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A quadrantal spherical triangle together with Napier's circle for use in his mnemonics. A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of π /2 radians at the centre of the sphere: on the unit sphere the side has length π /2.
A quadrantal spherical triangle together with Napier's circle for use in his mnemonics. A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of π /2 radians at the centre of the sphere: on the unit sphere the side has length π /2.
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices.
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, [1] orbifold notation, [2] and order.
The spherical octant itself is the intersection of the sphere with one octant of space. Uniquely among spherical triangles, the octant is its own polar triangle. [2] The octant can be parametrized using a rational quartic Bézier triangle. [3] The solid angle subtended by a spherical octant is π /2 sr, one-eight of the solid angle of a sphere. [4]
Geometries can also be computed by ab initio quantum chemistry methods to high accuracy. The molecular geometry can be different as a solid, in solution, and as a gas. The position of each atom is determined by the nature of the chemical bonds by which it is connected to its neighboring atoms.
Consider the projective (spherical) triangle at the point ; the vertices of this projective triangle are the three lines that join with the other three vertices of the tetrahedron. The edges will have spherical lengths α i , j , α i , k , α i , l {\displaystyle \alpha _{i,j},\alpha _{i,k},\alpha _{i,l}} and the respective opposite spherical ...
Another definition by Bernal (1964) is similar to the previous one, in which he was interested in the shapes of holes left in irregular close-packed arrangements of spheres. It is stated as a convex polyhedron with equilateral triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent ...