Tetrahedral chains and a curious semigroup
In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joined face to face cannot be closed. Świerczkowski gave a proof in 1959. Several other proofs are known, based on showing that the four reflections in planes though the origin parallel to the faces of the tetrahedron...
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| Format: | Article |
| Language: | English |
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University of Extremadura
2019-06-01
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| Series: | Extracta Mathematicae |
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| Online Access: | https://publicaciones.unex.es/index.php/EM/article/view/69 |
| Summary: | In 1957 Steinhaus asked for a proof that a chain of identical regular tetrahedra joined face to face cannot be closed. Świerczkowski gave a proof in 1959. Several other proofs are known, based on showing that the four reflections in planes though the origin parallel to the faces of the tetrahedron generate a group R isomorphic to the free product Z2 ∗ Z2 ∗ Z2 ∗ Z2 . We relate the reflections to elements of a semigroup of 3 × 3 matrices over the finite field Z3 , whose structure provides a simple and transparent new proof that R is a free product. We deduce the non-existence of a closed tetrahedral chain, prove that R is dense in the orthogonal group O(3), and show that every R-orbit on the 2-sphere is equidistributed. |
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| ISSN: | 0213-8743 2605-5686 |