Hypotrochoids, Kinematic Model by Martin Schilling, series 24, model 3, number 331
date: label: date made
content: ca 1900
name: label: maker
content: Schilling, Martin
notes: label: Description
content: Around 1900, American mathematicians introduced ideas to their students using physical models like this one. This model is the third in a series of kinematic models sold by the German firm of Schilling to show a mechanical method for generating mathematical curves.
label: Description
content: The name hypotrochoid comes from the Greek word hypo, which means under, and the Latin word trochus, which means hoop. Thus hypotrochoids are curves formed by tracing a point on the radius or extension of the radius of a circle rolling around the inside of another stationary circle.
label: Description
content: Hypotochoids are members of the family of curves called trochoids, curves that are generated by tracing the motion of a point on the radius of a circle as it rolls along another curve. They include the cycloids (see item 1982.0795.05) and epitrochoids (see item 1982.0795.01).
label: Description
content: An infinite number of hypotrochoids can be formed, depending on the distance of the tracing point from the center of the rolling circle. The ratio of the radius of the rolling disc to the radius of the outer ring will determine the number of nodes the hypotochoid will have. In this model, the curves each have five nodes. Hypotochoids, for which the tracing point is on the extension of the radius, form curves that resemble petalled flowers and are called roses. The Spirograph toy produces various types of hypotrochoids. (See item 2005.0055.2) In the 18th century, it was found that shaping the sides of gear teeth and the valley between teeth by using trochoidal curves reduced the torque of the rotating gears and allowed them to rotate more efficiently.
label: Description
content: This model consists of a stationary toothed metal ring (with teeth on the inner edge of the ring) of radius 80 mm. A toothed metal disc of radius 32 mm is attached to a brass arm of 7 cm that can be rotated by turning a crank below the baseplate. As the arm is rotated, the disc rolls around the inside of the ring. Three points lie along the radius of the disk and trace corresponding curves, or roulettes, on the glass overlay.
label: Description
content: The blue point on the circumference of the disc traces a blue five-pointed star shape referred to as a hypocycloid. The green point on the radius of the disc traces a green curve inside the ring, and the red point on the extension of the radius of the disc traces a curve that extends past the radius of the ring. The German title of this model is: Erzeugung der Hypotrochoiden als soche mit freiem Centrum.
label: Description
content: References:
label: Description
content: Schilling, Martin, <i>Catalog Mathematischer Modelle für den höheren mathatischen Unterricht</i>, Halle a.s., Germany, 1911, pp 56-57. Series 24, group 1, model 3.
label: Description
content: An online demonstration can be found at http://mathworld.wolfram.com/Hypotrochoid.html
label: Location
content: Currently not on view
place: label: place made
content: Germany
topic: label: Subject
content: Mathematics
label: Subject
content: Engineering
setName: label: See more items in
content: Medicine and Science: Mathematics
label: See more items in
content: Science & Mathematics
label: See more items in
content: Kinematic Models
creditLine: label: Credit Line
content: Gift of the Department of Mathematics, The University of Michigan
dataSource: label: Data Source
content: National Museum of American History
identifier: label: ID Number
content: 1982.0795.02
label: catalog number
content: 1982.0795.02
label: accession number
content: 1982.0795
objectType: label: Object Name
content: geometric model
physicalDescription: label: Physical Description
content: metal (overall material)
label: Physical Description
content: glass (overall material)
label: Physical Description
content: paper (overall material)
label: Physical Description
content: leather (overall material)
label: Measurements
content: overall: 5.8 cm x 22 cm x 27 cm; 2 9/32 in x 8 21/32 in x 10 5/8 in