Geometric Models - Models by Richard P. Baker

Around 1900, mathematicians across the world acquired physical models, both to illustrate concepts they taught and to demonstrate their familiarity with new ideas. They frequently purchased models from Europe, especially Germany. A few Americans also designed and made models. One of them was English-born Richard P. Baker (1866-1937), who began making models while he was a graduate student at the University of Chicago, publishing his first list of 100 models in 1905. These objects largely followed contemporary textbooks. By 1931. Baker was a professor of mathematics at the University of Iowa, and he had designed over 500 models, many on more abstract topics. Somme models were less expensive copies of those made in Germany. Baker’s designs also included surfaces associated with areas of physics such as thermodynamics, optics, and mechanics

Baker sold models to the University of Delaware, and a variety of other colleges and universities. After his death, his daughters sold part of his remaining stock to the University of Arizona, where it remains to this day. They also placed over one hundred models on exhibition at MIT, where they stayed from the late 1930s until 1956. From there, they came to the Smithsonian. A few other models in Baker’s style were given to the museum by Brown University, and are also included here.

Baker carefully numbered the designs for his models, and labeled examples with title and number. The objects shown here are in the order he assigned them, where this is known. Another dozen models have no number, and are listed afterward. Finally, the object group includes biographical materials relating to Baker and his career.

Model of a Riemann Surface by Richard P. Baker, Baker #410Z

This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa.
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 410 z is inscribed on an edge of the wooden base of this model and the typed part of a paper tag on the base reads: No. 410z, w (/) Riemann surface : (/) w2 = z5 - z (/) 2 models. The w is crossed but the 2 models refers to this model and model No. 410w (MA.211257.072) that are associated with the same equation. Both models are listed on page 17 of Baker’s 1931 catalog of models as w2 = z5 - z under the heading Riemann Surfaces. This means that both models represent a Riemann surface consisting of pairs of complex numbers, (z, w), for which w2 = z5 - 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 z after the number of this 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 sixteen sectors, pie-piece-shaped parts of a circle centered at 0, each of which has a central angle of 22.5 degrees.
If z = 0, ±1, or ±i, the equation w2 = z5 - z is satisfied by only one value w, i.e., w = 0. These five points on the z-plane are called branch points of the model and for all other points on the z-plane the equation w2 = z5 - 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, ±√30)). Thus there are two sheets representing 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 a sector or sectors on the base into which it is mapped.
The five dark blue points on the upper sheet of the model are marking the approximate locations of the five branch points of the model. The five branch points appear again on the lower sheet and the four colors on each sheet are represented in the regions surrounding the branch points on that sheet. 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 all of the branch cuts run out to infinity although two of them meet at a point that is not a branch point. One of the two branch cuts that meet runs from z = 0; the two branch cuts meet on the upper sheet at the point where two green regions meet two pink (possibly once purple) regions. All the branch cuts are represented by the horizontal edges of the vertical surfaces.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.073
accession number
211257
catalog number
211257.073

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

This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa.
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 : w3 = z. Model 411z is listed on page 17 of Baker’s 1931 catalogue of models as “w3 = 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 (211157.075) is associated with a more complicated equation involving w3.
The model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w3 = 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 w3 = 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 w3 = 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

