Mathematical Objects Relating to Charter Members of the MAA  Geometric Models  Richard P. Baker
Geometric Models  Richard P. Baker
Richard P. Baker (18661937) was born in England, studied mathematics and science at Oxford, and obtained a degree from London University in 1887. The next year he came to the United States, and practiced law for some years in Texas. By 1895, he had decided to take up a career in mathematics, and applied to graduate school at the newly established University of Chicago. His dissertation did not proceed rapidly, and he spent some years teaching both mathematics and music. From 1905, he was in the mathematics department at the University of Iowa.
Richard P. Baker, 1935. 
Even before the founding of the MAA, Baker took an active interest in posing and solving problems in the American Mathematical Monthly. He published a solution in 1912 and posed several problems over the years 19131919. From 19151916, he served as coeditor of the problems column. Baker also was one of the mathematicians who served on the editorial board of the Monthly from 1912 until soon after it became an MAA publication in 1916. He participated regularly in meetings, particularly of the Iowa Section.
Baker devoted much of his life to designing and making models relating to advanced topics in mathematics, statistics, and physics. Sometimes he followed European examples, sometimes he developed models to illustrate relatively recent mathematical results. He published his first catalog, which showed ninetyeight models, in 1905. His second, published in 1931, had models numbered as high as 542.

Model of a Cubic Cone with Nodal Line by Richard P. Baker, Baker #78 (a Ruled Surface)
 Description
 This string model was constructed by Richard P. Baker, possibly before 1905 when he joined the mathematics faculty at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.
 The typed part of a paper label on the bottom of the wooden base of this model reads: No. 78 (/) CUBIC CONE WITH NODAL LINE. Model 78 appears on page 7 of Baker’s 1931 catalog of models as “With nodal line” under the heading Cubic Cones . It also appears in his 1905 catalog of one hundred models.
 Baker’s string models always represent a special type of geometric surface called a ruled surface. A ruled surface, sometimes called a scroll, is one that is swept out by a moving line. This model shows two ruled surfaces. One of these surfaces is swept out by any of the threads connecting the curved vertical wooden sides of the model. The other ruled surface is swept out by any of the threads joining the curved horizontal piece of wood on the top of the model to the wooden base of the model. All the threads of this model pass through a point in the center of the model, which is the intersection of two special lines, one for each ruled surface.
 The special line for the surface joining the vertical sides is the line connecting the inflection points of the cubic curves, i.e. the points where the curve changes from concave upward to concave downward (for the curve y=x^{3}, it would be at the origin). This line is horizontal and passes over the center of the base.
 The special line for the other curve is the vertical line going through the center of the base. It is formed by connecting the point where the upper curve crosses itself with the center of the base, which is also the point where the curve on the base crosses itself. A point of curve where the curve crosses itself is called a node, so all points of this vertical line are nodes and this is the nodal line of the surface.
 Location
 Currently not on view
 date made
 ca 19001935
 maker
 Baker, Richard P.
 ID Number
 MA.211257.006
 accession number
 211257
 catalog number
 211257.006
 Data Source
 National Museum of American History, Kenneth E. Behring Center

