Model of a Cayley Diagram by Richard P. Baker, Baker #522 (3)

This, now damaged, 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 (3) (/) G.8 Elements at vertices of (/) cube. There are five models with the number 522 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.

Despite the damage to Baker 522 (3), it is possible to determine that it was constructed to represent the Cayley Diagram for Q8. This is because Q8 is one of two groups of order eight that has four bidirectional wires (the light color for this model) and it was possible to show that Baker 522 (4) represents the Cayley Diagram for C8. Since the group Q8 has three generators of order four, in addition to the four bidirectional wires this model should have three sets of eight wires of the same color with each color appearing on two paths of length four. While it looks as if those three colors are red, green, and black, the age of the model is such that it is hard to be sure that one is assigning the correct color to any given wire. Furthermore, there is at least one arrow whose connection appears to be damaged so that its intended direction is not certain.

The damaged model does not have the normal twenty-eight wires with seven wires at each vertex. Rather there are four vertices with eight wires, two with six wires, and two with five wires. In addition to missing one wire that would have been a diagonal of a face, there are two misplaced wires since each of a pair of parallel faces have two diagonal wires that both connect the same two points and indicate movement in the same direction.

The two paths of four along red wires are clearly visible on the model. Each path alternates between wires that are edges of a face and wires that are diagonals an adjacent face. While the paths of the other two colors appear to follow that pattern, only one can be seen to do so because of diagonals that are missing or misplaced and edges with damaged or misplaced arrows.

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