# Geometric Models - A. Harry Wheeler

• Contents

David Lindsay Roberts has already examined the career of A. Harry Wheeler (1873-1950). 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.

 A. H. Wheeler and Student

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 twenty-seven years, until his death.

### Geometric Model of a Deltahedron (Third Stellation of the Icosahedron) by A. Harry Wheeler and His Students

Polyhedra in which all faces are equilateral triangles are called deltahedra. The regular tetrahedron, octahedron, and icosahedron are the simplest deltahedra.
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

### Geometric Model of a Regular Icosahedron by A. Harry Wheeler or One of His Students

Greek mathematicians knew in ancient times that there are only five polyhedra that have identical faces with equal sides and angles.
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 20th-century model of a regular icosahedron. The sides are covered with sateen and brocade fabrics of various designs and colors, in the style of late 19th-century 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 Pre-1940 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

### Geometric Model by A. Harry Wheeler or One of His Students, Snub Cube

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 thirty-two equilateral triangles. It is called a snub cube. The model is made from plastic. A.
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 thirty-two 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
1915-1945
maker
Wheeler, Albert Harry
ID Number
MA.304723.060
accession number
304723
catalog number
304723.060

### Geometric Model by A. Harry Wheeler, Snub Dodecahedron

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.
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

### Geometric Model by R. Anderson, a Student of A. Harry Wheeler, Great Rhombicuboctahedron

The Archimedean solids are polyhedra with regular polygons for sides and edges of equal length. For example, this 26-faced model has twelve square sides, eight hexagonal sides, and six octagonal sides.
Description
The Archimedean solids are polyhedra with regular polygons for sides and edges of equal length. For example, this 26-faced 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 17th-century 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

### Geometric Model by A. Harry Wheeler, Great Stellated Dodecahedron

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 five-pointed star or pentagram.
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 five-pointed 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 Kepler-Poinsot 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
1915-1945
maker
Wheeler, Albert Harry
ID Number
MA.304723.084
accession number
304723
catalog number
304723.084

### Geometric Model by A. Harry Wheeler, Union of Four Cubes

Massachusetts high school teacher and model maker A. Harry Wheeler built several geometric models that represented the union of intersecting 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 six-sided 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

### Geometric Model by A. Harry Wheeler, Union of Three Cubes

Massachusetts high school teacher and model maker A. Harry Wheeler built several geometric models that represented the union of intersecting cubes, including this one.
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 six-sided 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

### Geometric Model by Robert Chaffe, a Student of A. Harry Wheeler, Hyperbolic Paraboloid

Suppose two opposite sides of a rectangle are joined by straight lines parallel to the other two sides.
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 saddle-shaped 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

### Model by Philip Malmberg, a Student of A.H. Wheeler, Cylinder Transformable into a Hyperboloid of One Sheet

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.
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 02 23
teacher of maker
Wheeler, Albert Harry
maker
Malmberg, Philip
ID Number
MA.304723.501
accession number
304723
catalog number
304723.501

### Geometric Model by Dick Holl, a Student of A.Harry Wheeler, Dodecadodecahedron

In this model, the twelve faces of a dodecahedron are replaced by twelve stars. Below and parallel to each of the stars is a regular pentagon. Only five rhombuses from the corners of a pentagon show beneath each star.
Description
In this model, the twelve faces of a dodecahedron are replaced by twelve stars. Below and parallel to each of the stars is a regular pentagon. Only five rhombuses from the corners of a pentagon show beneath each star. The pentagons intersect one another.
This model is cut and folded from tan paper. One star face and one three-sided dimple are shaded with pencil. A paper sticker gives Wheeler’s number for the model, 340. Another mark reads: Holl (/) Mar 23, 1934. A third mark reads: Dick (/) Holl (/) 34.
Richard B. Holl of Worcester, Massachusetts, is listed in U.S. Census records for 1920 and 1930, with a birth date of about 1917. Richard Bernhardt Holl (1916–1985) is listed in Masonic records as born in Worcester, Massachusetts, and dying in Waltham, Massachusetts. Richard Bernhardt Holl also is listed in the 1934 yearbook of North High School in Worcester, and his daughter confirms that he graduated from high school in 1934. She reports that he then attended Tufts College in Medford Massachusetts, majoring in physics and graduating with an engineering degree. Following his education, he went to work for Raytheon Company, working there for his entire career. WhileHoll was active in many social and community affairs, much of his work was classified and therefore not available for public presentation.
Compare MA.304723.186 and MA.304723.163.
References:
Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p.112.
A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
Ancestry Library Edition, accessed January 23, 2015.
Personal Communication, Liz (Holl) Michalenoick, May 8, 2020.
Location
Currently not on view
date made
1934
teacher of maker
Wheeler, Albert Harry
maker
Holl, Dick
ID Number
MA.304723.186
accession number
304723
catalog number
304723.186

