#
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

## 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's no. 405wn (MA*211257.069), while two have "405z" carved, Baker's no. 405z (MA*211257.070) and Baker's 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

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

- 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 zn (/) Riemann surface : w
^{2}= z^{3}- z. The label should have read "405 wn" and someone added a handwritten question mark after the "zn." Model 405wn 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. The spheres of the model are called complex, or Riemann, spheres. They are formed with north and south poles representing ∞ and 0, respectively. The equator represents the points x + yi with x^{2}+ y^{2}= 1. Thus the points ±1 and ±i are equally spaced along the equator. Riemann surfaces and spheres are named after the 19th-century German mathematician Bernhard Riemann.

- Baker explains in his catalog that the wn after the number of the model indicates that the model is made up of spheres representing w-values. These spheres are called the sheets of the model. There is no part of this model in which values of z or pairs (z,w) are represented. However, it is possible that the coloring on this model is related to the painted part of the wooden base of one of three other Baker models of Riemann surfaces that are associated with the equation of this model.

- 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 together with the point z = ∞ are called branch points of the model and for all other points on the w-sphere 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 complex w-sphere and together they represent what is called a branched cover of the complex w-sphere.

- On each of the sheets the equator is a thin circle and there are two great circles through the poles. On one of the great circles the values of w are purely imaginary while on the other they are real. Baker’s usual use of colors implies that the great circles facing the front and back represent imaginary numbers, while those facing the sides represent real numbers. Normally w = ∞ is at the north pole and w = 0 is at the south pole. However, as the four finite branch points of this model lie in the northern hemisphere, it appears that this model has that assignment of values reversed. The great circle facing the front and back has a thick white segment that connects the two imaginary branch points by way of w = ∞ at the south pole, while the other has a thick black segment connecting the two real branch points by way of w = 0 at the north pole. The parts of the great circles that connect two branch points are called branch cuts. This model has three, one is the black arc mentioned above and the others are the two halves of the white arc with ends at an imaginary branch point and infinity. These branch cuts are 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.

- There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, a model with Baker’s no. 405w (MA*211257.068), has “405w” carved on the base Two others, Baker’s no. 405z (MA*211257.070) and Baker’s no. 405zn (MA*211257.071), both have the mark “405z“ carved on them. 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*211257.069

- accession number
- 211257

- catalog number
- 211257.069

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

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

- 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 z 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 w (/) Riemann surface : w
^{2}= z^{3}- z. Someone corrected the error on the label by hand, crossing out the w and inserting a z. Model 405z is listed on page 17 of Baker’s 1931 catalogue 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 catalogue that the z after the number of the model indicates that the metal disks above the wooden base represent copies of a disk in the complex z-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 w-plane with the point w = 0 at its center and the real axis along the line between the yellow and dark green stripes.

- If z = 0 or z = ±1, the equation w
^{2}= z^{3}- z is satisfied by only one value of w, i.e., w = 0. These three points on the z-plane are called branch points of the model and for all other points on the z-plane the equation w^{2}= z^{3}- z is satisfied by two distinct values of w, each of which produces a different pair on the Riemann surface (if z = 2, the two distinct pairs on the Riemann surface are (2, ±√6)). Thus there are two sheets representing the same disk in the complex z-plane and together they represent part of what is called a branched cover of the complex z-plane. The color of a region on a sheet is chosen with the aim of indicating the stripe on the base in which the w coordinate lies.

- For each sheet, the center of the disc is the point z = 0 and the solid black line through that point is the real axis. The branch points of this model all lie on the real axis. The point z = –1 is the point inside the green and yellow oval where the real axis meets the small red circle representing the unit circle with center z = 0. The point z = 1 is the other point where the real axis meets the small red circle; it is inside the oval that includes all eight colors used in the model.

- The vertical surfaces between the two 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 z 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. In this model, one of the branch cuts connects z = 0 to z = 1 and the other runs from z = –1 to infinity; they 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, a model with Baker's number 405zn (MA*211257.071), has "405z" carved on the base. Two others, one with Baker's number 405w (MA*211257.068) and the other with Baker's number 405wn (MA*211257.069) have "405w" carved on the edge of the base. Baker carved a "z" or a "w" to indicate which variable is represented on the sheets of the model and added an "n" after the "z" or "w" 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*211257.070

- accession number
- 211257

- catalog number
- 211257.070

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

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

- Description

- The mark "405 z" 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 wn (/) Riemann surface : w
^{2}= z^{3}- z. The label is incorrect and should read "405 zn". Model 405zn is listed on page 17 of Baker’s 1931 catalogue 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. The spheres of the model are called complex, or Riemann, spheres. They are formed with north and south poles representing ∞ and 0, respectively. The equator represents the points x + yi with x^{2 + y}2 = 1. Thus the points ±1 and ±i are equally spaced along the equator. Riemann surfaces and spheres are named after the 19th-century German mathematician Bernhard Riemann.

- Baker explains in his catalog that the zn after the number of the model indicates that the model is made up of spheres representing z-values. These spheres are called the sheets of the model. It appears as if painted part of the wooden base of the model represents the Riemann surface as a torus, i.e., a donut, formed by pasting together the ends of the stripes to form a cylinder and then joining the ends of the cylinder.

- If z = 0 or z = ±1, the equation w
^{2}= z^{3}- z is satisfied by only one value of w, i.e., w = 0. These three points together with the point z = ∞ are called branch points of the model and for all other points on the z-sphere the equation w^{2}= z^{3}- z is satisfied by two distinct values of w, each of which produces a different pair on the Riemann surface (if z = 2, the two distinct pairs on the Riemann surface are (2, ±√6)). Thus there are two sheets representing the complex z-sphere and together they represent what is called a branched cover of the complex z-sphere. The color of a region on a sheet is chosen with the aim of indicating the stripe on the base into which it is mapped.

- On each of the sheets the equator is colored red and there are great circles through the poles that are colored yellow and black. The points on the yellow great circle are purely imaginary while those on the black great circle are real. Thus the real non-zero branch points, z = ±1, lie on the equator and on the black great circle, while the other two branch points are at the north and south poles. The darkened parts of the black great circle are called branch cuts. Assuming the pair (1,0) lies on the Riemann surface along edge shared by the center (yellow and green) stripes on the base and that the pair (–1,0) lies along the edges of the outer stripes on the base, one of the branch cuts runs between join z = 0 and z = 1 and other between z = –1 and z = ∞. These branch cuts are 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 z values into the equation. Thus one can construct the Riemann surface as a torus by cutting the spheres along the branch cut and sewing the two spheres together along those cuts while matching the four branch points.

- There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, with Baker's number 405z (MA*211257.070) has "405z" carved on the base. Two others, Baker's number 405 w (MA*211257.068) and Baker's number 405wn (MA*211257.069) have the mark "405w" on the base. Baker carved a "z" or a "w" to indicate which variable is represented on the sheets of the model and added an "n" after the "z" or "w" 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*211257.071

- accession number
- 211257

- catalog number
- 211257.071

- 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

## Model for the "Devil's Coffin" Diagram Relating to Computing the Volume of a Parallelepiped, Ross Solid

- Description
- This wooden model is one in a series illustrating the volume of solids designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The incomplete unpainted wooden model has two pieces. One is a cube, the second is part of a parallelepiped with one square face the same size as the cube. A paper label pasted to a square side of both pieces of the model reads: DEVIL’S COFFIN (/) Phillips & Fisher, p. 305 Van Velzer & Shutts, p. 300 (/) Wentworth, p. 303 Wells, p. 278. This is a reference to four American geometry textbooks published between 1894 and 1899.

- In the course of the 19th century, American geometry textbooks came to be more than reproductions of British works. By the 1890s, several texts discussing solid geometry used a figure demonstrating the volume of a parallelepiped that apparently arose in the United States.

