#
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 43 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

## Adjustable String Model

- Description
- From the early nineteenth century, mathematicians and engineers have studied surfaces generated by motion. The gold threads of this model form a cylinder, the red ones a double cone. Rotating the top circle of the frame twists the gold threads and untwists the red ones, forming surfaces called hyperboloids. The blue threads, which initially lie in a plane, become a hyperbolic paraboloid. This model was made in Germany and exhibited at the Columbian Exposition, the world's fair held in Chicago in 1893. It came to the Smithsonian from the mathematics department of Wesleyan University in Connecticut.

- Location
- Currently not on view

- Currently not on view

- Date made
- 1893

- maker
- Brill, L.

- ID Number
- 1985.0112.009

- accession number
- 1985.0112

- catalog number
- 1985.0112.009

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

## Model of An Oblong or Rectangle, Ross Surface Form #3

- Description
- In 1891, William Wallace Ross (1834–1906), the superintendent of schools in Fremont, Ohio, published a set of “dissected surface forms and geometrical solids” for teaching practical geometry and measurement in schools and colleges. He also prepared a manual that describes their use. Ross extended earlier work of Albert H. Kennedy, including a much larger number of surfaces. His models would be distributed at least as late as 1917, when they were listed in the catalog of the Atlas School Supply Company of Chicago, Illinois.

- In his manual, Ross listed eighteen “surface forms”, eighteen solids or volumes, and the five Platonic or regular solids. By the time of the 1917–1918 catalog, a set of the model reportedly contained fifty pieces. The Smithsonian collections include thirteen of the surface forms, ten of which correspond to objects in the 1891 list. They also contain all or part of twelve of the solid forms, at least five of which correspond to the 1891 list.

- This is the second of Ross’s surface forms, a rectangle (or, in Ross’s language, an oblong) that measures 6 inches by 1 inch. The first surface form was a square one inch on a side. Taking the area of this square to be one square inch, students were to observe that the area of the rectangle was six square inches. A paper label attached to the model reads: Oblong 1x6.

- Compare models 1985.0112.190 through 1985.0112.202.

- References:

- W. W. Ross,
*Mensuration Taught Objectively with Lessons on Form . . . Manual for the Use of the Author’s Dissected Surface Forms and Geometrical Solids*, Fremont, Ohio, 1891.

- Atlas School Supply Company,
*Catalog No. 39 1917-18*, Chicago, Illinois, 1917, p. 86.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.190

- accession number
- 1985.0112

- catalog number
- 1985.0112.190

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

## Model of a Rectangle or Oblong, Ross Surface Form #2

- Description
- This is the third in a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The model is a 6 inch by 4 inch rectangle, divided into 24 one inch by one inch squares. A paper label attached to the model reads: Oblong 4x6.

- Comparing its area to that of a 6 inch by 1 inch rectangle (1985.0112.191), Ross noted that the area was four times 6 square inches, or 24 square inches. He generalized to argue that the area of a rectangle equaled the number of square units corresponding to the product of the length times the breadth.

- Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.191.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.191

- accession number
- 1985.0112

- catalog number
- 1985.0112.191

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

## Model of a Rectangle Bisected into Two Right Triangles, Ross Surface Form #8

- Description
- This is the eighth in a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The unpainted rectangular wooden model is bisected along a diagonal. A paper label pasted to the model reads: Oblong 4x6 Bisected. According to Ross, this model demonstrates that a right-angled triangle with unequal sides adjacent to the right angle has half the area of a rectangle.

- Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.191.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.192

- accession number
- 1985.0112

- catalog number
- 1985.0112.192

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

## Model of a Dissected Trapezoid, Ross Surface Form #6

- Description
- This is the sixth in a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The unpainted rectangular wooden model is cut into two pieces at one corner. It may be arranged so that the pieces form either a rectangle or a trapezoid. A paper label attached to the model reads: Dissected Trapezoid 5x7.

- Ross argued that the area of the trapezoid equaled half the sum of its parallel sides, multiplied by its breadth.

- Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.191.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.193

- accession number
- 1985.0112

- catalog number
- 1985.0112.193

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

## Dissected Polygon, Probably a Ross Surface Form

- Description
- This unpainted wooden model consists of two doweled pieces that can be arranged as a quadrilateral. The model is incomplete. It resembles other Ross surface forms.

- Compare models 1985.0112.190 through 1985.0112.202, especially 1985.0112.193. For further information about Ross models, including references, see 1985.0112.191.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.194

- accession number
- 1985.0112

- catalog number
- 1985.0112.194

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

## Rectangle Transformable Into an Obtuse Triangle, Probably a Ross Surface Form

- Description
- This is apparently is one in a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The three doweled pieces of this unpainted wooden model can be arranged either as a rectangle or as an obtuse-angled triangle.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.195

- accession number
- 1985.0112

- catalog number
- 1985.0112.195

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

## Dissected Rhomboid, Ross Surface Form #5 (incomplete)

- Description
- This is the fifth in a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The unpainted wooden model is divided into two pieces, with the smaller piece missing.

- With the smaller piece, the model could be arranged either as a parallelogram or a rectangle. A paper label attached to the model reads: Dissected Rhomboid 4x6.

- Ross argued that the parallelogram (or, in his terminology, rhomboid), like the rectangle, was the product of its length and its altitude.

- Compare models 1985.0112.190 through 1985.0112.202.

- For further information about Ross models, including references, see 1985.0112.191.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.196

- accession number
- 1985.0112

- catalog number
- 1985.0112.196

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