#
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 75 items.

Page 1 of 8

## 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

## 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

## 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

## Geometric Model by Richard P. Baker, Thermodynamic Surface for Water, Ice, and Steam

- Description
- In the 1870s physicists in Scotland and the United States began to make three-dimensional models of the thermal properties of matter. The height of this wire surface corresponds to the volume of ice, water and steam as this changes with pressure (pressure increases coming toward the front of the model) and temperature (temperature increases going to the right). The solid state – with relatively slow changes in volume with temperature and pressure, and low temperatures - is represented by the relatively flat surface on the left. The part of the board under it is marked S. When ice melts, the volume decreases until melting is complete. Water expands more rapidly with temperature than ice, so that the surface rises more rapidly from the middle of the front going right. The part of the board under this section of the surface is marked L. As temperature rises, water turns to steam, and the volume increases even more rapidly. The section of the board under this part of the surface is marked V for vapor.

- A paper sticker glued to the underside of the base reads: No. 253 (/) Water Steam and Ice.

- This is one of a series of nine models Baker made that relate to thermodynamic surfaces. It was designed during his years at the University of Iowa under the supervision of his German-born colleague Karl Eugen Guthe (1866–1915), who taught in the physics department there from 1905 until 1909. Baker’s correspondence indicates that copies of the model were purchased by Columbia University and by the University of Michigan. The model remained in Baker’s catalog as late as 1931. A card catalog in the Baker papers indicates that the model sold at one time for $7.50.

- This particular example of the model was on loan for exhibition at MIT from 1939 until the mid-1950s. It, along with the other models in accession 211257, came to the Smithsonian from MIT in 1956.

- References:

- Accession file 211257.

- J. Willard Gibbs, “A Method Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces,”
*Transactions of the Connecticut Academy*, 2, 1873, pp. 382–404. Gibbs refers to earlier work of the Scottish engineer James Thomson, who devised a surface for representing the pressure volume and temperature of carbonic acid and carbon dioxide.

- H. Randall to Baker, January 15, 1908 and Columbia University to Baker, August 6, 1918, Richard P. Baker Papers, University Archives, University of Iowa, Iowa City, Iowa.

- Richard P. Baker,
*Mathematical Models*, Iowa City, 1931, p. 18.

- Location
- Currently not on view

- date made
- ca 1905-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.045

- accession number
- 211257

- catalog number
- 211257.045

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

## Geometric Model by Richard P. Baker, Wire Model of Clebsch's Diagonal Surface

- Description
- This geometric model was constructed by Richard P. Baker, probably in the 1920s while he was Associate Professor of Mathematics 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 taped to this wire model reads: No. 365 (/) THE DOUBLE SIX FROM 367. Model 365 appears on page 10 of Baker’s 1931 catalog of models as “The double six from 367” under the heading
*Clebsch’ Diagonal Surface*. Model 367 is the first model in this section and is listed as a plaster cast model title “Tetrahedral symmetry.*24 finite lines*.” While model 367 is not among the models in the Smithsonian collections, the second model in this section, Model 366 (MA*211257.055) “The lines of 367,” is.

- The double six in the title of this model is a Schlaefli Double Sixes structure, named after 19th-century Swiss mathematician Ludwig Schlaefli. A double sixes structure consists of two sets of six lines that satisfy the following three properties: no lines in the same set intersect, each line is paired with a line of the other set which it does not intersect, and each line intersects the five lines in the other set with which it is not paired.

- In this model, the twelve wire rods represent the twelve lines. It appears as if some of the rods have been repainted so it is no longer possible to distinguish six colors. However, it is likely that the paired rods were originally painted the same color. Labeling one of the sets of rods 1 through 6 and the other 1′ through 6′ as shown in one of the images, one can see that the yellow rod 6′ meets (left to right) rods 2, 4, 3, 1, and 5 but does not meet the yellow rod 6. Similarly the (clearly repainted) tan rod 4 meets (bottom to top) rods 5′, 6′, 1′, 3′, and 2′ but does not meet the purple rod 4′.

