No right angles?

Each edge is parallel to its antipode (the edge directly across). Thus the edges define 15 different directions in space. These can be divided into five sets of three mutually perpendicular directions.

One way to see this is by viewing the icosahedron along the perpendicular bisector of one of the vertices:

Marked in red are three pairs of edges in perpendicular directions. (The horizontal segment represents two edges, one directly behind the other. The red dots represent two edges that run directly "into the paper".)

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Alternatively, an icosahedron can be described by coordinates $(0,\pm 1, \pm \phi);\ (\pm \phi, 0, \pm 1);\ (\pm 1, \pm \phi, 0)$ , where $\phi = \tfrac12(-1+\sqrt 5)$ is the golden ratio. Two points are connected iff their distance equals two units. Thus, the points $(0, \pm 1, \phi)$ ; the points $(\phi, 0, \pm 1)$ ; and the points $(\pm 1, \phi, 0)$ describe edges that run parallel to the $y$ -, $z$ - and $x$ -axis, respectively.