Lennes' Polyhedron

Every simple polygon admits a triangulation, i.e., dissection into a finite number of triangles having exactly the same set of vertices as the polygon itself. This statement has no analog in the 3-dimensional space. This is not true that every polyhedron can be cut into tetrahedral pieces that have no other vertices except those of the polyhedron itself. This was established in 1911 by N. J. Lennes who constructed a counterexample. The applet below illustrates a simplified version discovered by E. Schönhard in 1928 [Eves, p. 211, or Devadoss & O'Rourke, p. 6].

Imagine a right prism with an equilateral triangle as the base. Let the bottom triangle be ABC and the upper A'B'C', with the natural correspondence of the vertices. Draw the side diagonals AB', BC', CA'. Think of all the line segments involved as rigid material pieces. Rotate the upper base π/6 degrees around the vertical axis through its center. The result is Schönhard's polyhedron. Any tetrahedron with vertices at the vertices of Schönhard' model contains an exterior piece of the latter.

In the illustration, the polyhedron has 6 vertices, 12 edges and 8 faces, exactly like octahedron. Euler's formula checks out: 6 - 12 + 8 = 2.

What if applet does not run?

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

(Acknowledgment: I have learned most of Java details from the implementation by Meiko Rachimow.)

References

  1. S. L. Devadoss, J. O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2012
  2. H. Eves, A Survey of Geometry, Allyn and Bacon, Inc. 1972

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