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exact dihedral angle (radians) dihedral angle – exact in bold, else approximate (degrees) Platonic solids (regular convex) Tetrahedron {3,3} (3.3.3) arccos ( 1 / 3 ) 70.529° Hexahedron or Cube {4,3} (4.4.4) arccos (0) = π / 2 90° Octahedron {3,4} (3.3.3.3) arccos (- 1 / 3 ) 109.471° Dodecahedron {5,3} (5.5.5) arccos ...
An angle of 0° means the face normal vectors are antiparallel and the faces overlap each other, which implies that it is part of a degenerate polyhedron. An angle of 180° means the faces are parallel, as in a tiling. An angle greater than 180° exists on concave portions of a polyhedron. Every dihedral angle in an edge-transitive polyhedron ...
A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In ...
This fact can be used to calculate the dihedral angles themselves for a regular or edge-symmetric ideal polyhedron (in which all these angles are equal), by counting how many edges meet at each vertex: an ideal regular tetrahedron, cube or dodecahedron, with three edges per vertex, has dihedral angles = / = (), an ideal regular octahedron or ...
Its dihedral angles are all right angles. One can use it as the basis for the construction of an infinite family of combinatorially distinct polyhedra with right dihedral angles, formed by gluing copies of Jessen's icosahedron together on their equilateral-triangle faces. [1]
This polyhedron is topologically related as a part of a sequence of cantellated polyhedra with vertex figure (3.4.n.4), which continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry .
The Dehn invariant is not a number, but a vector in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges. [44] Another of Hilbert's problems, Hilbert's 18th problem, concerns (among other things) polyhedra that tile space. Every such polyhedron must have Dehn invariant zero. [45]
The opposite angle, between long edges, is arccos(-5+9 √ 5 / 40 )≈67.783011547435° . The other two angles of each face, between a short and a long edge each, are both equal to arccos( 5-2 √ 5 / 10 )≈86.97415549104°. The dihedral angle between any pair of adjacent faces is arccos(-19-8 √ 5 / 41 )≈154.12136312578°.