For a regular tetrahedron with side length " a ", radius " R " of its circumscribing sphere, and distances " d i " from an arbitrary point in 3-space to its four vertices, we have
32.
The interior of the stella octangula is an octahedron, and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size ( i . e ., rectifying the tetrahedron ).
33.
In 1874, Jacobus Henricus van't Hoff and Joseph Achille Le Bel independently proposed that the phenomenon of optical activity could be explained by assuming that the chemical bonds between carbon atoms and their neighbors were directed towards the corners of a regular tetrahedron.
34.
Since the regular tetrahedron is self-dual, this means that this ratio is the same for any Platonic solid and its dual-is there any insight that makes this result obvious ?-- talk ) 01 : 45, 13 January 2017 ( UTC)
35.
In 1874, Jacobus Henricus van't Hoff and Joseph Achille Le Bel independently proposed that this phenomenon of optical activity in carbon compounds could be explained by assuming that the 4 saturated chemical bonds between carbon atoms and their neighbors are directed towards the corners of a regular tetrahedron.
36.
The "'tetrahedral pyramid "'is a special case of a "'5-cell "', a polyhedral pyramid, constructed as a regular tetrahedron base in a 3-space hyperplane, and an apex point " above " the hyperplane.
37.
The Reuleaux triangle can also be generalized into three dimensions in multiple ways : the Reuleaux tetrahedron ( the intersection of four spheres whose centers lie on a regular tetrahedron ) does not have constant width, but can be modified by rounding its edges to form the Meissner tetrahedron, which does.
38.
The highest possible symmetry is that of the regular tetrahedron as [ 3, 3 ], and this occurs in the prismatic point group [ 2, 2, 2 ] or [ 2 [ 3, 3 ] ] and the paracompact hyperbolic group [ 3 [ 3, 3 ] ].
39.
However, the bond angle between the two O H bonds is only 104.5? rather than the 109.5?of a regular tetrahedron, because the two lone pairs ( whose density or probability envelopes lie closer to the oxygen nucleus ) exert a greater mutual repulsion than the two bond pairs.
40.
Similarly in eight dimensions, the figure you get using all permutations of 4 out of 8 is the 3D face dual of the 7-simplex, or 3-rectified 7-simplex ( since 1111 scaled by 1 / 4 moves to the centre of the 3D regular tetrahedron face 1000 0100 0010 0001 ), and so on.
