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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)
Free energy diagram of n-butane as a function of dihedral angle. In stereochemistry, a torsion angle is defined as a particular example of a dihedral angle, describing the geometric relation of two parts of a molecule joined by a chemical bond. [4] [5] Every set of three non-colinear atoms of a molecule defines a half-plane. As explained above ...
The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by Ω = q θ − ( q − 2 ) π . {\displaystyle \Omega =q\theta -(q-2)\pi .\,} This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron { p , q } is a regular q -gon.
the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom is that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a regular octagon 135°. The dihedral angle between two adjacent squares on the edge where a square cupola is ...
For instance, for the ideal cube, the dihedral angles are / and their supplements are /. The three supplementary angles at a single vertex sum to 2 π {\displaystyle 2\pi } but the four angles crossed by a curve midway between two opposite faces sum to 8 π / 3 > 2 π {\displaystyle 8\pi /3>2\pi } , and other curves cross even more of these ...
The common physical model of a Klein bottle is a similar construction. The Science Museum in London has a collection of hand-blown glass Klein bottles on display, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by Alan Bennett. [3] The Klein bottle, proper, does not self-intersect.