Möbius Tetrahedra


Möbius tetrahedra, also called Möbius tetrads (Baker 1922, pp. 61-62) are a pair of tetrahedra, each of which has all the vertices lying on the faces of the other: in other words, each tetrahedron is inscribed in the other. As shown by Möbius in 1828, this apparently paradoxical geometric situation can be realized when some of the vertices lie not exactly on the surface of the polyhedron, but instead in the extensions of the facial planes.

The vertices A,B,C,D and F,G,H,I of the tetrahedra must be assigned to the faces as follows:

1. A to GHI

2. B to HIF

3. C to IFG

4. D to FGH

5. F to BCD

6. G to CDA

7. H to DAB

8. I to ABC.

It can be shown that each of the above eight rules is a consequence of the remaining seven.

See also

Möbius Tetrad Theorem, Pappus's Hexagon Theorem, Tetrahedron

Portions of this entry contributed by Margherita Barile

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Baker, H. F. Principles of Geometry, Volume 1: Foundations. Cambridge, England: pp. 61-62, 1922.Baker, H. F. Principles of Geometry, Volume 4: Higher Geometry. Cambridge, England: pp. 18-21, 1925.Möbius, F. A. "Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um- und eingeschrieben zugleich heissen?" J. reine angew. Math. 3, 273-278, 1828.

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Möbius Tetrahedra

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Möbius Tetrahedra." From MathWorld--A Wolfram Web Resource.

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