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"Wonky Trammel of Archimedes" An exploration of a generalized trammel.As a child, you may have played with a simple wooden toy called a "do-nothing" machine or simply a "grinder."Īs a mathematician, I find that name a very poor choice.Video of a Do-Nothing made from Lego bricks.
Trammel of archimedes how to#
Instructions on how to build a Kentucky Do-Nothing.Video of various trammel designs in action.Artobolevskii Mechanisms for the Generation of Plane Curves. Courier Dover 2003, ISBN:978-6-5, pp. 4–5 ( restricted online copy, p. Downs: Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas. American Mathematical Monthly 117 (2): 161–167. In these toys the drafting instrument is replaced by a crank handle, and the position of the sliding shuttles is usually fixed. Wooden versions of the trammel of Archimedes have been produced also as toys or novelty items, and sold under the name of Kentucky do-nothings, nothing grinders, do nothing machines, smoke grinders, or bullshit grinders. The history of such ellipsographs is not certain, but they are believed to date back to Proclus and perhaps even to the time of Archimedes. Usually the distances a and b are adjustable, so that the size and shape of the ellipse can be varied. An ellipsograph has the appropriate instrument (pencil, knife, router, etc.) attached to the rod. This is compounded by high forces owing to the short throw of the crank of only 1/4 the travel of the pivots.Īn ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. Thus, if a crank centred on the crossing point of the channels is used to engage the trammel at the midway point to drive it, the rotation of the crankpin and the trammel are equal and opposite, which in practical applications results in extra friction and accelerated wear. The direction of travel around the orbit is opposite to the sense of rotation of the trammel. The diameter of the circle is equal to the distance between the pivots.
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This circle is also a special case of an ellipse. The point midway between the pivots orbits in a circle around the point where the channels cross. This circle corresponds to the smaller circle in a Tusi couple. All points on a circle with a diameter defined by the two pivots reciprocate in such straight lines. The straight lines described by the pivots are special cases of an ellipse, where the length of one axis is twice the distance between the pivots and that of the other is zero. The semi-axes a and b of the ellipses have lengths equal to the distances from the point on the rod to each of the two pivots. The motion of the rod is termed elliptical motion. It consists of two shuttles which are confined ("trammeled") to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at fixed positions along the rod.Īs the shuttles move back and forth, each along its channel, all points on the rod move in elliptical paths. A trammel of Archimedes is a mechanism that generates the shape of an ellipse.