- The surface on which this model is based, the Clebsch Diagonal Surface, is defined by the cubic equation x
^{3}+y^{3}+z^{3}+w^{3}+v^{3}=0 assuming x+y+z+w+v=0; it contains thirty six double-six structures.

- Location
- Currently not on view

- date made
- ca 1915-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.054

- accession number
- 211257

- catalog number
- 211257.054

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

## Model of a Riemann Surface by Richard P. Baker, Baker #405w

- Description
- This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics 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 mark "405 w" is carved into one edge of the wooden base of this model and the typed part of a paper label on the base reads: No. 405 z (/) Riemann surface : w
^{2}= z^{3}- z. Someone corrected the error on the label by hand, crossing out the z and inserting a w. Model 405w is listed on page 17 of Baker’s 1931 catalog of models as “w^{2}= z^{3}- z” under the heading*Riemann Surfaces*. This means that the model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w^{2}= z^{3}- z. Complex numbers are of the form x + yi for x and y real numbers and i the square root of –1. A complex plane is like the usual real Cartesian plane but with the horizontal axis representing the real part of the number and the vertical axis representing the imaginary part of the number. Riemann surfaces are named after the 19th-century German mathematician Bernhard Riemann.

- Baker explains in his catalog that the w after the number of the model indicates that the metal disks above the wooden base represent copies of a disk in the complex w-plane. These disks are called the sheets of the model. The painted part of the wooden base of the model represents a square in the complex z-plane with the point z = 0 at its center and the real axis along the line between the yellow and light green stripes.

- If w = ±
^{4}√ (4/27) or w = ±i^{4}√ (4/27), the equation w^{2}= z^{3}- z is satisfied by two distinct values of z. For the two real values of w, ±^{4}√ (4/27), z takes the values –1/√3 and 2/√3, and for the two imaginary values of w, ±i^{4}√ (4/27), z takes the values 1/√3 and –2/√3. These four points on the w-plane are called branch points of the model and for all other points on the w-plane the equation w^{2}= z^{3}- z is satisfied by three distinct values of z, each of which produces a different pair on the Riemann surface (if w = 0, the three distinct pairs on the Riemann surface are (0,0) and (±1,0). Thus there are three sheets representing the same disk in the complex w-plane and together they represent part of what is called a branched cover of the complex w-plane. The color of a region on a sheet is chosen with the aim of indicating the stripe on the base in which the z coordinate lies.

- For each sheet, the point at the center is w = 0 and the line lying over the real axis of the base is the real axis of the sheet. The two points marked on the top sheet are the two imaginary branch points, w = ±i
^{4}√ (4/27); the two marked on the bottom sheet are the two real branch points, w = ±^{4}√ (4/27); and all four branch points are marked on the middle sheet.

- The vertical surfaces between the sheets are not part of the Riemann surface but call attention to what are called branch cuts of the model, i.e. curves on a sheet that produce movement to another sheet. This movement occurs when meeting a branch cut while following a path of the inputs of w values into the equation. While the defining equation determines the branch points, the branch cuts are not fixed by the equation but, normally, each branch cut goes through two of the surface’s branch points or runs out to infinity. All of the branch cuts of this model run to infinity and are represented by the horizontal edges of the vertical surfaces.

- There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One has "405w" carved on the base, Baker no. 405wn (MA*211257.069), while two have "405z" carved, Baker no. 405z (MA*211257.070) and Baker no. 405zn (MA*211257.071). Baker carved a "w" or a "z" to indicate which variable is represented on the sheets of the model and added an n after the w or z to indicate that the sheets of the model are spheres.

- Location
- Currently not on view

- date made
- ca 1915-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.068

- accession number
- 211257

- catalog number
- 211257.068

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

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