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.

The top and bottom faces of this model are regular hexagons. Four side faces are squares. Two opposite sides have square pyramids extending from them.
Description
The top and bottom faces of this model are regular hexagons. Four side faces are squares. Two opposite sides have square pyramids extending from them. Thus the model has a total of eight triangular faces, four square faces, and two hexagonal faces.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.075
accession number
1978.1065
catalog number
1978.1065.075
This model resembles a pentagonal prism, but a square pyramid has replaced one of the square sides. Its faces are four equilateral triangles, four squares, and two regular pentagons.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
This model resembles a pentagonal prism, but a square pyramid has replaced one of the square sides. Its faces are four equilateral triangles, four squares, and two regular pentagons.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.072
accession number
1978.1065
catalog number
1978.1065.072
This painted and varnished paper object is the first in a series of models of convex polyhedra with regular faces constructed by Martin Berman.
Description
This painted and varnished paper object is the first in a series of models of convex polyhedra with regular faces constructed by Martin Berman. The faces are four regular triangles.
In 1970 Berman (1938-1984), a physicist at the Pittsburgh Energy Technology Center, constructed a set of models of regular-faced convex polyhedra. A polyhedron is said to be uniform if its faces are regular and its vertices are all alike (so that it has the same arrangement of polygons at each vertex). A polyhedron (uniform or not) is convex if a line segment joining any two of its points lies entirely on or inside it.
A polygon (convex or not) is regular if it is uniform and its faces are all alike. The regular convex polyhedra are the five Platonic solids, which have been known since classical Greece. The ancient Greek mathematician Euclid proved in his Elements of Geometry that there are only five Platonic solids. These are the regular tetrahedron (four sides that are equilateral triangles), the cube (six sides that are squares), the regular octahedron (eight sides that are equilateral triangles), the regular dodecahedron (twelve sides that are regular pentagons), and the regular icosahedron (twenty sides that are equilateral triangles).
The term "semiregular" is used to describe polyhedra that are uniform but not regular. The semiregular convex polyhedra include thirteen solids associated with another ancient mathematician, Archimedes. He lived after Euclid and worked in Syracuse on the Mediterranean island of Sicily. These objects are called Archimedean solids. There also are an infinite number of semiregular prisms. These have like regular polygons on the top and bottom and straight lines joining the vertices of these to form the square sides. A second infinite group of semiregular solids are called antiprisms. These also have like polygons for top and bottom, but twisted so that each vertex of one polygon is joined to two vertices of the other to form an equilateral triangle.
Besides the regular and semiregular solids, there are just ninety-two other convex polyhedra with regular faces. In 1966 the American mathematician Norman W. Johnson, a student of H. S. M. Coxeter at the University of Toronto in Canada, enumerated them. These polyhedra are sometimes called the Johnson solids. In 1969 the Russian Viktor A. Zalgaller offered a computer-based computational proof that Johnson had completed the enumeration of convex polyhedra with regular faces.
Berman made paper models of the Platonic solids, the Archimedean solids, the Johnson solids, a prism with a triangular base, and an antiprism with a square base, for a total of 112 models. He published photographs of these and diagrams for making them in 1971, identifying the models with the same names as those used by Johnson. Berman gave the models to the Smithsonian in 1978.
References:
Accession File
Martin Berman, "Regular-faced Convex Polyhedra," Journal of the Franklin Institute, 291, 1971, pp. 329-352. Includes illustrations of models and of nets for making them.
Norman W. Johnson, "Convex Polyhedra with Regular Faces," Canadian Journal of Mathematics, 18, 1966, p. 169-200;
Viktor A. Zalgaller, “Convex Polyhedra with Regular Faces,” in Seminars in Mathematics, V. A. Steklov Math. Inst., Leningrad, vol. 2, English translation: Consultants Bureau, New York, 1969.
Online discussion of Johnson solids on Wikipedia and on Wolfram MathWorld (both accessed November 11, 2015).
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.001
accession number
1978.1065
catalog number
1978.1065.001
In this model, one of the hexagonal faces of a truncated tetrahedron has on it four equilateral triangles and three squares.
Description
In this model, one of the hexagonal faces of a truncated tetrahedron has on it four equilateral triangles and three squares. The faces of the model are thus three regular hexagons, three squares, and eight equilateral triangles.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.085
accession number
1978.1065
catalog number
1978.1065.085
The faces of the model are twelve regular pentagons.For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
The faces of the model are twelve regular pentagons.
For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.005
accession number
1978.1065
catalog number
1978.1065.005
This model has six equilateral triangles for faces, arranged like two regular tetrahedra facing in opposite directions.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
This model has six equilateral triangles for faces, arranged like two regular tetrahedra facing in opposite directions.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.032
accession number
1978.1065
catalog number
1978.1065.032
The faces of the model are twenty equilateral triangles.For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
The faces of the model are twenty equilateral triangles.
