Model of a Cayley Diagram by Richard P. Baker, Baker #522 (2)

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 (2) (/) 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).

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 (2) represents the Cayley Diagram for the dihedral group, D8, because it is the only one of the five models for which there are twenty bidirectional wires and eight wires with arrows on them. The wires with arrows are all the same color and appear on the edges of two faces of the cube that are parallel. None of the remaining five sets of four differently colored wires have arrows. Three of these sets are easy to see as each comprises the diagonals of a pair of parallel faces of the cube. The eight remaining wires include the four edges with no arrows and the four diagonals of the cube. Each set of four includes two edges without arrows that are not edges of any face of the cube, together with two diagonals of the cube that do not share any vertices with the edges in the set.

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