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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.
Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. Some tetrahedra that are not regular, including the Schläfli orthoscheme and the Hill tetrahedron, can tessellate.
An orthogonal polyhedron is one all of whose edges are parallel to axes of a Cartesian coordinate system. This implies that all faces meet at right angles, but this condition is weaker: Jessen's icosahedron has faces meeting at right angles, but does not have axis-parallel edges. Aside from the rectangular cuboids, orthogonal polyhedra are ...
This is a list of two-dimensional geometric shapes in Euclidean and other geometries. For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, see list of shapes.
There are 28 convex examples in Euclidean 3-space, [1] also called the Archimedean honeycombs. A honeycomb is called regular if the group of isometries preserving the tiling acts transitively on flags, where a flag is a vertex lying on an edge lying on a face lying on a cell. Every regular honeycomb is automatically uniform.
A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol {2}. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated digon, t{2} is a square, {4}.
A quadrilateral is a kite if and only if any one of the following conditions is true: The four sides can be split into two pairs of adjacent equal-length sides. [7] One diagonal crosses the midpoint of the other diagonal at a right angle, forming its perpendicular bisector. [9] (In the concave case, the line through one of the diagonals bisects ...
Within that plane, every triangle, irrespective of regularity, will tessellate. In contrast, regular pentagons do not tessellate. However, irregular pentagons, with different sides and angles can tessellate. There are 15 irregular convex pentagons that tile the plane. [6] Polyhedra are the three dimensional correlates of polygons.