- In this construction, the volume of an arbitrary parallelepiped is first compared to one constructed having the same altitude and rectangular bases equal in area to those of the original solid. This figure is then compared to a third parallelepiped, this with the same altitude and six rectangular sides. John Farrar, following A.-M. Legendre, proposed such a construction in his
*Elements of Geometry*. By the 1890s, the figure had taken a rather different form. Perhaps because it was difficult imagine from a two dimensional drawing, it was known as “the devil’s coffin.”

- Ross’s model of the construction had three parts, a parallelepiped with six sides in the shape of equilateral parallelograms, a parallelepiped with two square sides and four rhombic sides, and a cube. The parallelepipeds are dissected. The two models in the Smithsonian collections are the cube and one piece of one of the parallelepipeds.

- This model is not mentioned in Ross’s original manual for his surface forms and solids. The texts referred were published several times, but show the devil’s coffin construction on the pages indicated on the model on editions published between 1894 and 1899. Hence the date of about 1900 assigned to the model.

- References:

- A.-M. Legendre,
*Éléments de géométrie, avec des notes*, Paris: Didot, 1794, pp. 178–184, Plate 8.

- John Farrar,
*Elements of geometry, by A. M. Legrendre. Translated from the French for the use of the students at the University at Cambridge, New England*, Boston : Hilliard and Metcalf printers, 1819, pp. 134–139, plates IX and X.

- Thomas Heath, ed.,
*The Thirteen Books of Euclid’s Elements*, vol. 3, Book XI, propositions 29 and 30, especially the commentary on Proposition 30, New York: Dover, 1956, esp. pp. 333–336.

- Andrew Wheeler Phillips and Irving Fisher,
*Elements of Geometry*New York: American Book Company, 1896, p. 305–306.

- C. A. Van Velzer and George C. Shutts,
*Plane and Solid Geometry Suggestive Method*Madison, WI: Tracy Gibbs, 1894, p. 300.

- Webster Wells,
*The Elements of Geometry*, rev. ed., Boston: Leach, Shewell and Sanborn, 1894, p. 278.

- George A. Wentworth,
*Plane and Solid Geometry*, rev. ed., Boston: Ginn, 1899, p. 303.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Ross, W. W.

- ID Number
- 1985.0112.217

- catalog number
- 1985.0112.217

- accession number
- 1985.0112

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

## Dissected Circle Transformable into Parallelogram

- Description
- In the years following the Civil War, a handful of American educators designed and sold wooden solids or flat shapes hinged or doweled so that they could be transposed into other shapes that had areas known to students. One of them was Albert H. Kennedy (1848–1940), Superintendent of Schools in Rockport, Indiana. He sold this business to the Rockport School Desk Company. Modified forms of the solids would be sold by the Western School Supply House of Des Moines, Iowa, A. Cowles and Company of Chicago, Illinois, and the American School Furniture Company of Chicago.

- From ancient times, mathematicians sought to find a polygon with straight sides equal in area to the circle. This model represents Kennedy’s attempt to demonstrate that the area of a circle equaled half of the product of its circumference and its radius. It consists of a dissected circle, transformable into a parallelogram. The circle has of two semicircular portions. Each portion is divided into eight equal wooden segments, which are held together by cloth tape that is nailed to each segment around the circumference. Rearranging the pieces gives a rough parallelogram that has one side equal to half the circumference of the circle and a height equal to the radius. Multiplying the two factors together gives the desired area.

- In 1882, the German mathematician Lindemann demonstrated that no exact geometric squaring of the circle is possible. His work undoubtedly was unknown to Kennedy.

- The object has no maker's marks.

- Compare 2005.0054.01, 2005.0054.02, 2005.0054.03 and 2005.0054.04.

