Mathematical Objects Relating to Charter Members of the MAA  Geometric Models  A. Harry Wheeler
Geometric Models  A. Harry Wheeler
David Lindsay Roberts has already examined the career of A. Harry Wheeler (18731950). He has shown how Wheeler, a high school teacher and geometric model maker in Worcester, Massachusetts, moved between making geometric models with his students in the classroom, to attempting graduate work at Clark University, to teaching briefly as an adjunct at Brown University and Wellesley College. He remained a high school teacher in Worcester during his forays at Brown and Wellesley. He also corresponded with the dwindling number of research mathematicians – most notably H.S.M. Coxeter of the University of Toronto – who shared his interest in polyhedra.

Like all of those discussed here, Wheeler was a charter member of the MAA. He also joined the National Council of Teachers of Mathematics when it formed in 1920, and served on the Executive Committee of that organization during its first two years. However, he neither posed nor solved problems for the Monthly and was not terribly interested in publications of any sort. He apparently did not retain a long membership in either the MAA or the NCTM. Though not a research mathematician, Wheeler did join the American Mathematical Society in about 1923. He was planning to attend the International Congress of Mathematicians held in Toronto the next year. There he exhibited geometric models, an activity dear to his heart. Wheeler would remain a member of the AMS for twentyseven years, until his death.

Geometric Model of a Deltahedron (Third Stellation of the Icosahedron) by A. Harry Wheeler and His Students
 Description
 Polyhedra in which all faces are equilateral triangles are called deltahedra. The regular tetrahedron, octahedron, and icosahedron are the simplest deltahedra. It also is possible to replace each face of a regular dodecahedron with a “dimple” having five equilateral triangles as sides. This is a model of such a surface. It also may be considered as one of the polyhedra formed by extending the sides of—or stellating—a regular icosahedron.
 This deltahedron is folded from paper and held together entirely by hinged folds along the edges. Fifteen of the sixty faces have photographs of students of A. Harry Wheeler at North High School in Worcester, Massachusetts. All are boys. Another face reads: 1927 (/) Stanley H. Olson. A seventeenth face reads: Royal Cooper. Cooper is also shown on one of the sides with a photograph. There is a photograph of Lanley S. Olson, but not Stanley H. Olson. Yet another face of the model has a pencil mark that reads: June – 1927.
 Compare MA.304723.038, MA.304723.214, MA.304723.224, and MA.304723.308.
 Reference:
 Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 48.
 Location
 Currently not on view
 date made
 1927
 ID Number
 1979.0102.308
 accession number
 1979.0102
 catalog number
 1979.0102.308
 Data Source
 National Museum of American History

Geometric Model of a Regular Icosahedron by A. Harry Wheeler or One of His Students
 Description
 Greek mathematicians knew in ancient times that there are only five polyhedra that have identical faces with equal sides and angles. These five regular surfaces, called the Platonic solids, are the regular tetrahedron (four equilateral triangles as sides), the cube (six square sides), the regular octahedron (eight equilateral triangles as sides), the regular dodecahedron (twelve regular pentagons as sides) and the regular icosahedron (twenty equilateral triangles as sides). This is an early 20thcentury model of a regular icosahedron. The sides are covered with sateen and brocade fabrics of various designs and colors, in the style of late 19thcentury piece work. Catch stitches are along the edges.
 The model is unsigned, but associated with the Worcester, Massachusetts, schoolteacher A. Harry Wheeler. Wheeler taught undergraduates at Wellesley College, a Massachusetts women’s school, from 1926 until 1928. It is possible that one of his students there made the model.
 Reference:
 Judy Green and Jeanne LaDuke, Pioneering Women in American Mathematics: The Pre1940 PhD’s, Providence: American Mathematical Society, 2009, p. 21.
 Location
 Currently not on view
 date made
 ca 1926
 ID Number
 1979.0102.188
 accession number
 1979.0102
 catalog number
 1979.0102.188
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler or One of His Students, Snub Cube
 Description
 The Archimedean solids are polyhedra with regular polygons for sides and edges of equal length. For example, the faces of this surface are six squares and thirtytwo equilateral triangles. It is called a snub cube. The model is made from plastic. A. Harry Wheeler assigned this model the number 16, and referred to it as Archimedean solid XI. The model is undated and has no signature.
 Compare MA.304723.059 (plastic), MA.304723.060 (plastic), and MA.304723.061 (paper).
 References:
 Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 31.
 A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
 Location
 Currently not on view
 date made
 19151945
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.060
 accession number
 304723
 catalog number
 304723.060
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Snub Dodecahedron
 Description
 The Archimedean solids are polyhedra with regular polygons for sides and edges of equal length. For example, the faces of this surface are twelve regular pentagons and eighty equilateral triangles. It is called a snub dodecahedron. The model is cut and folded from paper. A mark on two faces reads: XII (/) 17. A. Harry Wheeler (/) Nov.1.1931 (/) Pat. 1292188. A paper sticker glued to another side reads: 17. Wheeler assigned the model the number 17, and referred to it as Archimedean solid XII.
 References:
 Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 32.
 A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
 Location
 Currently not on view
 date made
 1931
 patentee
 Wheeler, Albert Harry
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.062
 accession number
 304723
 catalog number
 304723.062
 Data Source
 National Museum of American History

