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Picture Name Schläfli symbol Vertex/Face configuration exact dihedral angle (radians) dihedral angle – exact in bold, else approximate (degrees) Platonic solids (regular convex)
A dihedral angle is the angle between two intersecting planes or half-planes. It is a plane angle formed on a third plane, perpendicular to the line of intersection between the two planes or the common edge between the two half-planes. In higher dimensions, a dihedral angle represents the angle between two hyperplanes.
This geometry also defines lunes of greater angles: {2} π-θ, and {2} 2π-θ. In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. [1] It is an example of a digon, {2} θ, with dihedral angle θ. [2] The word "lune" derives from luna, the Latin word for Moon.
The exsphere touches the face of the regular polyedron at the center of the incircle of that face. If the exsphere radius is denoted r ex, the radius of this incircle r in and the dihedral angle between the face and the extension of the adjacent face δ, the center of the exsphere is located from the viewpoint at the middle of one edge of the face by bisecting the dihedral angle.
The 12 face angles - there are three of them for each of the four faces of the tetrahedron. The 6 dihedral angles - associated to the six edges of the tetrahedron, since any two faces of the tetrahedron are connected by an edge. The 4 solid angles - associated to each point of the tetrahedron.
The dihedral angle of a triangular cupola between square-to-triangle is approximately 125°, that between square-to-hexagon is 54.7°, and that between triangle-to-hexagon is 70.5°. Therefore, the dihedral angle of a cuboctahedron between square-to-triangle, on the edge where the base of two triangular cupolas are attached is 54.7° + 70.5 ...
In pyritohedral pyrite, the faces have a Miller index of (210), which means that the dihedral angle is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the ...
This animation shows a transformation from a cube to a rhombic triacontahedron by dividing the square faces into 4 squares and splitting middle edges into new rhombic faces. The ratio of the long diagonal to the short diagonal of each face is exactly equal to the golden ratio , φ , so that the acute angles on each face measure 2 arctan ( 1 ...