Model of a Quartic Scroll by Richard P. Baker, Baker #84 (a Ruled Surface)
 Description
 This string model was constructed by Richard P. Baker, possibly before 1905, when he joined the mathematics faculty at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.
 The typed part of a paper label on the wooden base of this model reads: No. 84 Quartic Scroll, (/) with two nodal straight (/) lines. Model 84 appears on page 8 of Baker’s 1931 catalog of models as “Quartic Scroll , with two nodal straight lines.” The equation of the model is listed as (x^{2}/((z  1) ^{2})) + (y^{2}/((z + 1) ^{2})) = 1. It also appears in his 1905 catalog of one hundred models.
 Baker’s string models always represent a special type of geometric surface called a ruled surface. A ruled surface, sometimes called a scroll, is one that is swept out by a moving line. This model is swept out by any of the yellow threads joining the elliptically shaped horizontal piece of wood on the top of the model to the wooden base of the model.
 In addition to the yellow threads of the model, there are two horizontal red threads that run from the rods at near the edge of the base and are parallel to the lines connecting the midpoints of the opposite sides of the square of surface of the base. There is a segment of each of these red threads for which each point meets two different lines of the model and the points of these segments are called double points, or nodes, of the surface. Thus these line segments are the two nodal lines of the model. The horizontal plane z = 1 intersects the model at the upper horizontal thread, while the horizontal plane z = 1 intersects it at the lower horizontal thread. When z=1, the points of intersection are (0,y,1) for y between 2 and 2. When z=1, the points of intersection are (x,0,1) for x between 2 and 2. Thus the nodal lines are the line segments connecting (0,2,1) to (0,2,1) and (2,0,1) to (2,0,1).
 When z = 0 the equation of the surface becomes x^{2} + y^{2} = 1, so the horizontal plane z = 0 intersects the model at the unit circle with center at the origin. For any other value of z, the equation of the surface is of the form (x^{2}/a^{2}) + (y^{2}/b^{2}) = 1, where a does not equal b. This is the standard form for the equation of an ellipse.
 Location
 Currently not on view
 date made
 ca 19151935
 ca 19051935
 ca 19001935
 maker
 Baker, Richard P.
 ID Number
 MA.211257.010
 accession number
 211257
 catalog number
 211257.010
 Data Source
 National Museum of American History, Kenneth E. Behring Center

Model of Cones with a Common Vertex by Richard P. Baker, Baker #508 (a Ruled Surface)
 Description
 This string 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 typed part of a paper label on the bottom of the wooden base of this model reads: No. 508 (/) OSCULATING CONTACT; ROOTS 3. Model 508 appears on page 10 of Baker’s 1931 catalog of models as “Osculating contact” under the heading Cones with common vertex. It is explained on page 9 that the "3" listed for the model refers to “the multiplicity of the roots.”
 Baker’s string models always represent a special type of geometric surface called a ruled surface. A ruled surface, sometimes called a scroll, is one that is swept out by a moving line. This model shows two ruled surfaces, a circular cone whose lines are represented by red thread and an elliptical cone whose lines are represented by blue thread. These two cones share a vertex and each vertical side of the model shows a red circle and a blue ellipse. The cones meet and cross each other along two lines. The meeting is visible on each of the sides where there are two holes that share a red and a blue thread. The crossing is clearly visible at one of the holes, while the other is at a point of osculating contact.
 The term osculating derives from the Latin for kissing. In plane geometry, mathematicians use the term to refer to a point at which two curves (1) share a tangent line, (2) are on the same side of that common tangent line, and (3) have the same curvature. Since curvature measures how much a curve curves at a point, a straight line has zero curvature at every point. The curvature of a circle is also the same at every point and its numerical value is 1 divided by the radius of the circle so a circle with a small radius has a large curvature and a circle with large radius has a small curvature. An ellipse has a constantly changing curvature that is smallest where the curve looks straightest. On either vertical side of this model the point of osculating contact has multiplicity three as a point of intersection of the circle and the ellipse. Although these curves cross at that point of osculating contact, neither curve crosses the shared tangent line at that point.
 Location
 Currently not on view
 date made
 ca 19151935
 maker
 Baker, Richard P.
 ID Number
 MA.211257.101
 accession number
 211257
 catalog number
 211257.101
 Data Source
 National Museum of American History, Kenneth E. Behring Center

