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

National Museum of American History, Kenneth E. Behring Center
This painted plaster model showing contour lines fits in an open wooden box. In a catalog from about 1905, Baker described the surface shown as “An area which can be derived by deformation from 92 [e.g. model MA.211257.014] without losing the descriptive character of contours, except that the inloop curve becomes an outloop (which may occur in infinitesimal transformation). The contours are now curves of the type of equipotential lines, and the configuration is made as symmetrical as possible.” References: R. P. Baker, A List of Mathematical Models, [1905], p. 16. A copy of this document is in the Baker Papers at the University of Iowa Archives. R. P. Baker, >Mathematical Models>, Iowa City, Iowa, 1931, p. 5.

Plaster Model for Function Theory by L. Brill, No. 176, Ser. 14 No. 4

National Museum of American History, Kenneth E. Behring Center
This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. It was manufactured by the Darmstadt publishing company of Ludwig Brill and is model number 4 of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is both an R and an I inscribed on model 4, but on adjacent vertical faces, with the I on what looks like the back of the model. On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm. on model 4. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 4 is related to the level curves. The equation involving complex variables on which model 4 is based is w = 1/z. That equation defines a surface in 4 dimensions. Model 4 is a three-dimensional model and is defined by an equation using three real variables. That equation is found by replacing w by u + vi and z by x + yi, to get u + vi = (x – yi)/(x2 + y2), which is equivalent to the two equations u = x/(x2 + y2) and v = –y/(x2 + y2). . Normally this process produces two very different equations and three-dimensional models, but in this case, the two equations are identical except that switching from u to v, changes x to –y and y to –x. Therefore, these equations define the same model but with the x and y axes rotated 90 degrees. Only points on the curved surface of the solid model satisfy the equation that defines it; points in the solid plaster that supports that surface do not satisfy that equation. Computer generated versions show only the surface so are able to show details of the model lying below the complex z-plane. Plots of the surfaces produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Each has an R and an I superimposed to show the faces where they appear on model 4. The computer plots looking directly at the faces with the inscribed R and I show a hollow cropped spire and the same spire rotated 180 degrees around the origin to face downward. References: L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-71. G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 126, p. 123, and vol. 2 (Commentary), pp. 71-73.

Plaster Model for Function Theory by L. Brill, No. 175, Ser. 14 No. 3

National Museum of American History, Kenneth E. Behring Center
This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model was manufactured by the Darmstadt publishing company of Ludwig Brill and is number 3 of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is both an R and an I inscribed on the front of model 3, indicating that the vertical axis can represent either u or v. On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1/2 cm. on model 3. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 3 is related to the level curves on that model. The equation involving complex variables on which model 3 is based is w4 = 1 - z2. That equation defines a surface in four dimensions. Model 3 is a three-dimensional model and is defined by two almost identical equations, each using three real variables. Those, very complicated, equations are found by replacing w by u + vi and z by x + yi, and then eliminating u or v. Normally this process produces two very different equations and three-dimensional models, but in this case, the two equations are identical except that one includes u’s and the other v’s and they define the same model. The polynomial equations that define model 3 start with a term with a coefficient 256 and exponent of the variable 16. Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. Computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surfaces produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Each has an R and an I superimposed approximately where they appear on model 3. While model 3 does not show it, the computer generated plot positioned to look directly at the front shows a hole with a complicated boundary. References: L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-71. G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 125, p.122, and vol. 2 (Commentary), pp. 71-72, 75-76.

Group of Two Plaster Models for Function Theory by L. Brill, No. 173, Ser. 14 No. 1a and 1b

National Museum of American History, Kenneth E. Behring Center
This group of two plaster models was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The models were manufactured by the Darmstadt publishing company of Ludwig Brill and this group contains model numbers 1a and 1b of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. In this group, there is an R inscribed on the front of model 1a and an I inscribed on the front of model 1b. On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm. on models 1a and 1b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 1a is related to the level curves on model 1b. Similarly, the placement of the gradient curves model 1b is related to the level curves on the model 1a. The equation involving complex variables on which models 1a and 1b are based is w2 = z2 - 1. That equation defines a surface in 4 dimensions. Models 1a and 1b are 3-dimensional models, each of which is defined by an equation using three real variables. Those, much more complicated, equations are found by replacing w by u + vi and z by x + yi, and then eliminating u or v. Model 1a is defined by u4 - (x2 - y2 -1)u2 - x2y2 = 0 while model 1b is defined by v4 + (x2 - y2 -1)v2 - x2y2 = 0. Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. Computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surfaces produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Each has an R or an I superimposed approximately where it appears on the corresponding model. While model 1b does not show it, the computer generated plot positioned to look directly at the front does show a hole whose boundary is the unit circle centered at the origin and lying in the vertical plane through the x axis. References: L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-71. G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photos 123 (model 1a) and 124 (model 1b), pp. 120-121, and vol. 2 (Commentary), pp. 71-72.

