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}

What if applet does not run?

What if applet does not run?

Drag the mouse to rotate the prism. Use the right button to remove and put back individual faces.

Related material

  • Right Pentagonal Prism
  • Square Pyramid
  • Right Triangular Prism
  • Twisted Triangular Prism
  • Tetrahedron: an Interactive Model
  • Octahedron: an Interactive Model
  • Cube: an Interactive Model
  • Icosahedron: an Interactive Model
  • Dodecahedron: an Interactive Model
  • Three Pyramids are Better Than Two
  • Cube In Octahedron
  • Octahedron In Cube
  • Octahedron In Tetrahedron
  • Tetrahedron In Cube
  • Icosahedron In Cube
  • Great Stellated Dodecahedron
  • Lennes' Polyhedron
  • Triangulated Dinosaur
  • Volumes of Two Pyramids
  • Császár Polyhedron 1
  • Császár Polyhedron 4
  • Szilassi Polyhedron
  • Dissection of a Square Pyramid
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