Drawing Ovals with a Pencil, Three Nails, and Some String

A simple way to draw a circle is to use a pencil, a nail, and a loop of string. Stick the nail into a table—getting the owner’s permission first—and loop the string around the nail, and then move the pencil around the nail, keeping the string taut. This maintains a constant radius around the nail, resulting in a circle.

circle

If you do the same thing with two nails, the pencil point maintains a constant sum of distances from each of the two nails, and you end up with an ellipse.

ellipse animation

Now do the same thing with three nails. As you move the pencil around the triangle formed by the three nails, the far end of the loop gets caught on different combinations of nails, sometimes two nails and sometimes all three.

animation

Each combination corresponds to a different elliptical arc, where each arc is bounded by the lines formed by extending the segments joining each pair of nails. If the nails are arranged in an isosceles triangle, the final figure is an oval or egg shape.

oval

The individual ellipses that make up the oval:

showing all ellipses

Naturally, other arrangements give you different final shapes.

nails in a narrower triangle... ...in an equilateral triangle... ...in a scalene triangle

Figures created with Mathematica 7.

You might also enjoy this version, housed at the Wolfram Demonstrations Project:

"Drawing an Oval with a String and Three Nails" from the Wolfram Demonstrations Project

See Martin Gardner, The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications, pp. 63–64, 1997.

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