Model of a Riemann Surface by Richard P. Baker, Baker #412Z

This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa.
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 412 is inscribed on an edge of the wooden base of this model and the typed part of a paper tag on the base reads: No. 412z (/) Riemann surface : (/) z(1 + w3)2 = (1 – w3)2. This model is listed on page 17 of Baker’s 1931 catalog of models as z(1 + w3)2 = (1 – w3)2 under the heading Riemann Surfaces. According to the catalog Baker No. 411z (MA.211257.074), “411 is to serve as a first step to 412. ”That model is associated with the equation w3 = z.
This model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which z(1 + w3)2 = (1 – w3)2. 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 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 degrees. The front of the model is the edge on which 412 is inscribed, so the 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 a vertical rectangle on the polar axis of one or two sheets.
If z = 0, the equation z(1 + w3)2 = (1 – w3)2 is satisfied by only three values of w, i.e., the three complex roots of w3 = 1, 1 and (–1 ± √3 i) / 2, and (–1 + √3 i) / 2. If z = 1, the equation z(1 + w3)2 = (1 – w3)2 is satisfied by only one finite value of w, w = 0. Both z = 0 and z = 1 are called branch points of the model and for all other finite points on the z-plane the equation z(1 + w3)2 = (1 – w3)2 is satisfied by six distinct values of w, each of which produces a different pair on the Riemann surface (if z = –1, the six values of w are the six complex roots of w6 = –1). Thus there are six sheets representing 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 a sector or sectors on the base into which it is mapped. Baker’s use of red and black radii dividing the sheets into quadrants and white unit circles on all of the sheets indicated the continuous mapping of each sheet onto a 60 degree quadrant on the base. The top sheet is mapped into the sector defined by the polar lines (rays emanating from the origin) at 0 and 60 degrees. The subsequent sheets are defined by the five other 60 degree sectors defined by the polar lines at 60, 120, 180, 240 300, and 0 degrees.
On each sheet there is a circle of radius one drawn in white. These six circles are mapped to the six rays on the base emanated from the origin on which the three cube roots of i of –i lie. The circle on the upper sheet is mapped onto the ray with defining angle of 30 degrees. The subsequent circles are mapped onto rays at 90 (the positive imaginary axis), 150, 210, 270 (the negative imaginary axis), and 330 degrees.
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. In this model there are branch cuts along the positive real axis from z = 0 to z = 1 and from z = 1 to infinity. The movement produced when meeting an infinite branch cut is between the pair of sheets that lie above and below the vertical edges that define the branch cut, i.e., between sheets 1 and 2, 3 and 4, and 5 and 6. For branch cuts running from z=0 to z=1, the movement produced is also between consecutive sheets but is between sheets 1 and 6, 2 and 3, and 4 and 5.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.075
accession number
211257
catalog number
211257.075

Model of a Klein-Riemann Surface by Richard P. Baker, Baker #414

This metal and string model has a black wooden base. There are eight steel supports around the edges and another one up the middle. These supports hold eight horizontal metal sheets.
Description
This metal and string model has a black wooden base. There are eight steel supports around the edges and another one up the middle. These supports hold eight horizontal metal sheets. The top two and the bottom two sheets have four edges that are curved, while the middle four sheets are are octahedra. Gold-colored threads join some of the sheets. The sheets are painted blue, pink, tan and green. The circle painted on the base is divided into sections of different color. According to the accession memo, this is a Klein-Riemann surface, representing an inverse transformation of a function on the u, v plane.
A mark incised on the side of the base reads: 414.
Location
Currently not on view
maker
Baker, Richard P.
ID Number
MA.211257.129
accession number
211257
catalog number
211257.129

Model of a Klein-Riemann Surface by Richard P. Baker, Baker #415

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa.
Description
This geometric model was constructed by Richard P. Baker in the early twentieth century 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 415 and the initials R. P. B. are carved into one edge of the wooden base of this model. The typed part of a paper label on the base reads: “No. 415 (/) Klein-Riemann surface: v = y (/) 4u2 = 1 + 4(x + y) (1 − x) (1 − y).” Although Baker’s 1931 catalog, “Mathematical Models Made by R. P. Baker” lists the same equations, his handwritten notes on Model 415 and its inverse transformation, Model 415a (MA*211257.077), show that the actual equations that define these models are v = y and 4u2 = 1 − 4(x + y) (1 − x) (1 − y). The metal sheets of 415 represent parts of the x,y-plane and those of 415a represent parts of the u,v-plane.
While Baker did not define a Klein-Riemann surface, and that term does not appear to have been used except by him, the identification of points on opposite ends of the threads suggests that he is referring to a generalization of a Riemann surface known as a Klein surface.
The actual surface representing the equations is in four dimensional space with real coordinates (x,y,u,v), but Model 415 is made up of a pair of two dimensional sheets and does not show the coordinates (u,v) that produce the points satisfying those equations. By using the equation v=y, one can reduce the actual number of coordinates to three, x, y, and u. Since the defining equations of the surface are at most quadratic in u, the number of points on the model for a given value of (x,y) can only be 0, 1 or 2.
Baker’s model illustrates the curves on the model where the number of points is 1 by using vertical threads to connect the two sheets, thus illustrating that the points that lie above one another represent a single curve. These curves represent the boundary curve of the model and, therefore, none of the metal outside of the threads is actually part of the model of the surface. While one of these components of the boundary is represented by a small circle, the other three are asymptotic to the solid lines that run alongside them so these boundary curves run to infinity.
.The entry for Model 415 in Baker’s 1931 catalog is followed by the explanation “This model is closely related to Clebsch’ [sic] diagonal surface, giving a concise method of drawing 24 lines of the surface, 3 being principal lines at infinity.” All of the 27 lines of the Clebsch diagonal surface are real and 3 of them are lines at infinity. Baker numbered the 24 lines shown on the model using numbers between 1 and 27, omitting the three numbers assigned to the lines at infinity (13, 16, and 19).
As seen in image NMAH-AHB2019q017118, the number of each of the visible lines is written next to the point where the line leaves the model. Of the 24 numbered lines, 21 appear on both sheets of the model and only six lines appear on only one sheet (22, 23, and 24 on the upper sheet and 25, 26, and 27 on the lower sheet). Each of these six lines is asymptotic to the two different infinite components of the boundary curve. Each of the lines numbered 1 through 6 pass through a point of tangency of these infinite components of the boundary curve and move from one sheet to the other at that point as indicated by switching the representation of the line between solid and dashed on both sheets with the solid and dashed portions appearing above one another. Similarly, the lines numbered 7 through 12, 14, 15, 17, 18, 20, and 21 switch between solid and dashed representations and switch sheets at six different points of tangency to the finite component of the boundary curve.
References:
Richard P. Baker Papers, University Archives, Special Collections, The University of Iowa Libraries.
Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 17.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.076
accession number
211257
catalog number
211257.076

