Science & Mathematics

Chemical Balance

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.

Name: Wheeler, Albert Harry

Geometric Model of A. Harry Wheeler, Immersion of a Moebius Band (One-Sided Polyhedron)

Taking a long, thin rectangle and attaching the short sides with a half-twist produces a surface called a Moebius band.
Description
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.
Location
Currently not on view
date made
ca 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.416
accession number
304723
catalog number
304723.416

Geometric Model by A. Harry Wheeler, Cube Transformable into Four Regular Octahedra and a Faceted Cuboctahedron, Dissected Polyhedron

This paper model comes in a paper box folded in the shape of a cube.
Description
This paper model comes in a paper box folded in the shape of a cube. Inside the box, arranged in the shape of a cube, are a faceted cuboctahedron, six square pyramids that fit in the facets of the cuboctahedron and eight triangular pyramids that fit over the triangular faces of the cuboctahedron. The pieces separate into the faceted cubocatahedron, four square pyramids, a square pyramid hinged to four triangular pyramids, and another square pyramid hinged to another four triangular puramids. The four square pyramids can be arranged as two octahedra, as can both of the sets of hinged pieces. Hence the pieces of the model can be arranged either as a cube or as four regular octahedra and a faceted cuboctahedron.
A mark on the object reads: 763.
Compare MA.304723.294 (larger paper model), MA.304723.296 (middle-sized paper model), MA.304273.306 (smallest paper model), and MA.304723.132.
There is a drawing in 1979.3002.42 for a "cube with four removable octahedra." It is dated March 12, 1938, and is quite possibly for the plastic version model.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.296
accession number
304723
catalog number
304723.296

Geometric Model by A. Harry Wheeler, Fifteenth Stellation of the Icosahedron

A stellation of a regular polyhedron is a polyhedron with faces formed by extending the sides of the faces of the regular polyhedron. Extending the triangular sides of an icosahedron can produce a variety of complex polyhedra, including this one.
Description
A stellation of a regular polyhedron is a polyhedron with faces formed by extending the sides of the faces of the regular polyhedron. Extending the triangular sides of an icosahedron can produce a variety of complex polyhedra, including this one. The surface has sixty short three-sided spikes. These meet in groups of three—each meeting point might be considered as the vertex of a circumscribing regular dodecahedron.
The model is cut and folded from paper. It is Wheeler’s model 382, and number I21 in his series of icosahedra. Wenninger calls the surface the fifteenth stellation of the icosahedron.
References:
Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 62.
A. H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
Location
Currently not on view
date made
1927-05-12
maker
Wheeler, Albert Harry
ID Number
MA.304723.197
accession number
304723
catalog number
304723.197

Geometric Model by R. N. S. Merritt, a Student of A. Harry Wheeler, Four-Pointed Star

This tan paper model has twelve quadrilaterals, grouped in threes to form a four-pointed star.A mark on the model reads: R.N.S. Merritt (/) June 13, 1939.Currently not on view
Description
This tan paper model has twelve quadrilaterals, grouped in threes to form a four-pointed star.
A mark on the model reads: R.N.S. Merritt (/) June 13, 1939.
Location
Currently not on view
date made
1939 06 13
associated dates
1939 06 13 / 1939 06 13
maker's teacher
Wheeler, Albert Harry
maker
Merritt, R. N. S.
ID Number
MA.304723.647
accession number
304723
catalog number
304723.647

Geometric Model by A. Harry Wheeler, Spherical Triangles

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.
Description
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".
Location
Currently not on view
date made
ca 1945
maker
Wheeler, Albert Harry
ID Number
MA.304723.688
accession number
304723
catalog number
304723.688

Geometrical Model by A. Harry Wheeler, Regular Octahedron with Figures

This cut and folded paper model has eight equilateral triangles for sides. The vertices and sides are lettered and various figures are plotted in pencil.A mark on the model reads: Sept-5-1926.Currently not on view
Description
This cut and folded paper model has eight equilateral triangles for sides. The vertices and sides are lettered and various figures are plotted in pencil.
A mark on the model reads: Sept-5-1926.
Location
Currently not on view
date made
1926 09 05
maker
Wheeler, Albert Harry
ID Number
MA.304723.685
accession number
304723
catalog number
304723.685

