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  2. Euler characteristic - Wikipedia

    en.wikipedia.org/wiki/Euler_characteristic

    The number of vertices and edges has remained the same, but the number of faces has been reduced by 1. Therefore, proving Euler's formula for the polyhedron reduces to proving V − E + F = 1 {\displaystyle \ V-E+F=1\ } for this deformed, planar object.

  3. Face (geometry) - Wikipedia

    en.wikipedia.org/wiki/Face_(geometry)

    where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, and hence 6 faces.

  4. Platonic solid - Wikipedia

    en.wikipedia.org/wiki/Platonic_solid

    The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces.

  5. Polyhedral combinatorics - Wikipedia

    en.wikipedia.org/wiki/Polyhedral_combinatorics

    The dimension of a face is the dimension of this hull. The 0-dimensional faces are the vertices themselves, and the 1-dimensional faces (called edges) are line segments connecting pairs of vertices. Note that this definition also includes as faces the empty set and the whole polytope P.

  6. Steinmetz solid - Wikipedia

    en.wikipedia.org/wiki/Steinmetz_solid

    The generation of a bicylinder Calculating the volume of a bicylinder. A bicylinder generated by two cylinders with radius r has the volume =, and the surface area [1] [6] =.. The upper half of a bicylinder is the square case of a domical vault, a dome-shaped solid based on any convex polygon whose cross-sections are similar copies of the polygon, and analogous formulas calculating the volume ...

  7. Disdyakis triacontahedron - Wikipedia

    en.wikipedia.org/wiki/Disdyakis_triacontahedron

    It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place. If the bipyramids, the gyroelongated bipyramids, and the trapezohedra are excluded, the disdyakis triacontahedron has the most faces of any other strictly convex polyhedron where every face of the polyhedron has the same ...

  8. Uniform polyhedron - Wikipedia

    en.wikipedia.org/wiki/Uniform_polyhedron

    Some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. Double covers. Some non-orientable polyhedra have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces, edges and ...

  9. Boundary representation - Wikipedia

    en.wikipedia.org/wiki/Boundary_representation

    Boundary representation is essentially a local representation connecting faces, edges and vertices. An extension of this was to group sub-elements of the shape into logical units called geometric features, or simply features. Pioneering work was done by Kyprianou in Cambridge also using the BUILD system and continued and extended by Jared and ...