Model of a Klein-Riemann Surface by Richard P. Baker, Baker #415A

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa.
Description
This geometric model was constructed by Richard P. Baker in the early twentieth century 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 415 A and the initials R. P. B. are carved into one edge of the wooden base of this model. The typed part of a paper label on the upper sheet reads: “No. 415A (/) K. Riemann surface: the in-(/) verse transformation of (/) 415.” This means that the model is a Klein-Riemann surface for the same equations as for Baker’s Model 415 (MA*211257.076), i.e., v = y and 4u2 = 1 − 4(x + y) (1− x) (1 − y), but with metal sheets above the wooden base that represent parts of the u,v-plane rather than the x,y-plane. This equation is found in Baker’s handwritten notes on Models 415 and 415a and differs from what appears in Baker’s 1931 catalog, “Mathematical Models Made y R. P. Baker,” and on the label for Model 415.
While Baker did not define a Klein-Riemann surface, and that term does not appear to have been used except by him, the identification of points on opposite ends of the threads suggests that he is referring to a generalization of Riemann surface known as a Klein surface.
The actual surface representing the equations is in four dimensional space with real coordinates (u,v,x,y) but Model 415a is made up of a pair of two dimensional sheets and does not show the coordinates (x,y) that produce the points satisfying those equations. By using the equation v = y, one can reduce the actual number of coordinates to three, u, v, and x. Since the defining equations of the surface are at most quadratic in x, the number of points on the model for a given value of (u,v) can only be 0, 1 or 2.
Baker’s model illustrates the curves on the model where the number of points is 1 by using vertical threads to connect the sheets, thus illustrating that the points that lie above one another represent a single curve. The two metal sheets above the wooden base represent the part of the u,v-plane where that number is one or two and there are two curves that represent the points where it is one. The curve defined by the endpoints of the vertical threads represents the boundary curve of the model. The defining equations imply that one of these curves is the line v = 1 and the other is a cubic curve, 4u2 = v3 + v2 − v, which has two components, one of which is closed and does not meet the line v = 1. The other runs to infinity.
This model may have been restrung since, as can be seen in image NMAH-AHB2019q017117, the current placement of vertical threads does not show exactly what is described above. There are now two threads that do not correspond to the actual boundary curves of the model. These threads connect the component of the cubic curve that is not closed to the line v = 1, making it appear as if the boundary curves defined by the threads include two closed components, the smaller of which includes part of the line v = 1. One of these extra threads is above the label (23) Point and the other is above (26) Point. In fact, the two “Point” labels are referring to the points (±½,1) where the solid lines u = ±½ meet the line of threads v = 1. The numbers in parentheses refer to vertical lines on the surface that are not visible on the model but represent the equations in three dimensional space with real coordinates (u,v,x). Because none of the metal outside of the correctly placed threads is part of the model of the surface, the points (±½,1) are the only points that connect the parts of the sheets with v≤1 to those with v≥ 1.
As this model is the inverse transformation of Model 415, the lines seen in this model are also related to the lines of a Clebsch Diagonal Surface as noted in the catalog entry for Model 415, i.e., for both the Clebsch Diagonal Surface, and the surface represented in Models 415 and 415a, all of the 27 lines on the surface are real and 3 of them are lines at infinity. However, while on Model 415 there are 24 lines shown on the model, on Model 415a there are only 22 lines shown since, as mentioned above, lines 23 and 26 represent vertical lines through the points (−½,1) and (½,1), respectively. The numbers that are omitted on Model 415 (13, 16, and 19) also do not appear on this model.
All the lines shown on Model 415a appear in pairs that are located above one another on the two sheets and, with the exception of the solid lines u = ±½ (lines 22 and 24 are u = ½, and lines 25 and 27 are u = −½), one line of the pair is represented as a solid line and the other as a dashed line. All the lines that appear on the model pass from one sheet to the other when they are tangent to a component of the cubic boundary curve and, except for u = ±½, this movement is seen as a switch between a solid and a dashed line. The points of tangency of u = ±½ occur when v = −1 and are exceptional because they are double points and, therefore, cause the movement between sheets to be done twice so the lines remain solid when they pass the point of tangency.
References:
Richard P. Baker Papers, University Archives, Special Collections, The University of Iowa Libraries.
Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 17.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.077
accession number
211257
catalog number
211257.077

