Geometric Model of A. Harry Wheeler, Immersion of a Moebius Band (One-Sided Polyhedron)

Taking a long, thin rectangle and attaching the short sides with a half-twist produces a surface called a Moebius band. It has neither inside nor outside (that is to say, it is non-orientable), and has only one boundary component—tracing starting from one point on the edge takes one around both long edges of the rectangle. For most closed polyhedra, the Euler characteristic of the polyhedron, which equals the number of vertices, minus the number of edges, plus the number of faces the number, is 2. For a Moebius band, it is 0.

This model is an immersion of a Moebius band into three-dimensional space. That is, the surface passes through itself along certain lines. The model is dissected into three triangles and three four-sided figures (quadrilaterals). The triangles (colored black) have angles of 36, 72, and 72 degrees. The pass-through lines of the immersion meet the triangles only at their vertices. The quadrilaterals (colored yellow) are in the shape of isosceles trapezoids, and the diagonals of the trapezoids are the pass-through lines of the immersion. These diagonals divide a trapezoid into four regions. The region that abuts the longer parallel side of the trapezoid is visible from the front side of the model, and the regions that abut the non-parallel sides are hidden. One third of each of the regions abutting the shorter parallel sides of the trapezoids is visible. The boundary edge of the model is an equilateral triangle consisting of the longest sides of the three trapezoids.

Figure 1 is a rendering of the model with vertices (six), edges (twelve), and faces (six) labeled. Contrary to appearances, the edge labeled e4 separates T1 from Q3, the edge labeled e10 separates T1 from Q1, and the edge labeled e5 separates T1 from Q2, and similarly for the other two triangles. Each triangle shares one edge with each quadrilateral, and each quadrilateral has one edge along the boundary of the model and one edge in common with each triangle.

Figure 2 shows a rectangle that can be made into a Moebius band by identifying the vertical edges with a half-twist. The rectangle is dissected into three triangles and three quadrilaterals with the same pattern as this model. There is little distortion of T1 and Q1. T2 is only slightly distorted. However T2, Q2, and Q3 are required to go out one end and come back in the other.

Compare 1979.0102.416 (which has a full discussion of the surface), 1979.0102.197, 1979.0102.198, 1979.0102.199, 1979.0102.200, and MA.304723.718.

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