The 3D parallelo-hexagon

3D parallelo-hexagon

Theorem: In 3D space, if the opposite sides of a spatial 3D hexagon are parallel, then the opposite sides are equal; i.e. the formed spatial figure is a 3D parallelo-hexagon.

Corollary: Unlike the plane, where it's possible to construct a hexagon with opposite sides parallel, but not necessarily equal, the above theorem proves that this is impossible for a hexagon in 3D space.

Illustration: Above is a short video clip briefly & dynamically illustrating the 3D parallelo-hexagon.

For more info, read the paper by Heinz Schumann & myself in the July 2018 issue of 'The Mathematical Gazette', and is available at: A surprising 3D result involving a hexagon.

Note: The converse of the theorem above is false. Having opposite sides equal for a spatial 3D hexagon is a necessary, but not sufficient condition. One can easily construct a counter-example as shown below from the original 3D parallelo-hexagon. By reflecting A1A2A6 in the plane on which parallelogram A2A3A5A6 lies, a new spatial 3D hexagon A'1A2A3A4A5A6 is obtained, which clearly still has the pairs of opposite sides equal, but A'1A2 is no longer parallel to opposite side A4A5, and inclined at angle. Similarly, A'1A6 is also no longer parallel to opposite side A3A4.

3D hexagon counter

A 3D parallelo-hexagon is also formed by the centroids of the triangles subdividing a spatial 3D hexagon. See for example: A generalization of Varignon's Theorem to 2n-gons.

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Michael de Villiers, created 6 January 2019.