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The
symmetry of the tetrahedron |
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The tetrahedron is the least symmetrical of the Platonic solids. However it does |
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share
some of the cube's symmetry elements. As shown previously in Fig.
4a it can be |
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inscribed
in the cube to reveal its symmetry in a straightforward manner. As Fig. 4a |
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demonstrated,
the tetrahedron shares the cube's four [111] |
rotational axes
and (111) planes of 3-fold symmetry. And, |
like the cube,
it has six (110) mirror planes. But it has only |
three [110] axes and (110) planes of 2-fold rotational
symmetry |
and no
(100) planes - thirteen symmetry elements in all. |
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Likewise, the octahedron can also be inscribed in the tetrahedron to reveal its sym- |
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metry
by association. In fact a Polymorf model can be built of an octahedron inscribed |
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in
a tetrahedron inscribed in a cube to associate the symmetry of all three polyhedra. |
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Fig.
8a - Octahedron inscribed |
Fig.
8b - Octahedron inscribed in |
in
a tetrahedron |
a
tetrahedron inscribed in a cube |
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Required
parts |
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20
triangles |
20
large triangles, 12 small squares |
24
pinges |
24
right triangles, 72 pinges |
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Fig. 8 -
Associating the symmetry of the tetrahedron by inscription |
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Exercise: The (110) mirror
planes of the cube and tetrahedron have the same orienta- |
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tion but the [110] axes and (110)
planes of rotational symmetry do not. Explain why. |
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Page 7
Geometry rules! - Symmetry association by inscription |