Model of a Cayley Diagram by Richard P. Baker, Baker #522 (5)

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 (5) (/) G-8. Elements at vertices of cube. There are five models with this number because there are five distinct groups of order 5: 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 (5) represents the Cayley Diagram for the group C4 x C2 because that is the only group for which there are twelve bidirectional wires. Although this model only has eleven bidirectional wires, it is missing a directed wire that would have been a diagonal of one of the faces with four directed wires and the diagonals of the only other face with four such wires are both bidirectional.

Since the group C4 x C2 is the only group of order eight with five generators, this model is the only one whose wires were painted with five colors. One color can be seen on two sets of four wires with arrows that form the edges of two faces of the cube. A second color can be seen on the remaining two sets of four wires with arrows, i.e., those that are diagonals of the other four faces of the cube. Each of the remaining three colors appear on a set of four bidirectional wires. One of these is very easily seen as the four diagonals of the cube. A second is the diagonals of the faces of the cube that are formed with by wires with arrows and this, too, would be easily visible if the missing wire were not one of the four wires. The third set includes the four bidirectional edges of the cube. This set of four is not easily visible because the age of the model has changed some of the colors and it is now difficult to see they are all the same color and that color is different from the color of all the other wires.

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