Model of a Quartic Scroll by Richard P. Baker, Baker #84 (a Ruled Surface)

Model of a Quartic Scroll by Richard P. Baker, Baker #84 (a Ruled Surface)

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This string model was constructed by Richard P. Baker, possibly before 1905, when he joined the mathematics faculty at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.
The typed part of a paper label on the wooden base of this model reads: No. 84 Quartic Scroll, (/) with two nodal straight (/) lines. Model 84 appears on page 8 of Baker’s 1931 catalog of models as “Quartic Scroll , with two nodal straight lines.” The equation of the model is listed as (x2/((z - 1) 2)) + (y2/((z + 1) 2)) = 1. It also appears in his 1905 catalog of one hundred models.
Baker’s string models always represent a special type of geometric surface called a ruled surface. A ruled surface, sometimes called a scroll, is one that is swept out by a moving line. This model is swept out by any of the yellow threads joining the elliptically shaped horizontal piece of wood on the top of the model to the wooden base of the model.
In addition to the yellow threads of the model, there are two horizontal red threads that run from the rods at near the edge of the base and are parallel to the lines connecting the midpoints of the opposite sides of the square of surface of the base. There is a segment of each of these red threads for which each point meets two different lines of the model and the points of these segments are called double points, or nodes, of the surface. Thus these line segments are the two nodal lines of the model. The horizontal plane z = 1 intersects the model at the upper horizontal thread, while the horizontal plane z = -1 intersects it at the lower horizontal thread. When z=1, the points of intersection are (0,y,1) for y between -2 and 2. When z=-1, the points of intersection are (x,0,-1) for x between -2 and 2. Thus the nodal lines are the line segments connecting (0,-2,1) to (0,2,1) and (-2,0,-1) to (2,0,-1).
When z = 0 the equation of the surface becomes x2 + y2 = 1, so the horizontal plane z = 0 intersects the model at the unit circle with center at the origin. For any other value of z, the equation of the surface is of the form (x2/a2) + (y2/b2) = 1, where a does not equal b. This is the standard form for the equation of an ellipse.
Currently not on view
Object Name
geometric model
date made
ca 1915-1935
ca 1905-1935
ca 1900-1935
Baker, Richard P.
place made
United States: Iowa, Iowa City
Physical Description
thread (overall material)
wood (overall material)
metal (overall material)
yellow (overall color)
black (overall color)
red (overall color)
bolted, threaded and tied (overall production method/technique)
average spatial: 27.7 cm x 22.9 cm x 22.7 cm; 10 29/32 in x 9 1/32 in x 8 15/16 in
ID Number
accession number
catalog number
Credit Line
Gift of Frances E. Baker
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Mathematical Association of America Objects
Data Source
National Museum of American History
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