Out there

Brilli the Ant is tripping out in 7-dimensional space (don't ask me how she got there). She recalled looking at a kaleidoscope, and seeing pretty patterns through reflections, because the pairs of lines all met at \( 60^\circ\). This led her to wonder, what is the most number of (non-parallel) lines in \( \mathbb{R}^7 \) such that the angle between any two of them is the same?

Details and assumptions

Since we're only interested in the angle between the lines, you may assume that they all pass through the origin.

The angle between 2 intersecting lines in \(n\) dimensions can be determined by just looking at the plane containing both lines.

There is no requirement that the angle must be \( 60^\circ \).


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