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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.
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]
The truncated icosahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. [5] It has the same symmetry as the regular icosahedron, the icosahedral symmetry , and it also has the property of vertex-transitivity .
The convex polyhedron is well-defined with several equivalent standard definitions, one of which is a polyhedron that is a convex set, or the polyhedral surface that bounds it. Every convex polyhedron is the convex hull of its vertices, and the convex hull of a finite set of points is a polyhedron.
If a plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a cross-section of the solid. [1] 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.
The great icosahedron can be constructed as a uniform snub, with different colored faces and only tetrahedral symmetry: .This construction can be called a retrosnub tetrahedron or retrosnub tetratetrahedron, [1] similar to the snub tetrahedron symmetry of the icosahedron, as a partial faceting of the truncated octahedron (or omnitruncated tetrahedron): .
It can be constructed from the regular icosahedron, with one edge contraction, removing one vertex, 3 edges, and 2 faces. This contraction distorts the circumscribed sphere original vertices. With all equilateral triangle faces, it has 2 sets of 3 coplanar equilateral triangles (each forming a half- hexagon ), and thus is not a Johnson solid .
As the definition above, a Johnson solid is a convex polyhedron with regular polygons as their faces. However, there are several properties possessed by each of them. All but five of the 92 Johnson solids are known to have the Rupert property , meaning that it is possible for a larger copy of themselves to pass through a hole inside of them.