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Inscribing cubes and covering by rhombic dodecahedra via equivariant topology
Author(s) -
Hausel T.,
Makai E.,
Szűcs A.
Publication year - 2000
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015965
Subject(s) - equivariant map , mathematics , dodecahedron , inscribed figure , tetrahedron , cube (algebra) , topology (electrical circuits) , combinatorics , group (periodic table) , symmetry (geometry) , regular polygon , pure mathematics , geometry , chemistry , organic chemistry
First, a special case of Knaster's problem is proved implying that each symmetric convex body in ℝ 3 admits an inscribed cube. It is deduced from a theorem in equivariant topology, which says that there is no S 4 –equivariant map from SO (3) to S 2 , where S 4 acts on SO (3) on the right as the rotation group of the cube, and on S 2 on the right as the symmetry group of the regular tetrahedron. Some generalizations are also given.