Is there a crystal lattice possessing five-fold symmetry? (Q2883507)
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scientific article; zbMATH DE number 6032531
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Is there a crystal lattice possessing five-fold symmetry? |
scientific article; zbMATH DE number 6032531 |
Statements
10 May 2012
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semiregular tiling
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semiregular polytope
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Penrose tiling
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star-polyhedra
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five-fold symmetry
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0.73509336
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0.7211385
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0.71598417
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0.71500987
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0.71456355
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0.7115011
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0.7104285
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Is there a crystal lattice possessing five-fold symmetry? (English)
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In his treatise \textsl{Harmonices mundi} (1619) Johannes Kepler classified the 11 semiregular tilings of the Euclidean plane. None of them allowed 5-fold rotational symmetry. Kepler experimented with 5-fold rotational symmetry and had some planar tilings -- using several shapes -- with \textsl{local} 5-fold rotational symmetry. Bravais (1850) proved that in the Euclidean 3-space there is no crystal lattice with 5-fold rotational symmetry. It is easy to see that the hypercube lattice in the Euclidean 5-space has a 5-fold rotational symmetry.NEWLINENEWLINEThe paper exhibits a lattice in the Euclidean 4-space, which admits a 5-fold rotational symmetry.
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