- References:

*Arithmetic of Practical Measurements for Teachers' Instruction and Class Work in Mensuration. Published by Western School Supply House*, Des Moines: Iowa Printing Co., 1893. This reportedly was ”To accompany Kennedy’s improved dissecting mathematical blocks. 20th ed.” A copy of the sixteenth edition, which has the same date, is 2005.3099.01.

- C. L. F. von Lindemann, “Über die Zahl π,”
*Mathematische Annalen*, 20 (1882), 215.

- “Paintings Presented to Local Schools,”
*Rockport Journal*May 15, 1964.

- P. A. Kidwell, "American Mathematics Viewed Objectively: The Case of Geometric Models," in
*Vita Mathematica: Historical Research and Integration with Teaching*, ed. Ronald Calinger, Washington, D.C.: Mathematical Association of America, 1996, pp. 197–207.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Kennedy, Albert H.

- ID Number
- 2005.0054.01

- catalog number
- 2005.0054.01

- accession number
- 2005.0054

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

## Model of a Riemann Surface by Richard P. Baker, Baker #411z

- 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 100 of his models are in the museum collections.

- The mark 411 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. 411z (/) Riemann surface : w
^{3}= z. Model 411z is listed on page 17 of Baker’s 1931 catalogue of models as “w^{3}= z” under the heading*Riemann Surfaces*. The catalog description also notes that “411 is to serve as a first step to 412,” where Baker model 412z (MA*211157.075) is associated with a more complicated equation involving w^{3}.

- The model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w
^{3}= z where a complex number is 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 z after the number of the model indicates that the metal disks above the wooden base represent copies of a disk in the complex z-plane. These disks are called the sheets of the model. The painted disk on the wooden base of the model represents a disk in the complex w-plane with the point w = 0 at its center. The disk is divided into twelve sectors, pie-piece-shaped parts of a circle centered at 0, each of which has an angle of 30 degree. The front of the model is the edge on which 411 is inscribed so the two vertical rectangles lie above the polar axis, i.e. the ray emanating from the origin when the angle is 0 degrees, of the wooden base. This places every horizontal edge of the rectangles on a polar axis of a sheet.

- If z = 0, the equation w
^{3}= z is satisfied by only one value of w, i.e., w = 0. The point z = 0 is called a branch point of the model and for all other points on the z-plane the equation w^{3}= z is satisfied by three distinct values of w, each of which produces a different pair on the Riemann surface (if z = 1, the three distinct pairs on the Riemann surface are (1,1), and (1,(–1 ± √3 i)/2)). Thus there are three sheets representing the same disc in the z-plane and together they represent part of what is called a branched cover of the complex z-plane.

- Baker’s use of solid red circles, and dashed red and black circles indicates that each sheet is mapped continuously onto a different portion of the w-disk on the base. There are three radii of the disk on the base (the polar lines - rays emanating from the origin – for angles of 0, 120, and 240 degrees) that are the edges of sectors corresponding to quadrants on two different sheets. The order of the colors of the 30 degree sectors on the base starting at polar axis and proceeding counterclockwise correspond to the colors of the first through fourth quadrants of the top, middle, and then bottom sheets.

- The vertical rectangles mentioned above 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 the movement to another sheet. This movement occurs when meeting a branch cut while following a path of the inputs of z values into the equation. While the defining equation determines branch points, branch cuts are not fixed by the equation. However, the single branch cut for any surface with only one branch point must run from that point out to infinity. The branch cut of this model is represented on each sheet by the horizontal edges of the vertical surface or surfaces meeting that sheet.

- Location
- Currently not on view

- maker
- Baker, Richard P.

- ID Number
- MA*211257.074

- accession number
- 211257

- catalog number
- 211257.074

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

## Epitrochoid, Kinematic Model by Martin Schilling, series 24, model 1, number 329

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the first in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- Circles rolling around the outside of other circles, known as epicycles to the ancient Greeks, were used to describe the motions of the planets in a geocentric cosmology. These curves,called epitrochoids, are formed by tracing a point on the radius or the extension of the radius of a circle as it rolls around the outside of a second stationary circle.

- Epitrochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and hypotrochoids (see items 1982.0795.02 and 1982.0795.03). In the 18th century, it was found that when shaping the sides of gear teeth as the valley between teeth, using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

- Depending on the distance of the tracing point from the center of the rolling circle, an infinite number of curves can be formed. The three curves depicted in this model are bicyclic, meaning the smaller circle needs to rotate around the larger circle twice before returning to is original configuration. The ratio of the radii of the two circles will determine the number of nodes in the curve and how many rotations are required before the tracing point returns to its starting configuration. The Spirograph toy produces various types of epitrochoids. (See item 2005.0055.02)

- In this model, a toothed metal disc of radius 30 mm links to a smaller toothed metal disc of radius 12mm. Rotating a crank beneath the baseplate rolls the smaller disc around the outer edge of the larger disk. The model illustrates three curves that may be generated by the motion of a point at a fixed distance along the radius of a circle when the circle rolls around the outer edge of a larger circle.

- The green point within the smaller circle (at radius 4mm) produces the green curve on the glass overlay of the model. The blue point on the circumference of the smaller circle (in this special case, the curve is known as an epicycloid) produces the blue curve. The third, represented by a red curve on the glass, is on the extension of a radius of the smaller circle (20mm). As the smaller circle rolls, the point moves inside the larger circle. The German title of this model is: Erzeugung der Epitrochoiden als solche mit freiem Centrum.

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group 1, model 1.

- Chironis, Nicholas,
*Gear Design and Applications*, 1967, p. 160.

- Davis, W.O.,
*Gears for Small Machines*, 1953, p. 9.

- Grant, George,
*Teeth of Gears*, 1891, p. 69.

- Material for educators can be found online at the Durango Bill website.

- Online demonstrations can be found at the Wolfram website Mathworld.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.01

- catalog number
- 1982.0795.01

- accession number
- 1982.0795

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

## Hypotrochoids, Kinematic Model by Martin Schilling, series 24, model 3, number 331

- Description
- Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the third in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.

- The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop. Thus hypotrochoids are curves formed by tracing a point on the radius or extension of the radius of a circle rolling around the inside of another stationary circle.

- Hypotochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and epitrochoids (see item 1982.0795.01).

- An infinite number of hypotrochoids can be formed, depending on the distance of the tracing point from the center of the rolling circle. The ratio of the radius of the rolling disc to the radius of the outer ring will determine the number of nodes the hypotochoid will have. In this model, the curves each have five nodes. Hypotochoids, for which the tracing point is on the extension of the radius, form curves that resemble petalled flowers and are called roses. The Spirograph toy produces various types of hypotrochoids. (See item 2005.0055.2) In the 18th century, it was found that shaping the sides of gear teeth and the valley between teeth by using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.

- This model consists of a stationary toothed metal ring (with teeth on the inner edge of the ring) of radius 80 mm. A toothed metal disc of radius 32 mm is attached to a brass arm of 7 cm that can be rotated by turning a crank below the baseplate. As the arm is rotated, the disc rolls around the inside of the ring. Three points lie along the radius of the disk and trace corresponding curves, or roulettes, on the glass overlay.

- The blue point on the circumference of the disc traces a blue five-pointed star shape referred to as a hypocycloid. The green point on the radius of the disc traces a green curve inside the ring, and the red point on the extension of the radius of the disc traces a curve that extends past the radius of the ring. The German title of this model is: Erzeugung der Hypotrochoiden als soche mit freiem Centrum.

- References:

- Schilling, Martin,
*Catalog Mathematischer Modelle für den höheren mathatischen Unterricht*, Halle a.s., Germany, 1911, pp 56-57. Series 24, group 1, model 3.

- An online demonstration can be found at http://mathworld.wolfram.com/Hypotrochoid.html

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Schilling, Martin

- ID Number
- 1982.0795.02

- catalog number
- 1982.0795.02

- accession number
- 1982.0795

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