Group of Two Plaster Models for Function Theory by L. Brill, No. 174, Ser. 14 No. 2a and 2b

National Museum of American History, Kenneth E. Behring Center
This group of two plaster models was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The models were manufactured by the Darmstadt publishing company of Ludwig Brill and this group contains model numbers 2a and 2b of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. In this group, there is an R inscribed on the front of model 2a and an I inscribed on the front of model 2b. On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1/2 cm on models 2a and 2b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 2a is related to the level curves on model 2b. Similarly, the placement of the gradient curves model 2b is related to the level curves on the model 2a. The equation involving complex variables on which models 2a and 2b are based is w2 = z4 - 1. That equation defines a surface in 4 dimensions. Models 2a and 2b are 3-dimensional models, each of which is defined by an equation using three real variables. Those, much more complicated, equations are found by replacing w by u + vi and z by x + yi, and then eliminating u or v. Model 2a is defined by u4 - (x4 - 6x2y2 + y4 - 1)u2 - 4x2y2(x2 - y2)2 = 0 while model 2b is defined by v4 + (x4 - 6x2y2 + y4 - 1)v2 - 4x2y2(x2 - y2)2 = 0. Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. Computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surfaces produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Each has an R or an I superimposed approximately where it appears on the corresponding model. While model 2b does not show it, the computer generated plot positioned to look directly along the plane y = x does show a significant hole and that same shaped hole can be seen by rotating the plot 90 degrees and looking directly along the plane y = -x. References: L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-71. “Mathematische Modelle angefertigt im mathematischen Institut der k. technischen Hochschule in München unter Leitung von Professor Dr. Walther Dyck. Modelle zur Functionentheorie (Zu Serie XIV)” [“Mathematical Models Made in the Mathematical Institute of the Royal Technical College in Munich under the Direction of Professor Dr. Walther Dyck. Models for Function Theory (For Series 14)”], pp. 1-5 and fig. 2 plate I.

Group of Two Plaster Models for Function Theory by L. Brill, No. 179, Ser. 14 No. 7a and 7b

National Museum of American History, Kenneth E. Behring Center
This group of two plaster models was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The models were manufactured by the Darmstadt publishing company of Ludwig Brill and this group contains model numbers 7a and 7b of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. In this group, there is an R inscribed on the front of model 7a and an I inscribed on the front of model 7b. On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm. on models 7a and 7b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 7a is related to the level curves on model 7b. Similarly, the placement of the gradient curves model 7b is related to the level curves on the model 7a. Models 7a and 7b are based on a Weierstrass P-function. These complex valued functions are named after the nineteenth century German mathematician, Karl Weierstrass and each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the Weierstrass P-function is the same for corresponding points of the parallelograms of the tiling. The tiling associated with models 7a and 7b is made up of squares with sides parallel to the x and y axes. There are four such squares in each of the models so models 7a and 7b are both made up of four congruent sections each of which has a square base and has at its center a pair of cropped spires and a pair of narrowing holes. Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surfaces produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Each has an R or an I superimposed approximately where it appears on the corresponding model. For models 7a and 7b, the computer generated versions show the four congruent sections, each of which includes two spires that are hollow and two holes that are downward pointing versions of the hollow spires. References: L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-72. G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photos 129 (model 7a) and 130 (model 7b), pp. 126-127. and vol. 2 (Commentary), pp. 71-72, 75-76.

