Model, One-Sided Seven Face by Richard P. Baker, Baker #352

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 100 of his models are in the museum collections.

The typed part of a torn paper label on the bottom of the wooded base reads: “352 (/) One-sided seven-face.” The surface that Baker’s model represents is made up of two small black metal triangles, two larger dark pink metal triangles, and three sets of strings representing three quadrilaterals. One of the red triangles has strings attached to all of its edges while the each of the other ends of those strings run to the edge of the other three triangles. The result is that each edge of a quadrilateral coincides with an edge of a triangle. The dark pink triangles are difficult to see in Image NMAH2003-12919 and appear as black in Image NMAH2003-12920. This surface, which is quite small, is connected to the base with six thin metal rods, three of which are supported by a thicker vertical rod.

This small seven sided figure is based on Baker’s reading of a 1905 article by Ernst Steinitz, “Ueber ein merkwuerdiges Polyeder von einseitiger Gesamtflaeche” (Concerning a noteworthy polyhedron that is a one sided surface, in volume 130 of Crelle’s Journal. Figures 1 and 2 of the article are related to Baker’s surface and Figure 3 is related to the wooden base.

Steinitz’s Figure 1 (see the website cited in the references) shows a regular octahedron inside a cube. However Steinitz lists the faces of the surface as four triangles and three quadrilaterals. The diagram shows a triangle in each octant but those that are listed as being part of the surface are those in the first, third sixth and eight octants. Inside the octagon one can see dashed lines representing the linear diagonals of the octagon. These dashed lines also represent the diagonals of the three squares in the XY, YZ, and XZ planes that are the three quadrilaterals Steinitz lists as faces of the surface. Baker’s four triangles and three quadrilaterals are positioned similarly.

Since the vertices and edges of the surface are identical to the vertices and edges of the octagon there are 6 edges and 12 vertices. For any surface there is a number called the Euler characteristic that is defined to be V-E+F, where V is the number of vertices, E the number of edges and F the number of faces of the surface. While all the polyhedra people are familiar with have Euler characteristic 2, both the surface in Steinitz’s article and the one in Baker’s model has Euler characteristic 6-12+7=1, and any surface with Euler characteristic 1 is one-sided.

Steinitz’s Figure 2 represents a surface made up of three squares and four equilateral triangles, with each edge common to one square and one triangle. In Baker’s model the corresponding triangles and quadrilaterals are not regular polygons.

Steinitz’s figure 3 represents the more general choice of polygons seen in Baker’s model rather than what is seen in his figures 1 and 2, and is closely related to the design on the base of Baker’s model. Figure 3 shows two triangles and one quadrilateral that correspond to the pink and black triangles and the yellow quadrilateral painted on Baker’s wooden base. After identifying lines in Figure 3 using the labeling of the diagram, one can see two additional triangles and two additional quadrilaterals. Comparing Steinitz’s labeling of the polygons in Figure 2 with those of Figure 3 shows that the other two of Baker’s triangles are blue and red and the other two quadrilaterals are green and white. While on the base there are four separate edges to the portion of the surface that is green, Baker only included two edges of the white portion. The other half of the white quadrilateral is not shown on the base because it lies beyond the edge of the base opposite the edge of the base that includes the white triangle.

Reference:

Ernst Steinitz, “Ueber ein merkwuerdiges Polyeder von einseitiger Gesamtflaeche,” Crelle’s Journal, vol. 130, p. 282 (Figure 1), p. 283 (Figure 2), and Figure 3 (page 285). An electronic version of this article is presently available at the website with url: https://babel.hathitrust.org/cgi/pt?id=msu.31293022795946&view=1up&seq=309

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