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My boss claimed earlier today that its possible to draw 6 lines in 3-d space, all passing through the origin such that the angle between any pair of them is the same as that between any other pair.

As an example, the 3-d coordinate system with $x$, $y$ and $z$ axes is such a system with three lines where the angle between any pair is $\frac{\pi}{2}$.

I can't imagine what a picture like this (with the $6$ lines) would look like. Can anyone provide insight into this?

He also said this was an application of linear algebra.

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It's known:

Take icosahedron and 6 pairs of opposite vertices.

See also here: http://oeis.org/A002853

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