Plaster Model for Function Theory by L. Brill, No. 180, Ser. 14 No. 8

National Museum of American History, Kenneth E. Behring Center
This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model was manufactured by the Darmstadt publishing company of Ludwig Brill and is number 8 of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is both an R and an I inscribed on model 8. On adjacent faces of model 9 there is an R and an I inscribed, with the R on the face with the labels. On each model in series 14 there are two sets of curves that act much like the lines on 2-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm on model 8. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 8 is related to the level curves. Model 8 is based on the derivative of the Weierstrass P-function on which Brill models 7a and b (1985.0112.139) are based. Complex valued Weierstrass P-functions are named after the nineteenth century German mathematician, Karl Weierstrass and the derivative of each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the derivative of the Weierstrass P-function is the same for corresponding points of the parallelograms of the tiling. The tiling associated with models 7a, 7b, and 8 is made up of squares with sides parallel to the x and y axes and there are four such squares in each of the models. Model 8 is made up of four congruent sections each of which has a square base and three cropped spires alternating with three narrowing holes equally spaced around the center of each square. Only points on the curved surfaces of model 8 satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surface produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Although each plot has both an R and an I superimposed approximately where it appears on the model, if there is an R or an I facing the front, the x axis is parallel to that face and the vertical axis is labeled u if R is on that face and is labeled v if I is on it. For model 8, as well as models 7a and 7b, the computer generated versions show four congruent sections. In model 8 each section includes three hollow spires alternating with three downward pointing versions of the hollow spires. References L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-72. G. Fischer, ed. Mathematical Models: From the Collections of Universities and Museums, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 131, p. 128, and vol. 2 (Commentary), pp. 71-72. 75-76.

Group of Two Plaster Models for Function Theory by L. Brill, No. 181, Ser. 14 No. 9a and 9b

National Museum of American History, Kenneth E. Behring Center
This group of two plaster models was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The models were manufactured by the Darmstadt publishing company of Ludwig Brill and this group contains model numbers 9a and 9b of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. In this group, there is an R inscribed on a vertical face of model 9a and an I inscribed on a vertical face of model 9b. However, since there is no vertical face of either model that is parallel to the x axis, the R and I are inscribed on faces that meet the x axis in what appears to be a 30 degree angle. On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm on models 9a and 9b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 9a is related to the level curves on model 9b. Similarly, the placement of the gradient curves model 9b is related to the level curves on the model 9a. Models 9a and 9b are based on a Weierstrass P-function. These complex valued functions are named after the nineteenth century German mathematician, Karl Weierstrass and each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the Weierstrass P-function is the same for corresponding points of the parallelograms of the tiling. One tiling associated with the Weierstrass P-function defining models 9a and 9b is made up of rhombuses whose angles are 60 and 120 degrees and with one pair of sides parallel to the x axis. One can see one such rhombus in models 9a and 9b by joining the four points where the tops of each pair of cropped spires meet. The sides of the rhombus parallel to the x axis pass through the center of the cropped spires in model 9a and pass between a cropped spire and a hole in model 9b. The tiling by rhombuses leads to an alternate tiling by regular hexagons with one pair of sides parallel to the y axis and with the structures that were centered at each vertex of the tiling by rhombuses sitting at the center of each hexagon of the tiling. Models 9a and 9b are made up of four such hexagons that have been slightly trimmed. This tiling is difficult to see directly on the model but can be easily seen in two dimensional plots mimicking bird’s-eye views of the surfaces that were produced using the program Mathematica. These plots have been superimposed with an outline in red of four of the tiling hexagons and thicker black lines that show the models’ footprints. In addition, +’s and –‘s have been placed to indicate the location of the cropped spires (+’s) and the tapering holes (-‘s). These two-dimensional plots also show why the models do not have rectangular footprints and why the R and I could not be placed on faces parallel to the x axis. Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The computer generated versions show only the surfaces, and so are able to show details that would be difficult to portray on a plaster model. Plots of the surfaces produced using the program Mathematica show scales to indicate the direction of at least two of the variables. Each has an R or an I superimposed approximately where it appears on the corresponding model. For models 9a and 9b, the computer generated versions show that the spires are hollow and the pairs of tapering holes are downward pointing versions of the pairs of upward pointing hollow spires. These versions are produced by taking x and y values from a rectangle grid so extra portions of the surfaces are seen, including parts of single spires in the left front and right rear corners. References: L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-73. “Mathematische Modelle angefertigt im mathematischen Institut der k. technischen Hochschule in München unter Leitung von Professor Dr. Walther Dyck. Modelle zur Functionentheorie (Zu Serie XIV)” [“Mathematical Models Made in the Mathematical Institute of the Royal Technical College in Munich under the Direction of Professor Dr. Walther Dyck. Models for Function Theory (For Series 14)”], pp. 1-3, 7-8, 11-13 and fig. 6 plate III.

