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| Connect The Dots (Posted on 2005-02-06) |
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Take any four points in space. Draw all lines connecting pairs of them. Then draw all lines connecting pairs of points on those lines.
Can the resulting set of points cover all of space?
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Submitted by David Shin
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Rating: 3.6667 (3 votes)
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Solution:
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The answer is no.
If the four points are coplanar, then the resulting set of points lies in that plane and so cannot cover all of space.
Otherwise, it suffices to consider the case where the four points are the vertices of a regular tetrahedron. The reason for this is:
1. One can perform an affine transformation to map any set of four non-coplanar points onto the vertices of a regular tetrahedron.
2. Affine transformations preserve collinearity.
We shall therefore just consider the case where the four points are the vertices of a regular tetrahedron. Inscribe the tetrahedron in a cube via six of the face diagonals. Then, we claim that the four unused corners of the cube are missing from the final set.
To see this observe that points in the final set lie on lines connecting points on the extended edges of the tetrahedron. Now note that a line through points on adjacent (extended) tetrahedral edges lies in the plane of a tetrahedral face, and so misses the unused corners. And a line connecting one such corner to a point on a nearby edge lies in the plane of a face of the cube, and so misses the skew edge, which lies in the opposite plane.
It is interesting to note that the four given points are the only points in space missing from the final set. |
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