Geometric Model by Richard P. Baker, Axial Pencil and Transversals, Baker #235
 Description
 This geometric model was constructed by Richard P. Baker when he was Associate Professor of Mathematics at the University of Iowa, most likely some time before 1930. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.
 The typed part of a paper label taped to this wire model reads: No. 235 (/) Axial pencil (/) transversals. Model 235 appears on page 13 of Baker’s 1931 catalog of models as “Axial Pencil and transversals.”
 An axial pencil is a set of planes that pass through a line, called the axis of the pencil. The most obvious of the axial pencils represented in the model has its axis as the short rod parallel to the long rods of the base of the model. Each long rod of the base produces a plane that goes through the axis. One of the planes includes the yellow rods, the other the pink rods.
 The transversals in the title probably refer to the short rods connecting the long rods of the base. These short rods will be referred to as base transversals. There are two more rods parallel to these transversals, one at the top of the model and one, much shorter, slightly above the axis of the pencil described above, each of which is the axis of another axial pencil represented in the model. The colors of some of the rods are no longer very clear. However, it appears as if the original coloring would have been useful in describing these two additional axial pencils.
 Each base transversal meets two vertical rods and produces a plane that goes through one of the other two transversals. The triangles formed by each of the center three base transversals meet the upper nonbase transversal. Thus the plane of each of those three triangles is a part of the axial pencil with axis the transversal at the top of the model. The triangles formed by each of the outer two base transversals meet the very short nonbase transversal. Thus the plane of each of those triangles is part of the axial pencil with axis the very short transversal.
 Location
 Currently not on view
 date made
 19151930
 maker
 Baker, Richard P.
 ID Number
 MA.211257.041
 accession number
 211257
 catalog number
 211257.041
 Data Source
 National Museum of American History, Kenneth E. Behring Center

Geometric Model by Richard P. Baker, Thermodynamic Surface for Water, Ice, and Steam, Baker #253
 Description
 In the 1870s physicists in Scotland and the United States began to make threedimensional models of the thermal properties of matter. The height of this wire surface corresponds to the volume of ice, water and steam as this changes with pressure (pressure increases coming toward the front of the model) and temperature (temperature increases going to the right). The solid state – with relatively slow changes in volume with temperature and pressure, and low temperatures  is represented by the relatively flat surface on the left. The part of the board under it is marked S. When ice melts, the volume decreases until melting is complete. Water expands more rapidly with temperature than ice, so that the surface rises more rapidly from the middle of the front going right. The part of the board under this section of the surface is marked L. As temperature rises, water turns to steam, and the volume increases even more rapidly. The section of the board under this part of the surface is marked V for vapor.
 A paper sticker glued to the underside of the base reads: No. 253 (/) Water Steam and Ice.
 This is one of a series of nine models Baker made that relate to thermodynamic surfaces. It was designed during his years at the University of Iowa under the supervision of his Germanborn colleague Karl Eugen Guthe (1866–1915), who taught in the physics department there from 1905 until 1909. Baker’s correspondence indicates that copies of the model were purchased by Columbia University and by the University of Michigan. The model remained in Baker’s catalog as late as 1931. A card catalog in the Baker papers indicates that the model sold at one time for $7.50.
 This particular example of the model was on loan for exhibition at MIT from 1939 until the mid1950s. It, along with the other models in accession 211257, came to the Smithsonian from MIT in 1956.
 References:
 Accession file 211257.
 J. Willard Gibbs, “A Method Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces,” Transactions of the Connecticut Academy, 2, 1873, pp. 382–404. Gibbs refers to earlier work of the Scottish engineer James Thomson, who devised a surface for representing the pressure volume and temperature of carbonic acid and carbon dioxide.
 H. Randall to Baker, January 15, 1908 and Columbia University to Baker, August 6, 1918, Richard P. Baker Papers, University Archives, University of Iowa, Iowa City, Iowa.
 Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 18.
 Location
 Currently not on view
 date made
 ca 19051935
 maker
 Baker, Richard P.
 ID Number
 MA.211257.045
 accession number
 211257
 catalog number
 211257.045
 Data Source
 National Museum of American History, Kenneth E. Behring Center