Plaster Model for Function Theory by L. Brill, No. 182, Ser. 14 No. 10b

National Museum of American History, Kenneth E. Behring Center
This model is one of a group of two plaster models that was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model was manufactured by the Darmstadt publishing company of Ludwig Brill and is number 10b of the group that also contains model 10a of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886. The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later. Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is an I inscribed on a vertical face of model 10b but, since there is no vertical face of the model that is parallel to the x axis, the I is inscribed on a face that meets the x axis in what appears to be a 30 degree angle. On each model in series 14 there are two sets of curves that act much like the lines on 2-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm on model 10b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 10ba is related to the level curves on model 10a, which is not in the museum collections. Model 10b is based on the derivative of the Weierstrass P-function on which Brill models 9a and b (1985.0112.141) are based. Complex valued Weierstrass P-functions are named after the nineteenth century German mathematician, Karl Weierstrass and the derivative of each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the derivative of the Weierstrass P-function is the same for corresponding points of the One can see one such rhombus in model 10b by joining the four points where the tops of three cropped spires meet. At each vertex of the rhombus the sides of the rhombus parallel to the x axis pass in front of a cropped spire. parallelograms of the tiling. The tiling associated with models 9a, 9b, 10a, and 10b is made up of rhombuses whose angles are 60 and 120 degrees and with one pair of sides parallel to the x axis. One can see one such rhombus in model 10b by joining the four points where the tops of three cropped spires meet. At each vertex of the rhombus the sides of the rhombus parallel to the x axis pass in front of a cropped spire. The tiling by rhombuses leads to an alternate tiling by regular hexagons with one pair of sides parallel to the y axis and with the structures that were centered at each vertex of the tiling by rhombuses sitting at the center of each hexagon of the tiling. Model 10b is made up of four such hexagons that have been slightly trimmed. This tiling is difficult to see directly on the model but can be easily seen in a two dimensional plot mimicking a bird’s-eye view of the surface that were produced using the program Mathematica. This plot has been superimposed with an outline in red of four of the tiling hexagons and thicker black lines that show the model’s footprint. In addition, +’s and –‘s have been placed to indicate the location of the cropped spires (+’s) and the tapering holes (-‘s). These two-dimensional plots also show why the models do not have rectangular footprints and why the R and I could not be placed on faces parallel to the x axis. Only points on the curved surfaces of the model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The computer generated versions show only the surfaces, and so are able to show details that would be difficult to portray on a plaster model. Plots of the surface produced using the program Mathematica show scales to indicate the direction of at least two of the variables and each has an I superimposed approximately where it appears on the model. For model 10b, as well as models 9a and 9b, the computer generated versions show four congruent sections. In model 10b each section includes three hollow spires alternating with three downward pointing versions of the hollow spires. The plots are produced by taking x and y values from a rectangle grid so extra portions of the surface are seen, including parts of additional spires that are outside the footprint of the model. A version of this plot has been overlaid with the two sides of the rhombus that are parallel to the x axis and the long diagonal of the rhombus, which is parallel to the base of the vertical face with the inscribed I. References: L. Brill, Catalog mathematischer Modelle, Darmstadt, 1892, pp. 29-30, 70-73. “Mathematische Modelle angefertigt im mathematischen Institut der k. technischen Hochschule in München unter Leitung von Professor Dr. Walther Dyck. Modelle zur Functionentheorie (Zu Serie XIV)” [“Mathematical Models Made in the Mathematical Institute of the Royal Technical College in Munich under the Direction of Professor Dr. Walther Dyck. Models for Function Theory (For Series 14)”], pp. 1-3, 7-8, 11-13 and fig. 7 plate III.

A phylogenetic analysis of the Orchidaceae / Pamela Burns- Balogh and V.A. Funk

Smithsonian Libraries
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The Orchidaceae is the largest flowering plant family, with approximately 25,000 species. Sixty-eight apomorphies grouped into forty-two transformation series were used to construct a cladogram for the twenty-six tribes of the family. A detailed discussion of the characters is followed by an in-depth analysis of the cladogram. The cladogram was used to develop a classification and a natural key to the tribes. Seven subfamilies (Neuwiedioideae, Apostasioideae, Cypripedioideae, Spiranthoideae, Neottioideae, Orchidoideae, and Epidendroideae) are divided into 20 tribes, the majority of which are defined by synapomorphies. Of the intrasubfamily classifications, that of the Epidendroideae is the most tenuous. The phylogeny and classification presented here are hypotheses of relationship and are therefore subject to change as more information becomes available.
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