Mathematical Treasure: Hypotrochoid Kinematic Model by M. Schilling

Amy Shell-Gellasch (Montgomery College)

During the latter half of the 19th century and into the 20th, educators and engineers used models to visualize mathematical curves and surfaces. Models that depict geometric curves produced by constrained motion were called kinematic models. One of the largest publishers of mathematical models, both static and kinematic, was the firm of Martin Schilling of Leipzig, Germany. The Smithsonian houses ten of these models, most from the University of Michigan. They fall roughly into two categories: linkages and models that produce trochoids.

Hypotrochoids, Kinematic Model by Martin Schilling, series 24, model 3, number 331

Hypotrochoids (ca 1900), kinematic model by Martin Schilling, series 24, model 3, number 331, Smithsonian Institution negative number DOR2013-50214.

This model produces hypotrochoids, 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. Hypotrochoids 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—and include the well-known cycloids. In this model, three hypotrochoids are generated. 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.

By the early years of the 20th century, these types of models were losing popularity with educators. With the advent of computer aided design software, mathematical models are now museum pieces or curios in display cabinets in university mathematics or engineering departments.

The full set of the Smithsonian Schilling kinematic models can be found at

Index of Mathematical Treasures

Index of Mathematical Objects