Geometric Model by Richard P. Baker, Wire Model of Clebsch's Diagonal Surface, Baker #365
 Description
 This geometric model was constructed by Richard P. Baker, probably in the 1920s while he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.
 The typed part of a paper label taped to this wire model reads: No. 365 (/) THE DOUBLE SIX FROM 367. Model 365 appears on page 10 of Baker’s 1931 catalog of models as “The double six from 367” under the heading Clebsch’ Diagonal Surface. Model 367 is the first model in this section and is listed as a plaster cast model title “Tetrahedral symmetry. 24 finite lines.” While model 367 is not among the models in the Smithsonian collections, the second model in this section, Model 366 (211257.055) “The lines of 367,” is.
 The double six in the title of this model is a Schlaefli Double Sixes structure, named after 19thcentury Swiss mathematician Ludwig Schlaefli. A double sixes structure consists of two sets of six lines that satisfy the following three properties: no lines in the same set intersect, each line is paired with a line of the other set which it does not intersect, and each line intersects the five lines in the other set with which it is not paired.
 In this model, the twelve wire rods represent the twelve lines. It appears as if some of the rods have been repainted so it is no longer possible to distinguish six colors. However, it is likely that the paired rods were originally painted the same color. Labeling one of the sets of rods 1 through 6 and the other 1′ through 6′ as shown in one of the images, one can see that the yellow rod 6′ meets (left to right) rods 2, 4, 3, 1, and 5 but does not meet the yellow rod 6. Similarly the (clearly repainted) tan rod 4 meets (bottom to top) rods 5′, 6′, 1′, 3′, and 2′ but does not meet the purple rod 4′.
 The surface on which this model is based, the Clebsch Diagonal Surface, is defined by the cubic equation x^{3}+y^{3}+z^{3}+w^{3}+v^{3}=0 assuming x+y+z+w+v=0; it contains thirty six doublesix structures.
 Location
 Currently not on view
 date made
 ca 19151935
 maker
 Baker, Richard P.
 ID Number
 MA.211257.054
 accession number
 211257
 catalog number
 211257.054
 Data Source
 National Museum of American History, Kenneth E. Behring Center

Model of a Gaussian Surface by Richard P. Baker, Baker #448
 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 typed part of a paper label on the base of this plaster model reads: No. 448 (/) Gaussian Surface. Model 448 appears on page 21 of Baker’s 1931 catalog of models in the STATISTICS section as “Gaussian surface.” Baker named this model after the 19thcentury German mathematician Carl Friedrich Gauss. The surface is related to a statistical function whose graph is commonly referred to as a bell curve. While the bell curve is often referred to as a Gaussian curve, the formal name for a statistical function that produces a bell curve is a normal distribution function.
 Presumably Baker called this model a Gaussian surface because its vertical cross sections are Gaussian curves. The horizontal cross sections are ellipses and the formal name of this type of surface is a bivariate normal distribution surface.
 The term Gaussian surface is most commonly used in connection with electromagnetic fields. That usage is not related to this model.
 Location
 Currently not on view
 date made
 ca 19151935
 maker
 Baker, Richard P.
 ID Number
 MA.211257.095
 accession number
 211257
 catalog number
 211257.095
 Data Source
 National Museum of American History, Kenneth E. Behring Center

Model of a Riemann Surface by Richard P. Baker, Baker #405w
 Description
 This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.
 The mark "405 w" is carved into one edge of the wooden base of this model and the typed part of a paper label on the base reads: No. 405 z (/) Riemann surface : w^{2} = z^{3}  z. Someone corrected the error on the label by hand, crossing out the z and inserting a w. Model 405w is listed on page 17 of Baker’s 1931 catalog of models as “w^{2} = z^{3}  z” under the heading Riemann Surfaces. This means that the model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w^{2} = z^{3}  z. Complex numbers are of the form x + yi for x and y real numbers and i the square root of –1. A complex plane is like the usual real Cartesian plane but with the horizontal axis representing the real part of the number and the vertical axis representing the imaginary part of the number. Riemann surfaces are named after the 19thcentury 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 wplane. 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 zplane 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 wplane are called branch points of the model and for all other points on the wplane 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 wplane and together they represent part of what is called a branched cover of the complex wplane. 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 (211257.069), while two have "405z" carved, Baker's no. 405z (211257.070) and Baker's no. 405zn (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 19151935
 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 19thcentury 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 wvalues. 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 wsphere 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 wsphere and together they represent what is called a branched cover of the complex wsphere.
 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 (211257.068), has “405w” carved on the base Two others, Baker’s no. 405z (211257.070) and Baker’s no. 405zn (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 19151935
 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 19thcentury 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 zplane. 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 wplane 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 zplane are called branch points of the model and for all other points on the zplane 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 zplane and together they represent part of what is called a branched cover of the complex zplane. 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 19151935
 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
 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 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 19thcentury 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 zvalues. 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 zsphere 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 zsphere and together they represent what is called a branched cover of the complex zsphere. 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 nonzero 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 19151935
 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