Geometric Model by R. Anderson, a Student of A. Harry Wheeler, Great Rhombicuboctahedron
 Description
 The Archimedean solids are polyhedra with regular polygons for sides and edges of equal length. For example, this 26faced model has twelve square sides, eight hexagonal sides, and six octagonal sides. The surface is called a truncated cuboctahedron, a rhombitruncated cuboctahedron, or a great rhombicuboctahedron.
 Archimedean solids were known to the Hellenistic Greek mathematician Archimedes and studied by the 17thcentury mathematician and astronomer Johannes Kepler. This particular example was made from balsa wood by A. Harry Wheeler’s student R. Anderson, and is dated April 15, ’38. It is number 18 in Wheeler’s listing of models.
 For other examples of models of this surface, see MA.304723.063 (plastic) and MA.304723.064 (paper).
 References:
 Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 29.
 A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
 Location
 Currently not on view
 date made
 1938
 teacher of maker
 Wheeler, Albert Harry
 maker
 Anderson, R.
 ID Number
 MA.304723.065
 accession number
 304723
 catalog number
 304723.065
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Great Stellated Dodecahedron
 Description
 A stellation of a regular polyhedron is a polyhedron with faces formed by extending the sides of the faces of the regular polyhedron. For example, if one extends the sides of a regular pentagon, one can obtain a fivepointed star or pentagram. Considering the union of the twelve pentagrams formed from the twelve pentagonal faces of a regular dodecahedron, one obtains this surface, known as a great stellated dodecahedron. It also could be created by gluing appropriate triangular pyramids to the faces of a regular icosahedron – there are a total of sixty triangular faces.
 The great stellated dodecahedron was published by Wenzel Jamnitzer in 1568. It was rediscovered by Johannes Kepler and published in his work Harmonice Mundi in 1619. The French mathematician Louis Poinsot rediscovered it in 1809, and the surface and three related stellations are known as a KeplerPoinsot solids.
 This white plastic model of a great stellated dodecahedron is marked on a paper sticker attached to one side: 43 (/) DIV. A. Harry Wheeler assigned the model number 43 in his scheme, and considered it as the fourth species of a dodecahedron.
 Compare MA.304723.084, MA.304723.085, 1979.0102.016, and 1979.0102.253.
 References:
 Wenzel Jamnitzer, Perspectiva Corporum Regularium, Nuremberg, 1568.
 Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 40.
 A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
 Location
 Currently not on view
 date made
 19151945
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.084
 accession number
 304723
 catalog number
 304723.084
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Union of Four Cubes
 Description
 Massachusetts high school teacher and model maker A. Harry Wheeler built several geometric models that represented the union of intersecting cubes. He described this model as “four cubes wrapped about a regular octahedron.” The plastic pieces of the model are cut and glued to represent sections of red, yellow, black, and blue cubes.
 The model has no maker’s mark, but is numbered “94.” This corresponds to an entry in Wheeler’s handwritten catalog of his models. He also numbered the model H18, as it was based on a sixsided figure or hexahedron.
 Compare model MA.304723.117, which is the same surface, folded from paper.
 Reference:
 A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
 Location
 Currently not on view
 date made
 ca 1935
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.120
 accession number
 304723
 catalog number
 304723.120
 Data Source
 National Museum of American History

