| 词条 | Rhombicosidodecahedron | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 释义 |
In geometry, the rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, 60 vertices, and 120 edges. Names{{multiple image| align = left | total_width = 400 | image1 = Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png | image2 = Rhombicosidodecahedron in rhombic triacontahedron max.png | image3 = Nonuniform rhombicosidodecahedron as core of dual compound max.png }} Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a rhombicosidodecahedron, being short for truncated icosidodecahedral rhombus, with icosidodecahedral rhombus being his name for a rhombic triacontahedron.[1] There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron: Prominently its rectification (left), the one that creates the uniform solid (center), and the core of the compound with its dual (right). It can also be called an expanded or cantellated dodecahedron or icosahedron, from truncation operations on either uniform polyhedron. Geometric relationsIf you expand an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, and do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either. The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of six or twelve pentagrammic prisms. The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles. Twelve of the 92 Johnson solids are derived from the rhombicosidodecahedron, four of them by rotation of one or more pentagonal cupolae: the gyrate, parabigyrate, metabigyrate, and trigyrate rhombicosidodecahedron. Eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae. Cartesian coordinatesCartesian coordinates for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are all even permutations of:[2](±1, ±1, ±φ3), (±φ2, ±φ, ±2φ), (±(2+φ), 0, ±φ2), where φ = {{sfrac|1 + {{sqrt|5}}|2}} is the golden ratio. Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely {{sqrt|φ6+2}} = {{sqrt|8φ+7}} for edge length 2. For unit edge length, R must be halved, giving R = {{sfrac|{{sqrt|8φ+7}}|2}} = {{sfrac|{{sqrt|11+4{{sqrt|5}}}}|2}} ≈ 2.233. Orthogonal projections{{multiple image| align = right | perrow = 1 | total_width = 200 | image1 = Houghton Typ 520.43.454, crop solid and owl.jpg | image2 = Fotothek df tg 0003625, crop rhombicosidodecahedron.jpg | footer = Orthogonal projections in Geometria (1543) by Augustin Hirschvogel }} The rhombicosidodecahedron has six special orthogonal projections, centered, on a vertex, on two types of edges, and three types of faces: triangles, squares, and pentagons. The last two correspond to the A2 and H2 Coxeter planes.
Spherical tilingThe rhombicosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。
Copyright © 2023 OENC.NET All Rights Reserved
京ICP备2021023879号 更新时间：2024/9/24 20:34:00 |