayley Diagram by Richard P. Baker, Baker #522 (4)

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.

The entry for Baker 522 appears in his 1931 catalog in the section on Groups (/) Cayley Diagrams (/) General as “522. G8 The five types in wire. Elements at vertices of a cube. With independent generators.” A handwritten label attached to one of the wires of this model reads: #522 ( ) (/) G.8 Elements at vertices of cube. There are five models with this number because there are five distinct groups of order 8: C8 (the cyclic group of order 8), C4 x C2, C2 x C2 x C2, D8 (the dihedral group), and Q8 (the quaternion group).

For a description of the original 1878 Cayley Diagram, also known as a Cayley graph, and an overview of the five Baker models with Baker number 522, see MA.211257.103.

Baker 522 (4) represents one of two groups of order eight that has four bidirectional wires, C8 or Q8. Because this model has two monochromatic paths of length eight it represents the cyclic group of order eight, C8. In addition to the colors represented in the paths of length eight, there is one color that appears on two different paths of order four, and another that appears on four bidirectional wires. The last of these are easily seen the be the four diagonal wires of a light color that form the diagonals of a pair parallel faces of the cube. On the other four faces one can see wires colored red that represent C8’s generator of order four. The eight red wires appear in two sets of four wires, each of which consists of one diagonal from each of the faces for which the diagonals have arrows.

There are sixteen wires and two colors yet to be described. As the model was painted some time before 1931, the colors of these sixteen wires are difficult to determine, although we do know that each color appears on a set of eight wires defined by the two generators of C8 that have order eight. Each of these sets of wires must, therefore, appear in a monochromatic path of length eight. Because it is so hard to distinguish between these two remaining colors, it is difficult, but possible, to find these two paths.

References:

Arthur Cayley, ``Desiderata and Suggestions: No. 2. The Theory of Groups: Graphical Representation," American Journal of Mathematics, 1 (2): 1878, pp. 174-76.

R. P. Baker, ``Cayley Diagrams on the Anchor Ring,” American Journal of Mathematics, 53 (3): 1931, pp. 645-69.

Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 17.

Currently not on view