Model of a Potential Surface by Richard P. Baker, Baker #426

This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa. It has a black wooden base with a plaster surface atop it. The sides of the plaster are painted black, the top is white.
Description
This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa. It has a black wooden base with a plaster surface atop it. The sides of the plaster are painted black, the top is white. A typed paper tag attached to the sides reads at least in part: 426 (/) Potential surface z = (/) [log (x-1)2 + y2] + log [(x+1)2 + y2].
According to documents in the accession file, it is fact the surface associated with the function: z = log ((x-1) 2 + y2) + log ((x+1) 2 +y2) - log (x2 + y2).
Baker apparently made the model late in his career – it is not listed in the 1931 printed version of his catalog, but an annotation of the copy of the catalog in the accession file for the Baker collection lists it on p. 19 as a model for mathematical physics. Other annotations give the full equation represented.
The model was lent by the Baker family for exhibition at MIT in 1939 and came to the Smithsonian in 1956.
Reference:
Accession file 211257.
Location
Currently not on view
date made
1931-1935
1925-1935
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.078
accession number
211257
catalog number
211257.078

Model of a Spherical Representation of an Equipotential Surface by Richard P. Baker, Baker #427

This colorful model was made by Richard P. Baker in the early twentieth century when he was on the faculty in mathematics at the University of Iowa. It has a wooden base, painted black around the edges, with a painted grid showing polar coordinates atop this.
Description
This colorful model was made by Richard P. Baker in the early twentieth century when he was on the faculty in mathematics at the University of Iowa. It has a wooden base, painted black around the edges, with a painted grid showing polar coordinates atop this. Long bolts support three partitioned levels of painted metal circles above the base. Vertical rectangles join the circles.
A paper tag attached to the underside of the model indicates it is a model for Moebius's theorem, with Baker's model number 432d. On the other hand, a number etched into one of the edges of the base reads: 427.
According to Baker’s 1931 catalog, his model 427 is a model of the spherical representation of the equipotential surface represented by model 426 (an example of model 426 has museum number MA.211257.078). According the catalog, model 432d is of type his E, that is to say a wire and thread model with its projection shown a paper surface attached to a wooden base. This model is not of that type. Hence it seems likely that this is model 427, although no type for it is given in the catalog.
Reference:
Baker, R.P., Mathematical Models, Iowa City, 1931, p.1, 9, 13.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.079
accession number
211257
catalog number
211257.079