For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.004
accession number
1978.1065
catalog number
1978.1065.004
This model resembles a regular dodecahedron, but two opposite faces have been replaced by pentagonal pyramids. The faces total ten equilateral triangles and ten regular pentagons.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
This model resembles a regular dodecahedron, but two opposite faces have been replaced by pentagonal pyramids. The faces total ten equilateral triangles and ten regular pentagons.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.079
accession number
1978.1065
catalog number
1978.1065.079
The top and bottom faces of this model are regular pentagons. The remaining faces are ten squares and thirty equilateral triangles.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
The top and bottom faces of this model are regular pentagons. The remaining faces are ten squares and thirty equilateral triangles.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.066
accession number
1978.1065
catalog number
1978.1065.066
The faces of this model are twelve regular pentagons, thirty squares, and twenty equilateral triangles.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
The faces of this model are twelve regular pentagons, thirty squares, and twenty equilateral triangles.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.092
accession number
1978.1065
catalog number
1978.1065.092
This model has a square base, eight triangular sides, and four triangles arranged in a pyramid at the top. The faces thus are twelve equilateral triangles and one square.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
This model has a square base, eight triangular sides, and four triangles arranged in a pyramid at the top. The faces thus are twelve equilateral triangles and one square.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.030
accession number
1978.1065
catalog number
1978.1065.030
This model has a pentagonal top and bottom. The side faces are five squares, fifteen equilateral triangles, and five additional regular pentagons.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
This model has a pentagonal top and bottom. The side faces are five squares, fifteen equilateral triangles, and five additional regular pentagons.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.053
accession number
1978.1065
catalog number
1978.1065.053
The top and bottom faces of this model are regular pentagons and the side faces include pentagons, triangles, and squares.
Description
The top and bottom faces of this model are regular pentagons and the side faces include pentagons, triangles, and squares. The total number of faces includes thirty-five equilateral triangles, five squares, and seven regular pentagons.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.067
accession number
1978.1065
catalog number
1978.1065.067
This model is an extension of a truncated dodecahedron. On two of decagonal faces, the decagon serves as the base of a figure with a regular pentagon on top and five squares and five equilateral triangles below. These "augmented" sides have one decafgon between them.
Description
This model is an extension of a truncated dodecahedron. On two of decagonal faces, the decagon serves as the base of a figure with a regular pentagon on top and five squares and five equilateral triangles below. These "augmented" sides have one decafgon between them. The faces of the model total thrity equilateral triangles, ten squares, two regular pentagons, and ten regular decagons.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.090
accession number
1978.1065
catalog number
1978.1065.090
This model has a regular pentagon on the top, and a regular decagon on the bottom. There are ten equilateral triangles and five regular pentagons around the sides.
Description
This model has a regular pentagon on the top, and a regular decagon on the bottom. There are ten equilateral triangles and five regular pentagons around the sides. Hence the faces are ten equilateral triangles, six regular pentagons, and one regular decagon.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.026
accession number
1978.1065
catalog number
1978.1065.026
The faces of this model are seven regular pentagons, fifteen equilateral triangles, and fifteen squares. The top and bottom are pentagons. Five triangles are around the top pentagon. Five pentagons and five triangles are in the ring below this.
Description
The faces of this model are seven regular pentagons, fifteen equilateral triangles, and fifteen squares. The top and bottom are pentagons. Five triangles are around the top pentagon. Five pentagons and five triangles are in the ring below this. The next ring has ten squares, and the lowest (adjacent to the bottom pentagon) five triangles.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.060
accession number
1978.1065
catalog number
1978.1065.060
The faces of the model are twenty equilateral triangles and twelve regular pentagons.For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
The faces of the model are twenty equilateral triangles and twelve regular pentagons.
For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.007
accession number
1978.1065
catalog number
1978.1065.007
The faces of the model are eight equilateral triangles and six regular octagons.For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
The faces of the model are eight equilateral triangles and six regular octagons.
For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.010
accession number
1978.1065
catalog number
1978.1065.010
The faces of the model are twenty equilateral triangles and twelve regular decagons.For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
The faces of the model are twenty equilateral triangles and twelve regular decagons.
For further information about Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.012
accession number
1978.1065
catalog number
1978.1065.012
The base of this model is a pentagon, with five squares rising from it. The top is a pyramid with five triangular sides.
Description
The base of this model is a pentagon, with five squares rising from it. The top is a pyramid with five triangular sides. Hence the faces of the pyramid are a regular pentagon, five squares, and five equilateral triangles.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.029
accession number
1978.1065
catalog number
1978.1065.029
Twelve faces of this model are equilateral triangles and twelve are squares.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
Twelve faces of this model are equilateral triangles and twelve are squares.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.055
accession number
1978.1065
catalog number
1978.1065.055
Three faces of this model are pentagons and seven are equilateral triangles.On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
Three faces of this model are pentagons and seven are equilateral triangles.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.084
accession number
1978.1065
catalog number
1978.1065.084
This model has a regular decagon on the bottom and a regular pentagon on the top. Above the decagon are twenty triangles. Above the triangles are five pentagons alternating with five sets of two triangles.
Description
This model has a regular decagon on the bottom and a regular pentagon on the top. Above the decagon are twenty triangles. Above the triangles are five pentagons alternating with five sets of two triangles. This comes to a total of thirty equilateral triangles, six regular pentagons, and one regular decagon.
On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.045
accession number
1978.1065
catalog number
1978.1065.045
This model resembles a regular dodecahedron, but one side has a pentagonal pyramid attached to it. The faces total five equilateral triangles and eleven regular pentagons. On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.Currently not on view
Description
This model resembles a regular dodecahedron, but one side has a pentagonal pyramid attached to it. The faces total five equilateral triangles and eleven regular pentagons. On Berman's models of regular-faced convex polyhedra, see 1978.1065.01.
Location
Currently not on view
date made
1970
maker
Berman, Martin
ID Number
1978.1065.078
accession number
1978.1065
catalog number
1978.1065.078

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