### Geometric Model by A. Harry Wheeler, Ninth Stellation of the Icosahedron

This stellation of an icosahedron has twelve spikes, each with five sides. The paper model is cut and folded. Two paper tags give Wheeler’s number for the model, 368. A pencil mark reads: April (/) 8 (/) 1919. Another pencil mark reads: 366’8 (/) (368).
Description
This stellation of an icosahedron has twelve spikes, each with five sides. The paper model is cut and folded. Two paper tags give Wheeler’s number for the model, 368. A pencil mark reads: April (/) 8 (/) 1919. Another pencil mark reads: 366’8 (/) (368). Wheeler listed this model among his icosahedra as I5. Wenninger refers to it as the ninth stellation of the icosahedron.
References:
Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 55.
A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
Location
Currently not on view
date made
1919 04 08
maker
Wheeler, Albert Harry
ID Number
MA.304723.204
accession number
304723
catalog number
304723.204

### Geometric Model by A. Harry Wheeler, Stellation of the Icosadodecahedron, Described by Wheeler as a Stellated Icosahedron

Extending the sides of polygons can produce a variety of complex polyhedra, including this one. It has twelve relatively sharp points, each with five triangular edges converging. These points alone would form a twelve-pointed star that is itself a stellation of the icosahedron.
Description
Extending the sides of polygons can produce a variety of complex polyhedra, including this one. It has twelve relatively sharp points, each with five triangular edges converging. These points alone would form a twelve-pointed star that is itself a stellation of the icosahedron. In addition, there are twenty three-sided points—not as sharp. These faces alone are like those of a great stellated dodecahedron. Wenninger refers to this model as the thirteenth stellation of the icosidodecahedron.
The model is cut and folded from paper. Wheeler assigned it the number 371, and classified it as icosahedron I10. He also pointed out that the inner faces could be considered as parts of five intersecting tetrahedra.
A related model (304723.204) is dated 1919, hence the approximate date assigned to this model.
References:
Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 55, 73, 88.
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 1920
maker
Wheeler, Albert Harry
ID Number
MA.304723.205
accession number
304723
catalog number
304723.205

### Geometric Model by A. Harry Wheeler, Fifteenth Stellation of the Icosahedron

A stellation of a regular polyhedron is a polyhedron with faces formed by extending the sides of the faces of the regular polyhedron. Extending the triangular sides of an icosahedron can produce a variety of complex polyhedra, including this one.
Description
A stellation of a regular polyhedron is a polyhedron with faces formed by extending the sides of the faces of the regular polyhedron. Extending the triangular sides of an icosahedron can produce a variety of complex polyhedra, including this one. The surface has sixty short three-sided spikes. These meet in groups of three—each meeting point might be considered as the vertex of a circumscribing regular dodecahedron.
The model is cut and folded from paper. It is Wheeler’s model 382, and number I21 in his series of icosahedra. Wenninger calls the surface the fifteenth stellation of the icosahedron.
References:
Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 62.
A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
Location
Currently not on view
date made
1927-05-12
maker
Wheeler, Albert Harry
ID Number
MA.304723.197
accession number
304723
catalog number
304723.197

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

In the late 1930s and early 1940s, A. Harry Wheeler took great interest in polyhedra with interpenetrating sides, such as had been discussed by the German mathematician August F. Moebius. In this example, each of the two like-colored quadrilaterals (e.g.
Description
In the late 1930s and early 1940s, A. Harry Wheeler took great interest in polyhedra with interpenetrating sides, such as had been discussed by the German mathematician August F. Moebius. In this example, each of the two like-colored quadrilaterals (e.g. the two yellow sides) on the top pass through the model and appear as a white quadrilateral on the bottom. These three figures thus contribute only one side to the polygon.
A mark on the model reads: 695. This was Wheeler’s number for the model. Models MA.304723.413, MA.304723.397, and MA.304723.398 fit together. Model MA.304723.409 is a compound of four models like MA.304723.413.
Reference:
Kurt Reinhardt, “Zu Moebius’ Polyhedertheorie,” Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Classe, 37, pp. 106-125. Wheeler referred to this article.
Location
Currently not on view
date made
ca 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.413
accession number
304723
catalog number
304723.413

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

This plastic geometric model consists of four identical figures, each joined to the other three by a short edge. The middle is a large hole.
Description
This plastic geometric model consists of four identical figures, each joined to the other three by a short edge. The middle is a large hole. Wheeler envisioned this model as four examples of his model 695 (MA.304723.413), arranged in alternating quadrants of a regular octahedron.
The Wheeler’s number for the model, 699, is written on one face. Wheeler classified the model as a one-sided polyhedron.
Location
Currently not on view
date made
ca 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.409
accession number
304723
catalog number
304723.409

### 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.
Description
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.
Location
Currently not on view
date made
ca 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.416
accession number
304723
catalog number
304723.416

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

This self-intersecting polyhedron has twelve trapezoidal faces (made out of light turquoise plastic) and twelve triangular faces (made out of dark turquoise plastic).
Description
This self-intersecting polyhedron has twelve 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.
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 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.410
accession number
304723
catalog number
304723.410

### 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.
Description
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.
Location
Currently not on view
date made
ca 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.404
accession number
304723
catalog number
304723.404

