The dodecahedron and icosahedron are dual polyhedra. A regular dodecahedron has 12 faces and 20 vertices, whereas a regular icosahedron has 20 faces and 12 vertices. Both have 30 edges. When a regular dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an … See more A regular dodecahedron or pentagonal dodecahedron is a dodecahedron that is regular, which is composed of 12 regular pentagonal faces, three meeting at each vertex. It is one of the five Platonic solids. It has 12 faces, 20 … See more • The dihedral angle of a regular dodecahedron is 2 arctan(ϕ) or approximately 116.565° (where again ϕ = 1 + √5/2, the golden ratio). OEIS: A137218 Note that the tangent of the dihedral angle is exactly −2. • If the original regular dodecahedron has … See more Regular dodecahedral objects have found some practical applications, and have also played a role in the visual arts and in philosophy. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he … See more The surface area A and the volume V of a regular dodecahedron of edge length a are: Additionally, the … See more The following Cartesian coordinates define the 20 vertices of a regular dodecahedron centered at the origin and suitably scaled and oriented: (±1, ±1, ±1) (0, ±ϕ, ±1/ϕ) (±1/ϕ, 0, ±ϕ) (±ϕ, ±1/ϕ, 0) where ϕ = 1 + √5/2 is … See more The regular dodecahedron is the third in an infinite set of truncated trapezohedra which can be constructed by truncating the two axial … See more Regular dodecahedra fill space with cubes and bilunabirotundas (Johnson solid 91), in the ratio of 1 to 1 to 3. The dodecahedra alone make a lattice of edge-to-edge pyritohedra. The bilunabirotundas fill the rhombic gaps. Each cube meets six bilunabirotundas in … See more WebJan 1, 2011 · On the one hand, Plato lets Socrates explicitly say that he is convinced that the earth is round, meaning spherical, but on the other hand he compares the earth with a …
Dodecahedron - Definition, Formulas, Properties, Examples - Cuemath
WebThe Spherical Dodecahedron: The Stereographic Projection of the Dodecahedron: Volume and Surface Area of the Dodecahedron Volume = √5/2 φ 4 s 3 or V = ¼ (15 + 7√5)s 3 s = … WebJan 25, 2016 · Note that the N = 25 sphere and dodecahedron clusters do not actually share the same structure; the sphere cluster is a central particle surrounded by the N = 24 … mudd law firm
Homology sphere - Wikipedia
WebMar 31, 2014 · We have been developing electromagnetic spherical actuators capable of three-degree-of-freedom rotation. However, these actuators require complex control to realize simultaneous triaxial drive, because rotation around one axis interferes with rotation around another. In this paper, we propose a new three-degree-of-freedom actuator where … http://www.map.mpim-bonn.mpg.de/Poincar%C3%A9%27s_homology_sphere WebThe Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere, first constructed by Henri Poincaré. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphereitself) with a finite fundamental group. how to make toolbar larger windows 10