#
Science & Mathematics

The Museum's collections hold thousands of objects related to chemistry, biology, physics, astronomy, and other sciences. Instruments range from early American telescopes to lasers. Rare glassware and other artifacts from the laboratory of Joseph Priestley, the discoverer of oxygen, are among the scientific treasures here. A Gilbert chemistry set of about 1937 and other objects testify to the pleasures of amateur science. Artifacts also help illuminate the social and political history of biology and the roles of women and minorities in science.

The mathematics collection holds artifacts from slide rules and flash cards to code-breaking equipment. More than 1,000 models demonstrate some of the problems and principles of mathematics, and 80 abstract paintings by illustrator and cartoonist Crockett Johnson show his visual interpretations of mathematical theorems.

"Science & Mathematics - Overview" showing 217 items.

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## Flexible Polyhedron

- Description
- The mathematician Leonard Euler once wrote,"A closed spatial figure allows no changes, as long as it is not ripped apart." Proving the "rigidity" of polyhedra was another matter. In 1813, Augustin-Louis Cauchy showed that a convex polyhedral surface is rigid if its flat polygonal faces are held rigid. In 1974, Herman Gluck proved that almost all triangulated spherical surfaces were rigid. However, in 1977 Robert Connelly of Cornell University found a counterexample, that is to say a flexible polyhedron. He built this model of such a surface some years later. It is made of cardboard and held together with duct tape. Two cutout plastic windows allow the viewer to observe changes when the polyhedron is flexed. The top section has 12 large faces and a six-faced appendage. The bottom section has 12 corresponding faces but no appendage.

- Location
- Currently not on view

- date made
- 1985

- maker
- Connelly, Robert

- ID Number
- 1990.0492.01

- accession number
- 1990.0492

- catalog number
- 1990.0492.01

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Crocheted Model of the Hyperbolic Plane

- Description
- This model of the hyperbolic plane was crocheted by the Latvian-born mathematician Daina Taimina in about 2002. Although called a model of a plane, it is not flat like a Euclidean plane and its lines are not straight. However, lines on any plane, Euclidean or hyperbolic, are still the shortest paths along the plane connecting two points.

- The distinguishing difference between a hyperbolic plane and a Euclidean plane is that on a hyperbolic plane there are infinitely many lines parallel to a given line through a given point not on the given line. In this model lines are shown in yellow. The given line is the one closest to the top of the photograph and the given point is where the four other lines meet. None of those four lines will ever meet the given line, so they are all parallel to it.

- On page 27 of her book,
*Crocheting Adventures with Hyperbolic Planes*, (Wellesley, MA: A. K. Peters, 2009), Taimina has a photograph of a similar model, with only three yellow lines through the given point. On page 28 she has another photograph of that model with the caption: “The red line is a common perpendicular to only two of these yellow lines.” That photograph illustrates that on a hyperbolic plane, just as on a Euclidean plane, there is only one line through a given point not on a given line that is perpendicular to the given line.

- Location
- Currently not on view

- date made
- 2002

- maker
- Taimina, Daina

- ID Number
- 2002.0394.01

- catalog number
- 2002.0394.01

- accession number
- 2002.0394

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## 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, Kenneth E. Behring Center

## Geometric Model by A. Harry Wheeler, Musical Polyhedron, after Moebius

- 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.

- 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

- Data Source
- National Museum of American History, Kenneth E. Behring Center

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

- 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

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## 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 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

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Geometric Model of a Deltahedron (also a Form of Stellated 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.

- 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, Kenneth E. Behring Center

## Model of a Cubic Cone with Nodal Line by Richard P. Baker, Baker #78

- Description
- This string model was constructed by Richard P. Baker, possibly before 1905 when he joined the mathematics faculty at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

- The typed part of a paper label on the bottom of the wooden base of this model reads: No. 78 (/) CUBIC CONE WITH NODAL LINE. Model 78 appears on page 7 of Baker’s 1931 catalog of models as “With nodal line” under the heading
*Cubic Cones*. It also appears in his 1905 catalog of one hundred models.

- Baker’s string models always represent a special type of geometric surface called a ruled surface. A ruled surface, sometimes called a scroll, is one that is swept out by a moving line. This model shows two ruled surfaces. One of these surfaces is swept out by any of the threads connecting the curved vertical wooden sides of the model. The other ruled surface is swept out by any of the threads joining the curved horizontal piece of wood on the top of the model to the wooden base of the model. All the threads of this model pass through a point in the center of the model, which is the intersection of two special lines, one for each ruled surface.

