Geometric Model by A. Harry Wheeler, One-Sided Polyhedron

This self-intersecting polyhedron has 12 trapezoidal faces (made out of light turquoise plastic) and twelve triangular faces (made out of dark turquoise plastic). It has twelve vertices at which two trapezoids and two triangles meet and four vertices at which six trapezoids and three triangles meet. The polyhedron has a total of 42 edges. A mark on one face of the polyhedron reads: 710 (/) e = 16 (/) k = 42 (/) f = 24 (/) e – k + f = -2. The number 710 is that Wheeler assigned to the model. The other marks refer to the Euler characteristic of the polyhedron, which equals the number of vertices, minus the number of edges, plus the number of faces. Hence: 16 – 42 + 24 = -2.
Speaking more mathematically, this model consists of four copies of Wheeler’s model #708 (MA*304723.416) glued together in the pattern of a regular tetrahedron. It is a closed, non-orientable surface; that is to say it has neither inside nor outside. It has 4 x 6 = 24 faces. At first glance, there are 4 x 12 = 48 edges, but six are identified along the edges of the tetrahedron, leaving 42. At first glance, there are 4 x 6 = 24 vertices, Twelve of these (those like v4 – v6 in Figure 1) remain unidentified, but the others are amalgamated into the four vertices of the tetrahedron, for a total of 16 vertices. The Euler characteristic of the model is thus 16 – 42 + 24 = -2.
For a pattern related to this model, which is dated March 1945, see 1979.3002.104.
A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
Currently not on view
Object Name
geometric model
date made
ca 1940
Wheeler, Albert Harry
Physical Description
plastic (overall material)
blue (overall color)
blue, light (overall color)
cut and glued (overall production method/technique)
average spatial: 12 cm x 16 cm x 16 cm; 4 23/32 in x 6 5/16 in x 6 5/16 in
place made
United States: Massachusetts, Worcester
ID Number
accession number
catalog number
Science & Mathematics
Mathematical Association of America Objects
See more items in
Medicine and Science: Mathematics
Mathematical Association of America Objects
Data Source
National Museum of American History, Kenneth E. Behring Center
Credit Line
Gift of Helen M. Wheeler
general reference
Wheeler, A.H.. List of Wheeler Models
Additional Media

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