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When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.55%). [10] The resulting of both spheres' volumes initially began from the problem by ancient Greeks, determining which of two shapes has a larger volume: an icosahedron inscribed in a ...
The concave equilateral dodecahedron, called an endo-dodecahedron. [clarification needed] A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions. A regular dodecahedron is an intermediate case with equal edge lengths. A rhombic dodecahedron is a degenerate case with the 6 crossedges reduced to ...
The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume). The dodecahedron, on the other hand, has the smallest angular defect, the ...
As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%). [ 12 ] The dihedral angle of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism.
The surface area A and the volume V of the rhombic dodecahedron with edge length a are: [4] =, =. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron where the edges intersect perpendicularly.
Let φ be the golden ratio.The 12 points given by (0, ±1, ±φ) and cyclic permutations of these coordinates are the vertices of a regular icosahedron.Its dual regular dodecahedron, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points (±1, ±1, ±1) together with the 12 points (0, ±φ, ± 1 / φ ) and cyclic permutations of these coordinates.
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 ...
The volume of a flexible polyhedron must remain constant as it flexes; this result is known as the bellows theorem. ... depicts a stellated dodecahedron. [78] As the ...