One formula to rule them all!
P(d, F) = 4d / ((d+1) F)
In d-dimensional space, d≥2, given a regular polytope with F faces that come in opposite pairs, if you pick two points at random from the interior of the polytope and draw the line that contains them, the probability that the line will intersect both faces of any of the F/2 opposite pairs is P(d, F).
For d=2, this applies to all polygons with an even number of edges.
For d=3, the four Platonic solids apart from the tetrahedron.
For d=4, the hypercube, 16-cell, 24-cell, 120-cell and 600-cell.
For d>4, the hypercubes and the cross-polytopes.
Greg Egan
npub12c…gd38p
2025-08-26 03:28:14 UTC
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wss://relay.momostr.pinkPublished at
2025-08-26 03:28:14 UTCEvent JSON
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