Random Sections of Convex Octahedra

Sections of a convex octahedron by random planes are studied. The distribution of random planes is generated by a measure of planes that is invariant with respect to Euclidean motions. Using Ambartsumyan’s combinatorial formula, the probabilities of events that a quadrangle and a hexagon are formed in a section are expressed through the lengths of the edges and dihedral angles. For one family of octahedra, graphs of the dependence of these probabilities on the height of the octahedron are obtained. The asymptotic behavior of these probabilities is studied when the height of the octahedron tends to zero or infinity. An extremal property of a regular octahedron is obtained. The possibilities of applying the results in practice are given.