Geometric Model by A. Harry Wheeler, Union of Three Cubes
 Description
 Massachusetts high school teacher and model maker A. Harry Wheeler built several geometric models that represented the union of intersecting cubes, including this one. The plastic pieces of the model are cut and glued to represent sections of yellow, green, and turquoise cubes.
 The model has no maker’s mark, but corresponds to three other models numbered “95.” One of these was made by his student Lois M. Parker in April 1938—it is of balsa wood (see 1979.0102.040). A third version of the model is folded from paper and has only the model number on it (see 1979.0102.043). A fourth version of the model, also folded from paper, has notes in Wheeler’s hand indicating that the model is inscriptible in a cube, a rhombic dodecahedron, a tetrahedron, and an octahedron. These notes are dated 1931 and 1938.
 In Wheeler’s handwritten catalog of his models, this model is listed as 95 and also numbered H19, as it was based on a sixsided figure or hexahedron.
 Reference:
 A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
 Location
 Currently not on view
 date made
 ca 1935
 maker
 Wheeler, Albert Harry
 ID Number
 MA.304723.118
 accession number
 304723
 catalog number
 304723.118
 Data Source
 National Museum of American History

Geometric Model by Robert Chaffe, a Student of A. Harry Wheeler, Hyperbolic Paraboloid
 Description
 Suppose two opposite sides of a rectangle are joined by straight lines parallel to the other two sides. Lifting two opposite corners of the rectangle—and keeping the lines taut—one obtains a saddleshaped figure known as a hyperbolic paraboloid.
 This model of a hyperbolic paraboloid was made from balsa wood by Robert Chaffe, a high school student of A. Harry Wheeler in the class of 1937. It is likely that this person is Robert C. Chaffe (1918–1991) who was born in Connecticut, attended high school in Worcester, Massachusetts, graduated from the Worcester Polytechnic Institute in 1942, and seems to have spent his career as a salesman and sales engineer in Worcester and nearby Auburn.
 References:
 Gerd Fischer, Mathematical Models, vol. 2, Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 3–4.
 U.S. Census records.
 Massachusetts city directories.
 Location
 Currently not on view
 date made
 1937
 teacher of maker
 Wheeler, Albert Harry
 maker
 Chaffe, Robert
 ID Number
 MA.304723.180
 accession number
 304723
 catalog number
 304723.180
 Data Source
 National Museum of American History

Model by Philip Malmberg, a Student of A.H. Wheeler, Cylinder Transformable into a Hyperboloid of One Sheet
 Description
 Joining points along the radius of two circles generates a family of straight lines. If the circles are at their maximum separation, the lines form a cylinder. When rotated, the circles approach and the surface becomes a hyperboloid of one sheet. Further rotation (not possible on this model) yields a double cone.
 String models with elegant brass frames sold for engineering and mathematics education sold from the nineteenth century (see 1985.0112.009). Philip Malmberg, a high school student of A. Harry Wheeler, made this inexpensive version of the surface. He used disks cut from a cardboard box, leftover spools from thread, a wooden dowel, a bit of wire, and thread. Census records indicate that Malmberg went on to work as a draftsman.
 For a photograph of Malmberg, see 1979.0102.308.
 Location
 Currently not on view
 date made
 1927
 associated dates
 19270223
 teacher of maker
 Wheeler, Albert Harry
 maker
 Malmberg, Philip
 ID Number
 MA.304723.501
 accession number
 304723
 catalog number
 304723.501
 Data Source
 National Museum of American History
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