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A regular skew decagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew decagon and can be seen in the vertices and side edges of a pentagonal antiprism, pentagrammic antiprism, and pentagrammic crossed-antiprism with the same D 5d, [2 +,10] symmetry, order 20.
For mathematical objects in more dimensions, see list of mathematical shapes. For a broader scope, ... Decagon – 10 sides; Hendecagon – 11 sides;
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body. Vertex, a 0-dimensional element; Edge, a 1-dimensional element; Face, a 2-dimensional element; Cell, a 3-dimensional element; Hypercell or Teron, a 4-dimensional element; Facet, an (n-1)-dimensional element
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides (but no n) belongs to one of the facets.The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides (but no three) belongs to one of the faces. [1]
In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of an octagonal antiprism with the same D 8d, [2 +,16] symmetry, order 32. The octagrammic antiprism , s{2,16/3} and octagrammic crossed-antiprism , s{2,16/5} also have regular skew octagons.
Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (The icosidodecahedron is the equatorial cross-section of the 600-cell, and the decagon is the equatorial cross-section of the icosidodecahedron.)
Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N 3 and N 5; then if ordinates N 3 P 3, N 5 P 5 are drawn to the circle, the arcs AP 3, AP 5 will be 3/17 and 5/17 of the circumference." The point N 3 is very close to the center point of Thales' theorem over AF.
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