In [1, p. 16], Lewis Carroll posed the following problem:

"If a regular tetrahedron be placed, with one vertex downwards, in a socket which exactly fits it, and be turned round its vertical axis, through an angle of , raising it only so much as necessary, until it again fits the socket: find the locus of one of the revolting vertices."

The answer and solution are given [1, p. 26, p. 100]. The equations for the locus are ; , where the tetrahedron has edge length 1, altitude , and .

Reference

[1] L. Carroll, The Mathematical Recreations of Lewis Carroll: Pillow Problems and a Tangled Tale, 4th ed., New York: Dover, 1958.