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A Movable Pair of Tetrahedra

T. Tarnai and E. Makai
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 423, No. 1865 (Jun. 8, 1989), pp. 419-442
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/2398566
Page Count: 24
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A Movable Pair of Tetrahedra
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Abstract

When we draw all the diagonals of all the faces of a cube, we obtain the edges of two regular tetrahedra. Keeping one of these fixed, we move the other under the condition that the originally intersecting edges of the two tetrahedra should still remain coplanar (i.e. intersect, are parallel or coincide). We determine all such finite motions. These will be seen to constitute one-dimensional and two-dimensional smooth manifolds. We also deal with the infinitesimal degrees of freedom of the motions of our mechanism. In several positions the number of infinitesimal degrees of freedom is not one or two, but is three; this is connected with the bifurcation phenomenon of the solutions.

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