### Geometric Model by A. Harry Wheeler, Model for the Pythagorean Theorem

This is A. Harry Wheeler’s visual representation of the Pythagorean Theorem. The wooden base has a right triangle at the center with square holes cut out along each side of the triangle.
Description
This is A. Harry Wheeler’s visual representation of the Pythagorean Theorem. The wooden base has a right triangle at the center with square holes cut out along each side of the triangle. The two vertices of the triangle next to the long side are cut out so that metal filings can run from the smaller squares into the larger one. A clear plastic top and bottom hold the filings within the model. In theory, one can fill either the large square or the two smaller squares with filings—according to the Pythagorean Theorem, the area of the large square is the sum of the area of the small ones.
The model is unsigned and undated.
Location
Currently not on view
date made
1915-1945
maker
Wheeler, Albert Harry
ID Number
MA.304723.480
catalog number
304723.480
accession number
304723

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

This plastic model is in roughly the shape of a torus. All the faces are triangles. Twelve are turquoise and twelve white, with the colors alternating. The surface has thirty-six edges and twelve vertices.
Description
This plastic model is in roughly the shape of a torus. All the faces are triangles. Twelve are turquoise and twelve white, with the colors alternating. The surface has thirty-six edges and twelve vertices. This would give an Euler characteristic of vertices – edges + faces = 12 – 36 + 24 = 0, which is appropriate for a surface with one hole. Four of the white triangles are numbered. Face 1 also has a tag that reads: 739. Another tag on this side reads: A. Harry Wheeler. Another mark on this side reads: MP. Face 2 is a congruent white triangle on the lower left side, face 3 is a white triangle on the bottom of the back, and face 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, a paper version of the same size with museum number MA.304723.508 and a larger plastic version in yellow and white with museum number MA.304723.404. Musical notes are not indicated on this larger version of the model.
Wheeler’s model shows relationships between the twelve notes in a chromatic musical scale. In the Germanic system, going up by semitones, these are C, C#, D, D#, E, F, F#, G, G#, A , B, H (no flats are used). On a piano, C#, D#, F#, G#, and B would be black keys and the rest white.
If one raises pitches by a major third (four semitones) and keeps going until the original note returns (one octave higher), there are four cyclic sequences:
C E G# C, C# F A C#, D F# B D, D# G H D#
Each note of the chromatic scale appears in exactly one of these sequences.
Similarly, if one raises pitches by a minor third (three semitones), there are three cyclic sequences, each one note longer:
C D# F# A C, C# E G B C#, D F G# H D
Again, each note of the chromatic scale appears in one sequence.
Since three and four are divisors of twelve, the sequences of major and minor thirds all take place within one octave. The third musical interval studied is the perfect fifth, consisting of seven semitones. Since seven and twelve are relatively prime, raising the pitch by a fifth produces one multi-octave cycle:
C G D A E H F# C# G# D# B F C
Moebius and Wheeler sought to label the twelve vertices of the torus with notes of the chromatic scale in such a way that edges and triangles represent interesting musical relationships. Recall that two of the most common musical chords are the major triad (such as C E G) and the minor triad (such as C D# G). Any note of the chromatic scale can be the low note in a major triad or a minor triad, making a total of twenty-four triads, which are to be paired up with the twenty-four triangles of the model. The blue triangles of the model represent major triads and the white triangles represent minor triads.
In a major triad, the low and middle note are a major third apart and the middle and high note are a minor third apart, making the low and high note a perfect fifth apart. In a minor triad, the low and middle note are a minor third apart, and the middle and high note are a major third apart, again making the low and high note a perfect fifth apart. It follows that the thirty-six edges in the model need to be divided into three groups of twelve, one group representing a minor third, one group a major third, and the last group a perfect fifth. Each vertex should be incident to two edges of each type, and opposite edges should be of the same type.
We now discuss how the cycles of major thirds, minor thirds, and fifths discussed above are situated on the torus. For a topologist, one of the most significant features of a torus is that there are simple closed curves that cannot be shrunk to a point without leaving the torus. The four edge cycles representing major triads are of this type; they are commonly called meridians of the torus. (There are three edges in each cycle, but they do not bound a triangle on the torus.) The three cycles of minor thirds go the other way around the torus. The cycle of perfect fifths wraps itself around the torus in one continuous band that appears to form a trefoil knot in three-space.
Suppose the model is cut along the four meridians representing major triads (that is to say, cut into four parts at the corners). It is divided into four shapes, each with a six triangles around the edge in a zigzag pattern (an anti-prism). Gluing a triangle onto the top and bottom of a set of triangles would produce an octahedron. Thus the model can be thought of as four octahedra glued together in a ring.
Compare 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 for this model are labeled in Wheeler’s hand and dated July 1939.
References:
A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
August F. Moebius, Gesammelte Werke, vol. 2, ed. F. Klein, Leipzig: S. Hirzel, 1886, pp. 553–554.
Location
Currently not on view
date made
ca 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.405
accession number
304723
catalog number
304723.405

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