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A regular skew hexagon seen as edges (black) of a triangular antiprism, symmetry D 3d, [2 +,6], (2*3), order 12. A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes.
A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra, of which all but the smallest one (which is a regular dodecahedron) have mostly hexagonal faces.
For every convex polyhedron, there exists a dual polyhedron having faces in place of the original's vertices and vice versa, and; the same number of edges. The dual of a convex polyhedron can be obtained by the process of polar reciprocation. [21] Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space.Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex.
They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron.
The complete list of convex polygons that can tile the plane includes the above 15 pentagons, three types of hexagons, and all quadrilaterals and triangles. [5] A consequence of this proof is that no convex polygon exists that tiles the plane only aperiodically, since all of the above types allow for a periodic tiling.
Convex regular icosahedron A tensegrity icosahedron. In geometry, an icosahedron (/ ˌ aɪ k ɒ s ə ˈ h iː d r ən,-k ə-,-k oʊ-/ or / aɪ ˌ k ɒ s ə ˈ h iː d r ən / [1]) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) ' twenty ' and ἕδρα (hédra) ' seat '. The plural can be either ...
The polygon is the convex hull of its edges. Additional properties of convex polygons include: The intersection of two convex polygons is a convex polygon. A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices.