Geometric Model, L. Brill No. 187. Ser. 15 No. II, Model for Function Theory
date: label: date made
content: 1892
name: label: maker
content: L. Brill
notes: label: Description
content: 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.
label: Description
content: 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).
label: Description
content: 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.
label: Description
content: 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.
label: Description
content: 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.
label: Description
content: References:
label: Description
content: L. Brill, <I>Catalog</I>, 1892, p. 34, 72.
label: Description
content: M. Schilling, <I>Catalog</I>, 1911, p. 40, 42, 139.
label: Description
content: 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.
label: Location
content: Currently not on view
place: label: place made
content: Germany: Hesse, Darmstadt
topic: label: Subject
content: Mathematics
setName: label: See more items in
content: Medicine and Science: Mathematics
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content: Science & Mathematics
creditLine: label: Credit Line
content: Gift of Wesleyan University
dataSource: label: Data Source
content: National Museum of American History
identifier: label: ID Number
content: 1985.0112.147
label: catalog number
content: 1985.0112.147
label: accession number
content: 1985.0112
objectType: label: Object Name
content: Geometric Model
label: Object Name
content: geometric model
physicalDescription: label: Physical Description
content: plaster (overall material)
label: Measurements
content: 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