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Higher Dimensional Tetrahedrons (Simplexes) and the Star Tetrahedron - an Advanced Question
May 5, 2021
1:08 pm
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Narada Dan Vantari
Byron Bay NSW Australia
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Forum Posts: 58
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October 23, 2012
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This is a question for advanced higher dimensional students to contemplate and solve.

In 3D, one of the unique features of the Tetrahedron among the 5 Platonic Solids is that a line drawn from any of its vertexes directly through the centre of the form emerges in the centre of one of its faces. With all the other Platonics the line emerges at another vertex.

This makes the tetrahedron peculiarly unsymmetrical, especially in the way physical tetrahedrons relate to up and down in a gravitational field. The naturally sit on a face and have a vertex raised upwards, and are more 'pointy' than the other Platonics. In view of this, it seems natural to consider a tetrahedron to be only half of a complete form, the star tetrahedron, where two tetrahedra interpenetrate to create a more symmetrical structure, and the vertexes then form a cube.

My question is: Does this uniqueness have its analog in higher dimensional simplexes? (Simplex is the technical name for this triangular form in all dimensions). 

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