This cut and taped tan paper model shows two trihedral angles with the same center and radius but differeng angles facing opposite directions. The radii defining the upper triangle appear to be perpendicular to the lower triangle. The vertices of the upper triangle are labeled A, B, and C and those of the lower triangle are labled A", B", and C".
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.
This cut and glued transparent plastic model shows two perpendicular spherical discs which might be considered to be representations of the equator and a meridian disc of a sphere. Another half-disc also is perpendicular to the equator and rotates on the line joining the poles. Two symmetric congruent spherical triangles are on the same side of the model as the half-disc and pass through what might be considered the North Pole. A sticker on the model reads: C, S, SA. A mark next to the reads: PT.
For the pattern for this model, see 1979.3002.084. It is dated 1943.
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.
This bow was made by Albert H. Karr in Kansas City, Missouri, around 1940-1945. It is made of Pernambuco wood, with a plastic frog, and horsehair. The bow is stamped:
A.H.KARR
Albert Homer Karr (1885-1971) was an American bow and violin maker (1885-1971). For most of his career, Karr was the proprietor of musical instrument shop in Kansas City, Missouri. In addition to repairing and selling other instruments, Karr made over 1,300 violins during his career and several dozen handmade violin bows. During WWII, Karr was contracted by the U. S. government to produce quality student bows.
The Violinist magazine for January 1921 featured an article about Albert H. Karr as well as an advertisement of his shop:
ALBERT H. KARR Exclusive Violin Shop 306 East Tenth Str., Kansas City, Missouri.
The Albert H. Karr Handmade Violins, finest imported wood, sent at my expense on ten days’ trial to responsible parties. Large collections of old Violins including a Stradi- varius, a Guarnerius, an Amati, a Villaume and a Lupot.
One of the finest equipped shops in the United States for repair and adjustment of fine old instruments. Mr. Karr attends to this work per- sonally. All work guaranteed. Correspondence invited.
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.
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.
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.
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.
One adjustable tan paper spherical triangle is within a second one, both being on the same sphere. A pattern for this model indicates that it is a model for supplementary trihedral angles. The pattern is dated August 6, 1945, hence the date assigned to the model.
This cut and folded tan paper model consists of two discs and a ring, fit together to form lunes and spherical triangles. A sticker on the model reads: ST-LU*. Another sticker reads: Pat 1,192,483. Below this a sticker reads: A. Harry Wheeler.
Compare MA.304723.667 and MA.304723.670. For a pattern, see 1979.3002.095. Several patterns stored with this one are from 1945.
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.
This small plastic mode consists of three quadrants of a disc with a common center glued together at right angles to form a spherical triangle with three right angles. Such spherical triangles are called trirectangular. A sticker on the model reads: TR.ST.
This cut and folded tan paper model consists of two discs and a ring, fit together to form lunes and spherical triangles. A sticker on the model reads: ST-LU*.
Compare 304723.667 and 304723.670. For a pattern, see 1979.3002.095.
During World War II, A. Harry Wheeler made several models relating to spherical trigonometry. This one shows four quadrants of the celestial globe. One represents the equator. The other three pass are perpendicular to the equator through the pole. One passes through point T (Tokyo), another through point H (Honolulu), and the third through point S (San Francisco). A trihedral angle made from pieces of white plastic creates a spherical triangle joining these three points.
For the pattern for this model, see 1979.3002.084. The pattern is dated 1945.
Round solid wood cylinder with four concentric rings of drilled holes to hold test tubes. Used by Dr. Enders to roll test tubes containing tissue cultures of the polio viruses.