Model of a Cubic Surface by Richard P. Baker, Baker #430

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa.
Description
This geometric model was constructed by Richard P. Baker in the early twentieth century 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.
A typed label attached to a side of this model reads No. 430 (/) CUBIC SURFACE xyz-y+z=0. An entry for this model is in Baker's 1931 catalog in the section on "Analytic Geometry (/) Cyclides." Cyclides are quartic, not cubic, surfaces (e.g. they are of degree 2, not degree 3), which makes this grouping puzzling.
Although the model appears to be divided into four regions, the cubic surface has three separate components. In the photographs NMAH-DOR2014-00228 and NMAH-DOR2014-00230 one can see that one of the three components is colored pink and appears to have all z values positive and that another which is diagonally opposite it appears to have all z values negative. As seen in the same photographs, the third component is comprised of the part of the surface that is colored yellow and the part of the surface that appears above the label.
We can determine which is the first octant, i.e., where all three coordinates are either 0 or are positive, by looking at the vertical diagonals, y= x and y=-x. By substituting y for -x in the original equation we find that vertical diagonal is not defined for x=1 and x=-1. As can be seen on the two photographs listed above, this implies that y=-x is the vertical diagonal that intersects the two small components and, therefore, that the portion of the surface for which x=y must be contained in the union of the first octant and the octant in which all the coordinates are either 0 or are negative.
On that vertical diagonal the equation of the surface becomes x2z – x + z=0 so z=x/(x2 + 1). Since the denominator is always positive, z and x must have the same sign along the diagonal. It is clear from the first photograph listed above that the octant above the label satisfies the condition needed to be the first octant.
An entry describing this model appears in Baker’s 1931 catalog in an untitled subsection of the section on Analytic Geometry that follows the Cyclides subsection.
Reference:
Richard P. Baker, Mathematical Models, Iowa City, 1931, p. 10.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.080
accession number
211257
catalog number
211257.080

Model Relating to a Spherical Representation, by Richard P. Baker, Baker #430*

This painted plaster model is one of several hundred designed and made by Richard P. Baker, who taught mathematics at the University of Iowa in the early twentieth century. A typed paper label attached to one side reads: No. 430* (/) Spherical repr's (/) xyz + x - y = 0.
Description
This painted plaster model is one of several hundred designed and made by Richard P. Baker, who taught mathematics at the University of Iowa in the early twentieth century. A typed paper label attached to one side reads: No. 430* (/) Spherical repr's (/) xyz + x - y = 0. The surface has two clear local minimums and two local maximums.
The model apparently is not listed in Baker's 1931 catalog, but models with similar numbers are grouped in the category "analytic geometry."
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.081
accession number
211257
catalog number
211257.081

Model of Moebius's Theorem by Richard P. Baker, Baker #432a

This is a model relating to Moebius' theorem. It has a rectangular wooden base. A planar steel wire structure is stuck into holes in base. Black lines are drawn on the base, points labeled; and strings join wire and base.Currently not on view
Description
This is a model relating to Moebius' theorem. It has a rectangular wooden base. A planar steel wire structure is stuck into holes in base. Black lines are drawn on the base, points labeled; and strings join wire and base.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.082
accession number
211257
catalog number
211257.082

Model of Moebius's Theorem, by Richard P. Baker, Baker #432b

This is a model relating to Moebius' theorem. It is a painted wire structure.Currently not on view
Description
This is a model relating to Moebius' theorem. It is a painted wire structure.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.083
accession number
211257
catalog number
211257.083

Model of Culmann's Theorem by Richard P. Baker, Baker #432d

This geometric model was made by Richard P. Baker in the early twentieth century when he was on the faculty in mathematics at the University of Iowa.
Description
This geometric model was made by Richard P. Baker in the early twentieth century when he was on the faculty in 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 object has a rectangular wooden base covered with paper on which lines of projection are marked. A wire structure in two parts extends above the base, indicating the line segments projected (at least one wire is missing). A paper tag reads: Culmann's Theorem (/) THE TWO FUNICULARS (/) With the two dual polyhedra orthogonally (/) projected to the force polygon and funicular. In his catalog, Baker writes: The two funiculars with the dual polyhedra whose orthogonal projections are the force polygon and funicular. Baker grouped this model with three others associated with Moebius’ theorem.
Between 1864 and 1866, the German-born scholar Karl Culmann (1821-1881) of the Zurich Polytechnic Institute in Switzerland published a monograph on graphical statics from the point of view of projective geometry. He was particularly interested in connections between the funicular polygon (the figure assumed by a rope or cord with weights hanging from a number of points) and the force polygon (the diagram of the forces associated with the of the hanging weights).
References:
Baker, R. P., Mathematical Models, Iowa City, 1931, p. 13.
Culmann, K. Die graphische Statik, Zurich: Meyer & Zeller,1866.
Gerhardt, R., Kurrer, K., and Pichler, G., “The Methods of Graphical Statics and their Relation to the Structural Form,” Proceedings of the First International Congress on Construction History, Madrid, 20th-24th January 2003, ed. S. Huerta, Madrid: I. Juan de Herrera, SEdHC, ETSAM, A. E. Benvenuto, COAM, F. Dragados, 2003.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.084
accession number
211257
catalog number
211257.084

