| In 3-dimensional Euclidean space R3, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. There are only five solids meeting those criteria. An Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. There are only 13 Archimedean solids in R3. A convex body BcR3 is a compact convex set with non-empty interior. If the vertex set of a convex body B is finite, then B is called a plytope in R3 with non-empty interior. A convex body B has the Rupert property, if there are two orthogonal projections Bi, Bo of B such that Bi is a proper subset of the relative interior of Bo. The Nieuwland constant v(B) of a convex body B is defined by the superior of the positive real number v, which satisfies that there are projections of vB and B such that the first projection is contained in the relative interior of the second.In Chapter 1, we probe into the Rupert property of Archimedean solids. We prove that 8 of them, namely, the truncated cube, the cuboctahedron, the truncted octahe-dron, the rhombicuboctahedron, the icosidodecahedron, the truncated cuboctahedron, the truncated icosahedron, and the truncated dodecahedron have the Rupert property. Futhermore, we present a lower bound of the Nieuwland constant for each Archimedean solid listed above.A cage is the 1-skeleton of a 3-dimensional polytope in R3. A cage G is said to hold a convex body B if no rigid motion can bring B far away without meeting G on its way. Denote by L(B) the infimum of the 1-dimensional Hausdorff measure of the cage which can hold B. In Chapter 2, we investigate cages for the octahedron, the dodecahedron, and the icosahedron, and proved that for the regular octahedron O with unit edge lengths, L(O)≤(?); for the regular dodecahedron D with unit edge lengths, the regular icosahedron (?) with unit edge lengths, L((?))≤ (?). |
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