Model of a Cayley Diagram by Richard P. Baker, Baker #522 (1)

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 (1) (/) G. 8. Elements at vertices of cube. There are five models with this number because there are five distinct groups of order eight: C8 (the cyclic group of order 8), C4 x C2, C2 x C2 x C2, D8 (the dihedral group), and Q8 (the quaternion group).

A Cayley Diagram, also known as a Cayley graph or Cayley color graph, is named after the English mathematician Arthur Cayley who wrote about them in 1878 and is related to Cayley’s Theorem so the groups being considered should all be thought of as permutation groups. Cayley’s “Graphical Representation” of a group is based on a diagram including points that represent all the elements of the group and directed line segments that represent a set of generators of the group. The two-dimensional graph he displayed was for a non-abelian group of order 12 having 2 generators and arrows appear on every segment to indicate movement from one point to another. It is a Cayley graph because the vertices of the graph represent the elements of the group and directed segments going from x to y if (1/x)y is a generator. While all of the line segments of Cayley’s graph show a single arrow, it is possible to have segments that are bidirectional, i.e., when (1/x)y = (1/y)x.

In the same year that Baker published his catalog, he also published an article, ``Cayley Diagrams on the Anchor Ring,” where Anchor Ring is an archaic name for a torus. In that article he lists five groups of order eight but mistakenly lists two dihedral groups rather than D8 and Q8. He does not discuss representing the five groups as three-dimensional models based on cubes.

All five of the Baker 522 models display each of the eight elements of the group it represents as a vertex of a cube. The wires connecting the vertices represent directed segments associated with the generators and are colored accordingly. Each model should have seven wires at each vertex. Had Baker not included the phrase, “With independent generators,” then there would have been seven colors shown on every model and for every wire with an arrow there would have been an additional wire between the same two vertices but oppositely directed and differently colored.

Not every wire includes an arrow, so it appears as if Baker is representing bidirectionality by showing no arrow rather than two arrows pointing in opposite directions. For each model there is at least one color representing bidirectional wire, and every such color is represented by four such wires, no two of which meet at a vertex. By looking only at the number of bidirectional wires of the model, one can determine which groups three of the models represent. However, because each of C8 and Q8 has only one bidirectional generator, one must take the colors into account in order to distinguish between them.

Baker 522 (1) represents the Cayley Diagram for C2 x C2 x C2 because it is the only one of the five models for which all the wires are bidirectional. However, the model has one peculiarity, eight wires meet at two of vertices. What can be seen is that the four diagonals of the cube are of the same color, each of the three sets of four parallel wires, and two of the three sets of diagonals of parallel faces. While the three wires on the same face appear to have the same color, the two wires on the parallel face definitely do not. It is not clear why Baker included the extra wire or why the colors on those parallel faces are not all the same.

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

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