Geometric Model by A. Harry Wheeler, Moebius Polyhedron (Polyhedron of Musical Chords)

This plastic model is in roughly the shape of a ring or torus. All the faces are triangles,twelve yellow and twelve white, with the colors alternating. The surface has thirty-six edges and twelve vertices. This would give an Euler characteristic of vertices minus edges plus faces equal to 12 – 36 + 24 = 0. This is appropriate for a surface with one hole. Four of the white triangles are numbered. Face 1 is on the side along with Wheeler’s 739 model number, 2 is a congruent white triangle on the left side, 3 is a white triangle on the bottom of the back, and 4 is a triangle on the bottom of the right side.

Wheeler called the surface a “polyhedron of musical chords,” following the German mathematician August F. Moebius, who described the surface in the second volume of his collected works. Wheeler made two other versions of the model, on which a musical note is indicated at each vertex of the models. For a fuller description of the mathematics of the model, and of its relationship to musical chords, see MA.304723.405. For a paper version of the model, see MA.304723.508. For patterns, see 1979.3002.060. For an undated English translation of the relevant pages from Moebius, see 1979.3002.110.

Some patterns of a smaller version of this model are labeled in Wheeler’s hand and dated July 1939. They are clipped together with patterns showing this version. A label glued to the cover of these patterns reads: Dr. Shook, Mar. 45. It is possible that Dr. Shook was Clarence Albert Shook (June 11, 1895–1957?), who taught mathematics at Lehigh University in Pennsylvania, or Glenn Alfred Shook (July 16, 1882–1954) who taught physics and mathematics at Wheaton College in Massachusetts and published Sound and Musical Instruments (1944).

Reference:

A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.

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