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In some cases they have geometric realizations. An example is the Szilassi polyhedron, a toroidal polyhedron that realizes the Heawood map. In this case, the polyhedron is much less symmetric than the underlying map, but in some cases it is possible for self-crossing polyhedra to realize some or all of the symmetries of a regular map.
Most Goldberg polyhedra can be constructed using Conway polyhedron notation starting with (T)etrahedron, (C)ube, and (D)odecahedron seeds. The chamfer operator, c, replaces all edges by hexagons, transforming GP(m,n) to GP(2m,2n), with a T multiplier of 4.
The Petrie dual of a regular polyhedron is a regular map whose vertices and edges correspond to the vertices and edges of the original polyhedron, and whose faces are the set of skew Petrie polygons. [ 12 ]
In cartography, R. Buckminster Fuller used the net of a regular icosahedron to create a map known as Dymaxion map, by subdividing the net into triangles, followed by calculating the grid on the Earth's surface, and transferring the results from the sphere to the polyhedron.
Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property.
It is one of the Platonic solids, a set of polyhedrons in which the faces are regular polygons that are congruent and the same number of faces meet at a vertex. [2] This set of polyhedrons is named after Plato. In Theaetetus, a dialogue of Plato, Plato hypothesized that the classical elements were made of the five uniform regular solids. Plato ...
The n dimensional real projective space is the quotient of the n sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere – either 0 or 1. The n dimensional torus is the product space of n circles. Its Euler characteristic is 0, by the product property.
3 constructions for a {3,5+} 6,0 An icosahedron and related symmetry polyhedra can be used to define a high geodesic polyhedron by dividing triangular faces into smaller triangles, and projecting all the new vertices onto a sphere.