# Geometric Models - Minimal Surfaces as Soap Films

• Contents

A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces greatly interested a few nineteenth century physicists and mathematicians, fascinated by their connection to soap films. In the 1880s, students at the technical high school in Munich, Germany, under the direction of their professor, Alexander Brill, designed a series of wire models that, when dunked in appropriate soapy water, produced intriguing surfaces.

Brill arranged that these and other mathematical models be manufactured and distributed by his brother Ludwig Brill. Examples were exhibited at the world’s fair held in Chicago in 1893. These and the other Brill models shown at the fair were purchased by Wesleyan University, which later donated them to the Smithsonian. The mathematics associated with soap films continues to fascinate both scholars and the general public.

### Geometric Model, L. Brill No. 148. Ser. 10 No. 1a, Minimal Surface in Two Parts

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model.
This, the first model in the series, is in two parts. One is a wire circle with three wire supports and the second circle of the same diameter with a handle. If the circles are held together, a soap film spans both of them. Separating the circles produces a film in the shape of a catenoid and then, when the circles are separated further, circles span the two discs separately.
This example was exhibited at the Columbian Exposition, a world's fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
A. T. Fomenko, The Plateau Problem, Part I, Historical Survey, New York: Gordon and Breach, 1990, pp. 22-34.
J. Plateau, Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires, vol. 1, Paris: Gauthier-Villars, 1873, pp. 93-103. This work is mentioned in the Brill catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.106
catalog number
1985.0112.106
accession number
1985.0112

### Geometric Model, L. Brill No. 148 . Ser. 10 No. 1b, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the second model in the series, has a straight vertical axis with a spiral on the outside. The form of the soap film is a helicoid.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
J. Plateau, Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires, vol. 1, Paris: Gauthier-Villars, 1873, pp. 216-217. This work is mentioned in the Brill catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.107
catalog number
1985.0112.107
accession number
1985.0112

### Geometric Model, L. Brill No. 148. Ser. 10 No. 1c, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the third model in the series, is in the shape of a regular octahedron with a handle rising from one vertex.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
H.A. Schwarz, Bestimmung einer speciellen Minimalfläche, Berlin: F. Dümmler's Verlags-Buchhandlung, 1871. This source is mentioned in Brill’s catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.108
catalog number
1985.0112.108
accession number
1985.0112

### Geometric Model, L. Brill No. 148. Ser. 10 No. 1d, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the fourth model in the series, is in the shape of a four-sided pyramid with a handle attached to one vertex.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
H.A. Schwarz, Bestimmung einer speciellen Minimalfläche, Berlin: F. Dümmler's Verlags-Buchhandlung, 1871. This source is mentioned in Brill’s catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.109
catalog number
1985.0112.109
accession number
1985.0112

### Geometric Model, L. Brill No. 148. Ser. 10 No. 1e, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the fifth model in the series, is in the shape of a three-sided prism, with a handle attached at one corner. The dimensions given are for when model is placed upright.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
H.A. Schwarz, Bestimmung einer speciellen Minimalfläche, Berlin: F. Dümmler's Verlags-Buchhandlung, 1871. This source is mentioned in Brill’s catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.110
catalog number
1985.0112.110
accession number
1985.0112

### Geometric Model, L. Brill No. 148. Ser. 10 No. 1f, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the sixth model in the series, is in the shape of a regular tetrahedron with a handle at one vertex.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
H.A. Schwarz, Bestimmung einer speciellen Minimalfläche, Berlin: F. Dümmler's Verlags-Buchhandlung, 1871. This source is mentioned in Brill’s catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.111
catalog number
1985.0112.111
accession number
1985.0112

### Geometric Model, L. Brill No. 148 g Ser. 10 No. 1g, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the seventh model in the series, is in the shape of a cube. A handle is attached to a wire joining the midpoints of two opposite sides.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
H.A. Schwarz, Bestimmung einer speciellen Minimalfläche, Berlin: F. Dümmler's Verlags-Buchhandlung, 1871. This source is mentioned in Brill’s catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.112
catalog number
1985.0112.112
accession number
1985.0112

### Geometric Model, L. Brill No. 148. Ser. 10 No. 1h, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the eighth model in the series, is in the shape of a six-sided prism with alternate edges missing on the top and the bottom faces . A handle extends from the top.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
H.A. Schwarz, Bestimmung einer speciellen Minimalfläche, Berlin: F. Dümmler's Verlags-Buchhandlung, 1871. This source is mentioned in Brill’s catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.113
catalog number
1985.0112.113
accession number
1985.0112

### Geometric Model, L. Brill No. 148 (?). Ser. 10 No. 1i?, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. It is one of a series. The model is in the shape of a four-sided prism with near rectangular sides. Half of its edges are missing, and the edges at the bottom are curved. There is a handle at the top. Either model 1985.0112.114 of model 1985.112.115 may be Brill Ser. 10 No. 1i. That model was designed to illustrate one of the surfaces proposed by the German mathematician Heinrich Scherk (1798-1885) in a paper of 1835.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
Scherk, H.F., “Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen,” Journal fuer die reine und angewandte Mathematik, 13, 1835, pp. 185-208. This article is mentioned in the Brill catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.114
catalog number
1985.0112.114
accession number
1985.0112

### Geometric Model, L. Brill No. 148 (?). Ser. 10 No. 1i?, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. It is one of a series. The model is in the shape of a rhombus, bent along a diagonal. A handle rises from one of the corners not at the bend. Either model 1985.0112.114 or model 1985.112.115 may be Brill Ser. 10 No. 1i. That model was designed to illustrate one of the surfaces proposed by the German mathematician Heinrich Scherk (1798-1885) in a paper of 1835.
This example was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
Scherk, H.F., “Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen,” Journal fuer die reine und angewandte Mathematik, 13, 1835, pp. 185-208. This article is mentioned in the Brill catalog.
Location
Currently not on view
1892
maker
L. Brill
ID Number
1985.0112.115
catalog number
1985.0112.115
accession number
1985.0112

### Geometric Model, L. Brill No. 148. Ser. 10 No. 1k, Minimal Surface

Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885.
Description
Students at the technical high school in Munich, working under the direction of Alexander Brill, developed a series of wire models of minimal surfaces that was first published by Ludwig Brill in 1885. A minimal surface is the surface of smallest area of all the surfaces bounded by a closed curve in space. Its mean curvature is zero. Minimal surfaces are often represented by soap films, as was the intention with this model. This, the final model of the series, is in the shape of two rectangles intersecting at a right angle, with a handle extending from one point of intersection. It is designed to illustrate one of the surfaces proposed by the German mathematician Heinrich Scherk (1798-1885) in a paper of 1835.
This example was exhibited at the Columbian Expositoin, a world's fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 21, 85.
G. Fischer, Mathematical Models: Commentary, Braunschweig / Wiesbaden: Friedr. Vieweg & Sohn, 1986, pp. 41-43.
Scherk, H.F., “Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen,” Journal fuer die reine und angewandte Mathematik, 13, 1835, pp. 185-208. This article is mentioned in the Brill catalog.
Location
Currently not on view