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

This colorful cut and glued plastic model has tetrahedral symmetry. It might be considered as the union of four congruent pentahedra, each with one square side, two triangular sides, and two sides that are irregular quadrilaterals.
Description
This colorful cut and glued plastic model has tetrahedral symmetry. It might be considered as the union of four congruent pentahedra, each with one square side, two triangular sides, and two sides that are irregular quadrilaterals. The quadrilaterals and one triangle of each pentahedra meet at a vertex – the four vertices would be those of a regular tetrahedron. One quadrilateral side is coplanar with a triangle in the adjacent pentahedra and the sides are of the same color. Numerous other sides are coplanar and colored alike. A mark on the model reads: 649. Wheeler also assigned it number M8.
According to Wheeler, the model has fourteen vertices (e), twenty-four edges (k) and eleven faces (f), for Euler characteristic e – k + f = 1. He reports that it was described by K. Merz in an article of June, 1938. In the Vierteljahresscrhtift der Naturforschenden Gesellschaft in Zurich..
Reference:
A.H. Wheeler, Catalog of Models, A. H. Wheeler Papers, Mathematics Collections, National Museum of American History.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.418
catalog number
304723.418
accession number
304723

Geometric Model by A. Harry Wheeler, Pentagonal Dodecahedron

This cut, folded and glued tan paper model has six isosceles triangles on top, twelve larger isoscleles triangles aroun the sides, and six isosceles triangles on the bottom.
Description
This cut, folded and glued tan paper model has six isosceles triangles on top, twelve larger isoscleles triangles aroun the sides, and six isosceles triangles on the bottom. Each large triangle is in the same plane as another large triangle and each small triangle is in the same plane as another small triangle. A mark on the model reads: Pentagonal (/) Dodecahedron (/) Dec 28-29 1933.
Location
Currently not on view
date made
1933 12 29
maker
Wheeler, Albert Harry
ID Number
MA.304723.652
accession number
304723
catalog number
304723.652

Geometric Model by E. Johnson, a Student of A. Harry Wheeler, Union of Five Tetrahedra

This cut and folded tan paper model represents a compound of five regular tetrahedra inscriptible in a rhombic dodecahedron. A mark on the model reads: No 394 (/) I33 (/) E. Johnson '27B.Compare MA.304723.212, MA.304723.218, and MA.304723.233.Currently not on view
Description
This cut and folded tan paper model represents a compound of five regular tetrahedra inscriptible in a rhombic dodecahedron. A mark on the model reads: No 394 (/) I33 (/) E. Johnson '27B.
Compare MA.304723.212, MA.304723.218, and MA.304723.233.
Location
Currently not on view
date made
1927
unspecified
Wheeler, Albert Harry
maker
Johnson, E.
ID Number
MA.304723.218
accession number
304723
catalog number
304723.218

Geometric Model by A. Harry Wheeler, Filler for Bisected Cube, Dissected Polyhedron (Portion)

This cut and folded paper model forms a hollow cube.It is part of Wheeler's model 767 ( MA.304723.295), a bisected cube, and is labeled as a filler for a bisected cube.
Description
This cut and folded paper model forms a hollow cube.
It is part of Wheeler's model 767 ( MA.304723.295), a bisected cube, and is labeled as a filler for a bisected cube. The pieces of the bisected cube can be arranged around this cube to form a rhombic dodecahedron.
Location
Currently not on view
date made
1931 04
maker
Wheeler, Albert Harry
ID Number
MA.304723.309
accession number
304723
catalog number
304723.309

Geometric Model by A. Harry Wheeler, Union of Ten Tetrahedra

Wheeler made several models of intersecting polyhedra, including this tan paper cut and glued model showing the union of ten tetrahedra.A mark reads: I 35 (/) A. Harry Wheeler. Another mark reads: No. 396 (/) I35.Compare 304723.223, MA.304723.229, and MA.304723.119.
Description
Wheeler made several models of intersecting polyhedra, including this tan paper cut and glued model showing the union of ten tetrahedra.
A mark reads: I 35 (/) A. Harry Wheeler. Another mark reads: No. 396 (/) I35.
Compare 304723.223, MA.304723.229, and MA.304723.119. For a different compound of ten tetrahedra, see MA.304723.108.
Reference:
Magnus J. Wenninger, Polyhedron Models, Cambridge: The University Press, 1971, p. 45.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.223
accession number
304723
catalog number
304723.223

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

This self-intersecting polyhedron has twelve trapezoidal faces (made out of light turquoise plastic) and twelve triangular faces (made out of dark turquoise plastic).
Description
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.
Location
Currently not on view
date made
ca 1940
maker
Wheeler, Albert Harry
ID Number
MA.304723.410
accession number
304723
catalog number
304723.410

