# Geometric Model, L. Brill No. 187. Ser. 15 No. II, Model for Function Theory

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Description
This model was among those sold by Ludwig Brill and exhibited at the Columbian Exposition, a World’s Fair held in Chicago in 1893. It was published by Brill in 1886 and was based on a result in the Göttingen University dissertation written by Karl Reinbeck (1859-1939), Über diejenigen Flächen auf welche die Flächen zweiten Grades durch parallele Normalen conform abgebildet werden ("On those surfaces on which the surfaces of second degree are mapped by parallel normals"). This model represents the case when the surface of second degree is an ellipsoid.
While both the dissertation title and the title of the model are clearly technical, they do not give a hint as to the restrictive result to which they refer. That result ties together mappings that preserve angles of corresponding intersecting curves (conformal maps) and those that preserve the direction of corresponding normals (the normal at a point is the unique line that is perpendicular to the surface at that point).
If one surface is simply a translation in space of the other (the original surface is moved without any rotation), the mapping between them is both normal preserving and conformal. Reinbeck showed that being normal preserving and conformal is so restrictive that if the mapping is not simply a translation, there is only one other possibility. Moreover, in that case, the two surfaces have curvature of opposite signs where surfaces such as spheres and ellipsoids have positive curvature while hyperboloids have negative curvature.
This model shows a part of the second surface when the first one is an ellipsoid. The displayed surface clearly has negative curvature although it is not a hyperboloid.
The white plaster object has a rectangular base and top, with two vertical holes in the center. A paper tag reads: 187. Another paper tag reads: Bestimmung der Fläche, (/) auf welche des Ellipsoid durch parallele (/) Normale conform abgebildet wird. (/) Verl. v. L. Brill. 15. Ser. Nr. II. The first part of series 15 consisted of projections of four-dimensional surfaces. Schilling would later sell the model as number 193, Ser. 17 No. 13. Schilling did not include the projections in the series, and called it simply plaster models of different kinds.
References:
L. Brill, Catalog, 1892, p. 34, 72.
M. Schilling, Catalog, 1911, p. 40, 42, 139.
Stefan Neuwirth discusses this model and Reinbeck’s result under the section “Surface sur laquelle l’ellipsoïde est représenté de manière conforme par normales parallèles” of the web page entitled “Modèles Mathématiques du Laboratoire de Mathématiques de Besançon,” accessed April 13, 2018.
Location
Currently not on view
Object Name
Geometric Model
geometric model
1892
maker
L. Brill
Physical Description
plaster (overall material)
Measurements
overall: 20.2 cm x 19.5 cm x 12.5 cm; 7 15/16 in x 7 11/16 in x 4 29/32 in
ID Number
1985.0112.147
catalog number
1985.0112.147
accession number
1985.0112
Credit Line
subject
Mathematics
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Data Source
National Museum of American History
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