Mikhalev's Octahedrons
Two octahedrons composed of exactly same set of faces - one of them convex, the other concave. Which one has a greater volume?
Surprisingly, there is a pair of such octahedron in which the concave one has a greater volume. The pair has been discovered in 2002 by S. N. Mikhalev. Mathematical Institute of the Russian Academy of Sciences spawned a Laboratory of Propaganda and Popularization of Mathematics. The laboratory prepares short animated movies that explain various mathematical concepts and constructions. Mikhalev's polyhedra is featured in one of those etudes.
The ratio of the volumes is about 1.163.
The vertices of the convex octahedron are
\begin{align} N(0, 0, 1), & \\ A(10, 1, 0), B(0, 6, 0), C(-10, & 1, 0), D(0, -10, 0), \\ S(0, 0, -1) & \end{align}
And here are the vertices of the concave octahedron:
\begin{align} N(0, 0, \sqrt{\frac{61}{3}}), & \\ A(\sqrt{71}, 4\sqrt{\frac{2}{3}}, 0), B(0, -5\sqrt{\frac{2}{3}}, 0), C(-\sqrt{71}, &4\sqrt{\frac{2}{3}}, 0), D(0, -11\sqrt{\frac{2}{3}}, 0), \\ S(0, 0, -\sqrt{\frac{61}{3}}) & \end{align}
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