Geometric Model by A. Harry Wheeler, Rhombic Pyramid

This paper model has a rhombus for base and four triangular faces that meet at a point.Compare MA.304723.524, MA.304723.542, and MA.304723.543.Currently not on view
Description
This paper model has a rhombus for base and four triangular faces that meet at a point.
Compare MA.304723.524, MA.304723.542, and MA.304723.543.
Location
Currently not on view
date made
1930 02 05
maker
Wheeler, Albert Harry
ID Number
MA.304723.524
accession number
304723
catalog number
304723.524

Geometric Model by A. Harry Wheeler, Triangular Pyramid

Models 1979.0102.205, MA.304723.715, MA.304723.716, and MA.304723.717 are identical triangular pyramids. Each has three plastic transparent sides and a white plastic triangular base.
Description
Models 1979.0102.205, MA.304723.715, MA.304723.716, and MA.304723.717 are identical triangular pyramids. Each has three plastic transparent sides and a white plastic triangular base. The four models can be joined to form a square pyramid with white sides and a transparent base.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.715
accession number
304723
catalog number
304723.715

Geometric Model by A. Harry Wheeler, Faceted Icosidodecahedron

This colorful plastic model is derived from the icosidodecahedron, with its twenty triangular faces and twelve pentagonal faces. All of the pentagonal faces have been replaced by inward-facing pyramids as have ten of the triangular faces.Currently not on view
Description
This colorful plastic model is derived from the icosidodecahedron, with its twenty triangular faces and twelve pentagonal faces. All of the pentagonal faces have been replaced by inward-facing pyramids as have ten of the triangular faces.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.530
accession number
304723
catalog number
304723.530

Geometric Model by A. Harry Wheeler, Pentagonal Trapezoidal Dodecahedron

This cut and folded tan paper model represents the union of two star pyramids with pentagrams as bases. The bases face opposite directions and the vertices of the pyramids extend atop the pentagrams.
Description
This cut and folded tan paper model represents the union of two star pyramids with pentagrams as bases. The bases face opposite directions and the vertices of the pyramids extend atop the pentagrams. Faces include two pentagrams with star-shaped hols, twenty trapezoids, and twenty triangles. A faint mark reads: No 289 (/) DT3. Another faint mark reads: June-30-1921 (/) July-2-1931 (/) Third species (/) No. 289 (/) DT3 Pentagonal (/) Trapezoidal (/) Dodecahedron
Compare models MA.304723.593 and MA.305723.594.
Location
Currently not on view
date made
1921 06 30
maker
Wheeler, Albert Harry
ID Number
MA.304723.594
accession number
304723
catalog number
304723.594

Geometric Model by A. Harry Wheeler, Quadrilateral Inside Out, Plane Dissection

This hinged dissection has four pieces which may be arranged as a quadrilateral in two ways. An inscribed parallelogram transforms inside out. The model has a wooden base, metal hinges and a plastic top.This appears to be Wheeler's model 4S, a Quadrilateral Inside Out.
Description
This hinged dissection has four pieces which may be arranged as a quadrilateral in two ways. An inscribed parallelogram transforms inside out. The model has a wooden base, metal hinges and a plastic top.
This appears to be Wheeler's model 4S, a Quadrilateral Inside Out. See his list of decomposition of polygons.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.756
accession number
304723
catalog number
304723.756

Geometric Model by A. Harry Wheeler, Tetrakaidekahedron Fourth Species

Wheeler designed several models that extended a model with fourteen faces. This is what he called the “fourth species” of the tetrakaidekahedron. It has eight regions that are small triangular pyramids facing inward and hence contribute three triangular sides each.
Description
Wheeler designed several models that extended a model with fourteen faces. This is what he called the “fourth species” of the tetrakaidekahedron. It has eight regions that are small triangular pyramids facing inward and hence contribute three triangular sides each. It also has six regions that contribute eight triangular sides each. Finally, twelve regions have two pentagonal and two triangular sides. Many sides appear to be coplanar.
A mark on the model reads: Tetrakaidodecahedron (/) Fourth Species. A paper tag reads: RS 21. Another mark reads: 9 / 27.
Compare MA.304723.072 and MA.304723.073. They are not identical.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.072
accession number
304723
catalog number
304723.072

