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Icosahedron
and dodecahedron symmetry |
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The icosahedron and the dodecahedron are duals of each other and therefore are |
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symmetrically
identical. Both possess fifteen mirror planes, fifteen axes and planes of |
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 |
 |
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Fig. 10 -
Icosahedron |
Fig.
11a - Dodecahedron - Fig. 11b |
Required
parts |
20 triangles |
60 triangles |
30 isosceles
triangles |
30 pinges |
90 pinges |
20 pinges, 4
rubber bands |
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Fig. 10 and
11 - Icosahedron and dodecahedron* |
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2-fold
rotational symmetry, ten axes and planes of 3-fold symmetry, and six axes and |
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planes of
5-fold symmetry for a total of 46 elements. In Fig. 11b rubber bands are |
used to outline
the shape of a cube inserted inside a dodecahedron to show that the |
dodecahedron
shares the four (100) mirror planes of the cube as well as its four [111] |
axes of 3-fold
symmetry. |
*Note: Fig. 11a is a representation
of a dodecahedron that has concave pentagonal |
faces replacing
the classic pentagons. Fig. 11b is a dodecahedron nolid (the negative |
of a solid).
It is constructed from 30 isosceles triangles that converge on the center of |
the shape. Actually it is
mathematically impossible to construct with the dimensions |
of the isosceles triangles used
in Polymorf, but there is enough "give" in the model to |
make it work. (This is
why the Greeks did not rely on models to prove their theorems!) |
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Exercise: Build Fig. 11b
(including the rubber bands). Then compare |
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the symmetry elements of the
dodecahedron to those of the cube. |
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Geometry rules! - Symmetry association by inscription |