The dihedralangles for the edge-transitive polyhedra are: Coxeter, Regular Polytopes (1963), Macmillan Company Regular Polytopes, (3rd edition, 1973)...
number ofangles associated with each Platonic solid. The dihedralangle is the interior angle between any two face planes. The dihedralangle, θ, of the...
infinite-dimensional vector space, determined from the lengths and dihedralanglesof a polyhedron's edges. Another of Hilbert's problems, Hilbert's 18th problem, concerns...
vertices of a convex regular polyhedron all lie on a sphere. All the dihedralanglesof the polyhedron are equal All the vertex figures of the polyhedron are...
plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedralangles. Two...
global symmetries of the polyhedron rather than local neighborhoods. Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally...
is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal...
octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral...
instead of Archimedean, so their duals are Platonic instead of Catalan. (They are shown with brown background in the table below.) All dihedralanglesof a...
sense: the exact angles and edge lengths may differ. Hart introduced the reflection operator r, that gives the mirror image of the polyhedron. This is not...
almost all of the uniform polyhedra from the acute and obtuse Schwarz triangles. The numbers that can be used for the sides of a non-dihedral acute or obtuse...
a ratio of 2:1. The Coxeter-Dynkin diagram of the generated polyhedron contains three nodes representing the three mirrors. The dihedralangle between...
octahedron. It has cubical or octahedral symmetry, and is the only convex polyhedron whose faces are all squares. Its generalization for higher-dimensional...
prism, the dihedral group D 3 h {\displaystyle D_{3\mathrm {h} }} of order twelve. Its dihedralangles can be calculated by adding the anglesof the component...
coloring with dihedral symmetry as a truncated triangular antiprism. The constructional names are given for each. Their Conway polyhedron notation is given...
models: the angles A, B, C determine the sides a, b, c. The Law of Cosines can be generalized to all polyhedra by considering any polyhedron with vector...
regular, all faces of triangular bipyramid are equilateral. It is an example of a deltahedron and of a Johnson solid. Many polyhedrons are related to the...
Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedralangle on a...
tessellation of projective space or lens space yields a distinct notion ofpolyhedron. List of spherical symmetry groups List of character tables for chemically...
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting...
conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. This polyhedron is topologically...
quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex. A regular polyhedron is a uniform polyhedron which...
In geometry, an octagon (from Ancient Greek ὀκτάγωνον (oktágōnon) 'eight angles') is an eight-sided polygon or 8-gon. A regular octagon has Schläfli symbol...
sides of a triangle with angles π/l, π/m and π/n (measured in radians). The product of the reflections in two adjacent sides is a rotation by the angle which...
consequently its vertices can be repositioned by folding (changing the dihedralangle) at edges and face diagonals. The cuboctahedron's kinematics is noteworthy...
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