Geometric Model by A. Harry Wheeler, Trapezohedron

The twenty-four faces of this tan paper model are kite-shaped quadrilaterals known as trapezia. The edges are not joined around the center.A mark reads: Nov. 20, 1937 (/) Trapezohedron (/) 601.Currently not on view
Description
The twenty-four faces of this tan paper model are kite-shaped quadrilaterals known as trapezia. The edges are not joined around the center.
A mark reads: Nov. 20, 1937 (/) Trapezohedron (/) 601.
Location
Currently not on view
date made
1937 11 20
maker
Wheeler, Albert Harry
ID Number
MA.304723.357
accession number
304723
catalog number
304723.357

Geometric Model by R. Tibeotts, a Student of A. Harry Wheeler, Pyramid with Circumscribed Prisms

This wooden model shows a triangular pyramid with three small triangular prisms emerging from one side. These form, together with the pyramid itself, three stacked triangular prisms. A paper tag on the model reads: 236. Another mark reads: R.
Description
This wooden model shows a triangular pyramid with three small triangular prisms emerging from one side. These form, together with the pyramid itself, three stacked triangular prisms. A paper tag on the model reads: 236. Another mark reads: R. Tibbott, (/) May 38.
George Wentworth, in his textbook Plane and Solid Geometry
Compare MA.304723.179, MA.304723.589 and MA.304723.662.
Reference:
George Wentworth with David E. Smith,, Plane and Solid Geometry, Boston: Ginn and Company, 1906, p. 342 .
Location
Currently not on view
date made
1938 05
maker's teacher
Wheeler, Albert Harry
math student
Tibbetts, R.
ID Number
MA.304723.179
accession number
304723
catalog number
304723.179

Geometric Model by A. Harry Wheeler, Pentagonal Dodecahedron

The faces of this cut, folded and glued tan paper model include twelve triangles, twelve smaller triangles, and six irregular hexagons. Some of the triangles appear to be coplanar, either with the hexagons or with other triangles.
Description
The faces of this cut, folded and glued tan paper model include twelve triangles, twelve smaller triangles, and six irregular hexagons. Some of the triangles appear to be coplanar, either with the hexagons or with other triangles. A mark on the model reads: Pentagonal (/) Dodecahedron (/) Dec. 14, 1935.
Location
Currently not on view
date made
1935 12 14
maker
Wheeler, Albert Harry
ID Number
MA.304723.028
accession number
304723
catalog number
304723.028

Geometric Model by A. Harry Wheeler,Two Cubes Transformable into a Rhombic Dodecahedron, Dissected Polyhedron (Portion)

This tan paper model is half of Wheeler's model 790. It consists of a cube dissected into four hinged six-sided figures that fit in a paper box folded in the shape of a cube. Model MA.304723.317 is identical.
Description
This tan paper model is half of Wheeler's model 790. It consists of a cube dissected into four hinged six-sided figures that fit in a paper box folded in the shape of a cube. Model MA.304723.317 is identical. The pieces of these two models can be combined to form a rhombic dodecahedron..
Location
Currently not on view
date made
1931 05 05
maker
Wheeler, Albert Harry
ID Number
MA.304723.326
accession number
304723
catalog number
304723.326

Geometric Model by A. Harry Wheeler, Three Great Circles on the Surface of a Sphere

This cut and fit tan paper model shows three perpendicular rings of equal size, representing great circles on a sphere.Compare MA.304723.178, MA.304723.598, MA.304723.672, MA.304723.721, MA.304723.722, and MA.304723.724.Currently not on view
Description
This cut and fit tan paper model shows three perpendicular rings of equal size, representing great circles on a sphere.
Compare MA.304723.178, MA.304723.598, MA.304723.672, MA.304723.721, MA.304723.722, and MA.304723.724.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.598
accession number
304723
catalog number
304723.598

Geometric Model by A. Harry Wheeler, Irregular Hexahedron

This tan paper model is cut, glued, and folded. It is a irregular hexahedron has six trapezoids as faces. a mark in pencil reads: 4.Models MA.304723.015 and MA.304723.541 fit together to form a rectangular parallelopiped.
Description
This tan paper model is cut, glued, and folded. It is a irregular hexahedron has six trapezoids as faces. a mark in pencil reads: 4.
Models MA.304723.015 and MA.304723.541 fit together to form a rectangular parallelopiped. So do models MA.304723.033 and MA.304723.523.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.523
accession number
304723
catalog number
304723.523

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