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A central cross section of a regular tetrahedron is a square. The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle. [11] When the intersecting plane is near one of the edges the rectangle is long and skinny.
A plane containing a cross-section of the solid may be referred to as a cutting plane. The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid. For instance, while all the cross-sections of a ball are disks, [2] the cross-sections of a cube depend on how the cutting plane is related to the ...
Vertex, edge and face of a cube. The Euler characteristic χ was classically defined for the surfaces of polyhedra, according to the formula = + where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron. [2]
The 600-cell has icosahedral cross sections of two sizes, and each of its 120 vertices is an icosahedral pyramid; the icosahedron is the vertex figure of the 600-cell. Another polytope with regular icosahedrons as its cell is the semiregular 4-polytope of snub 24-cell .
Two identically formed Rubik's Snakes can approximate an octahedron. Especially in roleplaying games, this solid is known as a "d8", one of the more common polyhedral dice. If each edge of an octahedron is replaced by a one-ohm resistor, the resistance between opposite vertices is 1 / 2 ohm, and that between adjacent vertices 5 / 12 ...
Nevertheless, some polyhedrons may not possess one or two of those symmetries: A polyhedron with vertex-transitive and edge-transitive is said to be quasiregular, although they have regular faces, and its dual is face-transitive and edge-transitive. A vertex- but not edge-transitive polyhedron with regular polygonal faces is said to be semiregular.
The faces of the icosahedron are eight congruent equilateral triangles with the short side length, and twelve congruent obtuse isosceles triangles with one long edge and two short edges. [8] Jessen's icosahedron is vertex-transitive (or isogonal), meaning that it has symmetries taking any vertex to any other vertex. [9]
In this respect, ideal polyhedra are different from Euclidean polyhedra (and from their Euclidean Klein models): for instance, on a Euclidean cube, any geodesic can cross at most two edges incident to a single vertex consecutively, before crossing a non-incident edge, but geodesics on the ideal cube are not limited in this way. [26]