Model of Vector Addition by Richard P. Baker, Baker #433

This painted wire model is one of several hundred designed by Richard P. Baker, a British-born mathematics professor at the University of Iowa. It illustrates the addition of vectors - quantities that have both magnitude and direction.
Description
This painted wire model is one of several hundred designed by Richard P. Baker, a British-born mathematics professor at the University of Iowa. It illustrates the addition of vectors - quantities that have both magnitude and direction. Vectors are commonly associated with such physical variables as velocity and force, and have been used under that name from the late nineteenth century.
To find the sum of two vectors, one commonly places them head to tail, as in this model. The sum is represented by the magnitude and direction of the vector joining the tail of the first vector to the head of the second. As the order in which vectors are added doesn't matter, a parallelogram represents two ways of finding the sum, tie the diagonal of the parallelogram representing the sum itself.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.085
catalog number
211257.085
accession number
211257

Model for Dynamics by Richard P. Baker, Baker #435

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa.
Description
This geometric model was constructed by Richard P. Baker in the early twentieth century 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.
This painted metal and wire structure has a tag that reads: No. 435 (/) Displacement; angle Bisector. It has a wooden base painted black, a painted metal wire extending up for the base, and a metal representation of a bisecting plane between two planes. The metal piece is presently not joined to the supporting wire.
Baker built nine models to correspond to figures described by the German mathematician Eduard Study in a book on the geometry of dynamics (shortened by Baker to Dynamen). This model is one of them, which he described in his 1931 catalog as representing the “Construction of proper angle bisector for two planes which are transforms [sic] by a screw.”
References:
Baker, R.P., Mathematical Models, Iowa City, 1931, p. 16.
Study, E., Geometrie der Dynamen: Die Zusammensetzung von Kräften und Verwandte Gegenstände der Geometrie Bearb., Leipzig: B.G. Teubner, 1903, esp. p. 16, Fig. 2.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.086
accession number
211257
catalog number
211257.086

Model of Normal Planes of Chord of Screw by Richard P. Baker, Baker #436

This geometric model was made by Richard P. Baker in the early twentieth century when he was on the faculty in mathematics at the University of Iowa.
Description
This geometric model was made by Richard P. Baker in the early twentieth century when he was on the faculty in 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 model has a wooden base painted black, with a painted metal wire extending up for the base. Two metal pieces that would attach to this support are disconnected. A mark reads: No. 436 (/) Normal Planes, Chord of Screw .
Baker built nine models to correspond to figures described by the German mathematician Eduard Study in a book on the geometry of dynamics (shortened by Baker to Dynamen). This model is one of them, which he described in his 1931 catalog as representing “normal planes of chord of screw.”
References:
Baker, R.P., Mathematical Models, Iowa City, 1931, p. 16.
Study, E., Geometrie der Dynamen: Die Zusammensetzung von Kräften und Verwandte Gegenstände der Geometrie Bearb., Leipzig: B.G. Teubner, 1903, esp. p. 18, Fig. 3.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.087
accession number
211257
catalog number
211257.087

Model of a Keil Trapezoid by Richard P. Baker, Baker #437

This is a model of trapezoidal wedge, a surface with two triangular and three trapezoidal faces.Here two triangular faces and two trapezoidal faces of a wedge are shown in sheet metal.
Description
This is a model of trapezoidal wedge, a surface with two triangular and three trapezoidal faces.Here two triangular faces and two trapezoidal faces of a wedge are shown in sheet metal. Two interior intersecting triangles that bisect the wedge and join opposite sides of the triangles are also shown in sheet metal - the trapezoidal face that would cover them is not shown. Wires rising from two edges of one of these bisecting triangles aand from the line of intersection of the bisectors form two sides and the diagonal of a parallelogram.
Baker based this model on a diagram from a volume on dynamics by the German Eduard Study.
References:
Baker, R. P., Mathematical Models, Iowa City, Iowa, 1931, p. 16.
Study, E., Geometrie der Dynamen: Die Zusammensetzung von kräften und verwandte Gegenstände der Geometrie, Leipzig: B.G. Teubner, 1901, esp. Figure 5 , p. 33.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.088
catalog number
211257.088
accession number
211257

