# Geometric Model for String Construction of Ellipsoids; L. Brill No. 109. Ser. 10.1 No. 2b.

Description:

This model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model, labeled 109, was manufactured by the Darmstadt publishing company of Ludwig Brill under the direction of the German mathematician Otto Staude.

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.

This white plaster model shows an ellipsoid and a hyperboloid. The visible part of the ellipsoid is the thickening around the hyperboloid. Looking at the front of the model (see AHB2016q011675.jpg) there are thin metal rods extending from the lower right and the underside of the upper left. The model originally had small eyes at the ends of the rods through each of which was a string that looped tightly around the plaster model; the two strings were of equal length.

The model is listed in Brill’s 1892 catalog as “Model of Staude’s string construction of the ellipse from two confocal second order surfaces.” Otto Staude wrote a short article about the construction in Alexander von Brill’s ca. 1892 Discussions on the models for higher mathematical teaching published by the publishing house of L. Brill in Darmstadt. The confocal surfaces referred to are an ellipsoid and a hyperboloid of one sheet.

This string, or thread, construction does not produce a string model in the usual sense, i.e., it is not a model of a ruled surface in which every point of which is on a straight line contained entirely in the surface. While the hyperboloid is such a surface, the ellipsoid, to which the thread construction refers, is not. Rather, the construction is analogous to the pencil and string construction of an ellipse. In that construction the two ends of a string are fixed at two points (the foci of the intended ellipse) and the string is kept taut by a pencil while drawing a curve with that pencil. In this model, the the analog of the piece of string whose ends are anchored at the foci of the ellipse is a loop of string that passes around the plaster model. The two eyes at the ends of the rods represent two different possibilities for the location of the analog of the pencil point that pulls the string tight.

The short article by Staude describes the model as having a red piece of string attached to the rod on the lower right side and looped around the model as well as a yellow piece of string of the same length attached to the rod on the upper left side also looped around the model. Both strings are taut. A photograph in the mathematical models collection at the University of Groningen in Holland shows the placement of the threads looking at the back of the model (Series X, 2b).

Staude describes the properties of two differently defined points that lie on the ellipsoid to be constructed. They are represented by the place on the rods where the red and the yellow thread would be anchored. All points on that ellipsoid must be constructed using a piece of string of the same length as the red and yellow pieces of string and must follow a path composed in the same way as one of the paths described below for the red or the yellow thread. In both cases the thread would leave the rod in two straight line segments, each of which is part of a line tangent to both the ellipsoid and the hyperboloid.

When the red thread left the rod it ran in two line segments both of which first met the plaster on the ellipse at a point of tangency (the points of tangency with the hyperboloid occurs further along those lines). The line segment running from the right hand rod and to the front of the plaster met the ellipsoid near the upper curve of intersection of the ellipsoid and the hyperboloid, while the one that ran to the back met the ellipsoid near the lower curve of intersection. The line segments were continued by a geodesic (minimal length) curve on the ellipsoid that ran to the closer curve of intersection and was continued along the curve of intersection until the two parts of the thread were joined along another geodesic that ran between the two lines of intersection.The path of the thread that ran between the lines of intersection is visible on the left side of the ellipse in AHB2016q011676.jpg.

When the yellow thread left the rod it also ran in two line segments. Starting from the left hand rod it ran to the back of the plaster model, met the hyperboloid, continued as a line segment until it met the ellipsoid, traveled on a geodesic curve seen on the ellipsoid (visible on the left side of the model in AHB2016q011678.jpg) to the lower curve of intersection of the ellipsoid and the hyperboloid, traveled along the lower curve of intersection onto another geodesic curve as it ran along a geodesic curve on the ellipsoid until it met the line segment that took it back to the rod.

Reference:

“Modell zur Fadenconstructionen des Ellipsoids aus zwei gegeben confocalen Flächen zweiten Grades” by Dr. O. Staude, 2 pp. In Alexander von Brill, Abhundlungen zu den durch die Verslagshandlung von L. Brill in Darmstadt veröffentlichten Modellen für den höheren matheatischen Unterricht. Darmstadt: Brill, ca. 1892.

Date Made: 1892

Maker: L. Brill

Location: Currently not on view

Place Made: Germany: Hesse, Darmstadt

Subject: Mathematics

Subject:

See more items in: Medicine and Science: Mathematics, Science & Mathematics

Exhibition:

Exhibition Location:

Credit Line: Gift of Wesleyan University

Data Source: National Museum of American History

Id Number: 1985.0112.077Catalog Number: 1985.0112.077Accession Number: 1985.0112

Object Name: Geometric Modelgeometric model

Physical Description: metal (overall material)plaster (overall material)Measurements: overall: 12 cm x 20 cm x 8.5 cm; 4 23/32 in x 7 7/8 in x 3 11/32 in

Record Id: nmah_693953

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