- The special line for the surface joining the vertical sides is the line connecting the inflection points of the cubic curves, i.e. the points where the curve changes from concave upward to concave downward (for the curve y=x
^{3}, it would be at the origin). This line is horizontal and passes over the center of the base.

- The special line for the other curve is the vertical line going through the center of the base. It is formed by connecting the point where the upper curve crosses itself with the center of the base, which is also the point where the curve on the base crosses itself. A point of curve where the curve crosses itself is called a node, so all points of this vertical line are nodes and this is the nodal line of the surface.

- Location
- Currently not on view

- date made
- ca 1900-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.006

- accession number
- 211257

- catalog number
- 211257.006

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Model of a Quartic Scroll by Richard P. Baker, Baker #84

- Description
- This string model was constructed by Richard P. Baker, possibly before 1905, when he joined the mathematics faculty at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

- The typed part of a paper label on the wooden base of this model reads: No. 84 Quartic Scroll, (/) with two nodal straight (/) lines. Model 84 appears on page 8 of Baker’s 1931 catalog of models as “
*Quartic Scroll*, with two nodal straight lines.” The equation of the model is listed as (x^{2}/((z - 1)^{2})) + (y^{2}/((z + 1)^{2})) = 1. It also appears in his 1905 catalog of one hundred models.

- Baker’s string models always represent a special type of geometric surface called a ruled surface. A ruled surface, sometimes called a scroll, is one that is swept out by a moving line. This model is swept out by any of the yellow threads joining the elliptically shaped horizontal piece of wood on the top of the model to the wooden base of the model.

- In addition to the yellow threads of the model, there are two horizontal red threads that run from the rods at near the edge of the base and are parallel to the lines connecting the midpoints of the opposite sides of the square of surface of the base. There is a segment of each of these red threads for which each point meets two different lines of the model and the points of these segments are called double points, or nodes, of the surface. Thus these line segments are the two nodal lines of the model. The horizontal plane z = 1 intersects the model at the upper horizontal thread, while the horizontal plane z = -1 intersects it at the lower horizontal thread. When z=1, the points of intersection are (0,y,1) for y between -2 and 2. When z=-1, the points of intersection are (x,0,-1) for x between -2 and 2. Thus the nodal lines are the line segments connecting (0,-2,1) to (0,2,1) and (-2,0,-1) to (2,0,-1).

- When z = 0 the equation of the surface becomes x
^{2}+ y^{2}= 1, so the horizontal plane z = 0 intersects the model at the unit circle with center at the origin. For any other value of z, the equation of the surface is of the form (x^{2}/a^{2}) + (y^{2}/b^{2}) = 1, where a does not equal b. This is the standard form for the equation of an ellipse.

- Location
- Currently not on view

- date made
- ca 1915-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.010

- accession number
- 211257

- catalog number
- 211257.010

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Geometric Model by Richard P. Baker, Axial Pencil and Transversals

- Description
- This geometric model was constructed by Richard P. Baker when he was Associate Professor of Mathematics at the University of Iowa, most likely some time before 1930. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

- The typed part of a paper label taped to this wire model reads: No. 235 (/) Axial pencil (/) transversals. Model 235 appears on page 13 of Baker’s 1931 catalog of models as “
*Axial Pencil*and transversals.”

- An axial pencil is a set of planes that pass through a line, called the axis of the pencil. The most obvious of the axial pencils represented in the model has its axis as the short rod parallel to the long rods of the base of the model. Each long rod of the base produces a plane that goes through the axis. One of the planes includes the yellow rods, the other the pink rods.

- The transversals in the title probably refer to the short rods connecting the long rods of the base. These short rods will be referred to as base transversals. There are two more rods parallel to these transversals, one at the top of the model and one, much shorter, slightly above the axis of the pencil described above, each of which is the axis of another axial pencil represented in the model. The colors of some of the rods are no longer very clear. However, it appears as if the original coloring would have been useful in describing these two additional axial pencils.

- Each base transversal meets two vertical rods and produces a plane that goes through one of the other two transversals. The triangles formed by each of the center three base transversals meet the upper non-base transversal. Thus the plane of each of those three triangles is a part of the axial pencil with axis the transversal at the top of the model. The triangles formed by each of the outer two base transversals meet the very short non-base transversal. Thus the plane of each of those triangles is part of the axial pencil with axis the very short transversal.

- Location
- Currently not on view

- date made
- 1915-1930

- maker
- Baker, Richard P.

- ID Number
- MA*211257.041

- accession number
- 211257

- catalog number
- 211257.041

- Data Source
- National Museum of American History, Kenneth E. Behring Center