Model of the Geometric Sum of Spherical Steps by Richard P. Baker, Baker #438

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa.
Description
This geometric model was constructed by Richard P. Baker in the early twentieth century 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.
This painted wire structure has a tag that reads: No. 438 (/) Geometric sum spherical (/) steps.
Baker built nine models to correspond to figures described by the German mathematician Eduard Study in his book on the geometry of dynamics (shortened by Baker to Dynamen). Examples with numbers MA.211257.089 (this one) and MA.211257.090 relate to finding the sum of spherical wedges.
References:
Study, E., Geometrie Der Dynamen: Die Zusammensetzung Von Kräften Und Verwandte Gegenstände Der Geometrie Bearb., Leipzig: B.G. Teubner, 1903, esp. pp. 34-35.
Baker,R.P., Mathematical Models,, Iowa City, 1931, p. 16.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.089
accession number
211257
catalog number
211257.089

Model by Richard P. Baker, Dual of Model 438, Baker #438a

This geometric model was constructed by Richard P. Baker in the early twentieth century when he was Associate Professor of Mathematics at the University of Iowa.
Description
This geometric model was constructed by Richard P. Baker in the early twentieth century 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.
This painted wire structure includes six directional arrows. It has a paper tag that reads: No. 438* (/) Dual of 438. Although the inscription denotes the model as 438*, Baker's catalog lists it as 438a.
Baker built nine models to correspond to figures described by the German mathematician Eduard Study in his book on the geometry of dynamics (in his catalog, Baker shortened the title to Dynamen). Examples with numbers MA.211257.089 and MA.211257.090 (this one) relate to finding the sum of spherical wedges.
References:
Study, E., Geometrie Der Dynamen: Die Zusammensetzung Von Kräften Und Verwandte Gegenstände Der Geometrie Bearb., Leipzig: B.G. Teubner, 1903, esp. pp. 34-35.
Baker,R.P., Mathematical Models, Iowa City, 1931, p. 16.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.090
accession number
211257
catalog number
211257.090

Model for the Petersen-Morley Theorem by Richard P. Baker, Baker #441

This geometric model was made by Richard P. Baker in the early twentieth century when he was on the faculty in mathematics at the University of Iowa.
Description
This geometric model was made by Richard P. Baker in the early twentieth century when he was on the faculty in 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.
Now in stored in a cardboard box with lid, the painted wire structure is presently in four pieces.
This is one of nine models Baker built to correspond to figures described by the German mathematician Eduard Study inthe book cited below (Baker shortened the title to Dynamen). Baker described it in his 1931 catalog as representing the “Petersen-Morley theorem.”
The Petersen-Morley theorem was developed independently by the Danish mathematician Johannes Petersen (1873-1950) and the English-born American mathematician Frank Morley (1860-1937). {Johannes Petersen would change his last name to Hjelmslev}. The theorem relates to the properties of line segments of shortest distance joining three skew lines. Morley presented a model relating to the theorem to the London Mathematical Society in 1898 and, after he had moved to the United States, an expanded version of it to the American Mathematical Society in 1899. Study also cites Petersen’s work from 1898.
References:
Baker, R.P., Mathematical Models, Iowa City, 1931, p. 16.
Lützen, J. "The Mystery of Ten Wooden Blocks: Hjelmslev’s Geometry of Reality," Math Semesterber,2020, vol. 67, pp. 161-167.
Study, E., Geometrie der Dynamen: Die Zusammensetzung von Kräften und Verwandte Gegenstände der Geometrie Bearb., Leipzig: B.G. Teubner, 1903, esp. pp. 106-107.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.091
accession number
211257
catalog number
211257.091

Model of Stereographic Projection by Richard P. Baker, Baker #443

A stereographic projection maps a sphere (or, in this case, a portion of a sphere) onto a plane.
Description
A stereographic projection maps a sphere (or, in this case, a portion of a sphere) onto a plane. One assumes that the eye of an observer is one of the poles of the sphere, one draws line segments between that pole and points of the sphere, and the points where the line segments intersect the plane are the points of the projection. Baker’s wire model on a wooden base shows a sphere with two poles, five latitude circles, and eight longitude circles. The observer is supposed to be at the top pole. The lower three latitude circles project onto circles penciled on the base (the upper latitude circles also would project into circles, but these would be outside the base of the existing model). The longitude circles project onto lines on the base (only part of these lines shows – they extend indefinitely from the lower pole). Also shown is the projection of a point on the sphere onto the base and a projection of a triangle on the sphere onto a triangle on the base.
A typed paper tag on the bottom of the base of the model reads: Mx No. 443 (/) Stereographic Projection.
Scientific instruments known as astrolabes – which were outdated by Baker’s time - used stereographic projection. In them, the eye of the observer is presumed to be at the celestial south pole. The center of the projection is the celestial north pole, and the circles are those on a celestial sphere. To explain the projection in surviving astrolabes, it would be easiest to invert Baker’s model.
Reference:
R. P. Baker, Mathematical Models, Iowa City, Iowa, 1931, p. 17.
Location
Currently not on view
date made
ca 1930-1935
maker
Baker, Richard P.
ID Number
MA.211257.092
accession number
211257
catalog number
211257.092

Model of the Real Part of the Function Cos Z by Richard P. Baker, Baker #444

This model is one of several hundred designed by Richard P.
Description
This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa.
Particularly from the nineteenth century, mathematicians and physicists have found it useful to consider functions of complex numbers, that is to say numbers of the form Z = a + bi, where i represents the square root of negative one and a and b are real numbers. Baker’s 1931 catalog listed seven models which he grouped as relating to the field of complex analysis; examples of four of these survive at the National Museum of American History. These are his #443 (MA.211257.092), #310 (MA.211257.050), #444 (MA.211257.093 – this model), and #446 (MA.211257.094). The models came to the Smithsonian in 1956 after exhibition at MIT.
Just as complex numbers have real and imaginary parts, so do functions of them. This model represents the real part of the function cos (Z). If, as before, Z = a + bi, it represents the values of the product cos a cosh b, where cosh b is the hyperbolic cosine of b.
The model has a wooden base and sides, with a white plaster surface representing values of cos (Z). A tag on the side reads: No. 444 (/) The real part of Cos Z.
Values of the cosine, which run from -1 to +1 and back to -1, form the upper edge of the two higher opposite wooden sides. The upper edge of the other two opposite sides are values of -cosh b, with values of this function running from -1 to 0 to -1 (-coshb is the value of the function cos a coshb whien cosa = -1, as it does in the planes that contains these sides). Values of the product of the two functions within these intervals are represented by the plaster surface.
Baker also designed a model for the imaginary part of the function cos (Z), but it does not survive in the collections.
Reference:
Richard P. Baker, Mathematical Models, Iowa City, 1931, p.17.
Accession file 211257.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.093
accession number
211257
catalog number
211257.093

Model of the Absolute Value of Cos Z by Richard P. Baker, Baker #446

This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa. Plaster on top, it has wooden sides and base, both painted black.
Description
This model is one of several hundred designed by Richard P. Baker, a mathematics faculty member at the University of Iowa. Plaster on top, it has wooden sides and base, both painted black. The plaster is color-coded, with a scale on one side explaining the meaning of the spectrum of colors used. A paper tag taped to one side reads: No. 446 The absolute value (/) of Cos Z.
Particularly from the nineteenth century, mathematicians and physicists have found it useful to consider functions of complex numbers, that is to say numbers of the form Z = x + yi, where i represents the square root of negative one and x and y are real numbers. Just as complex numbers have real and imaginary parts, so do functions of them. This model represents the absolute value of the function cos (Z) where, as before, Z = x + yi. This consists of positive values of the equation (cos 2x cosh2 y + sin2x sinh2y) 1/2. This equation can be reduced to (cos2x + sinh2y)1/2. In the model, values of the angle x run from 0 to 2 pi , while those of y run from roughly -3 to 0 to roughly +3. The colored scale indicates regions corresponding to different values of x. For smaller and for larger values of y, values of sinh2y become much larger and the small fluctuations associated with the first term in the equation would not be visible.
Baker’s 1931 catalog listed seven models which he grouped as relating to the field of complex analysis; examples of four of these survive at the National Museum of American History. These are his #443 (MA.211257.092), #310 (MA.211257.050), #444 (MA.211257.093), and #446 (MA.211257.094 – this model). The models came to the Smithsonian in 1956 after exhibition at MIT.
References:
Baker, R.P., Mathematical Models, Iowa City, 1931, p.17.
Accession file 211257
http://math.pugetsound.edu/~martinj/courses/spring2010/m352/ComplexVariables_Sec33.pdf , accessed August 27, 2020.
https://functions.wolfram.com/ElementaryFunctions/Cos/visualizations/5/, accessed August 27, 2020.
Location
Currently not on view
date made
ca 1906-1935
maker
Baker, Richard P.
ID Number
MA.211257.094
accession number
211257
catalog number
211257.094

Model of a Gaussian Surface by Richard P. Baker, Baker #448

This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa.
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 19th-century 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 1915-1935
maker
Baker, Richard P.
ID Number
MA.211257.095
accession number
211257
catalog number
211257.095

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