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The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper [ 1 ] though it seems that the special case, in which the map is defined for pentagons only, goes ...
A geodesic polygon is a polygon whose sides are geodesics. It is analogous to a spherical polygon, whose sides are great circles. The area of such a polygon may be found by first computing the area between a geodesic segment and the equator, i.e., the area of the quadrilateral AFHB in Fig. 1 (Danielsen 1989). Once this area is known, the area ...
The regular 65537-gon (one with all sides equal and all angles equal) is of interest for being a constructible polygon: that is, it can be constructed using a compass and an unmarked straightedge. This is because 65,537 is a Fermat prime , being of the form 2 2 n + 1 (in this case n = 4).
Given: Ellipsoid x 2 / a 2 + y 2 / b 2 + z 2 / c 2 = 1 and the plane with equation n x x + n y y + n z z = d, which have an ellipse in common. Wanted: Three vectors f 0 (center) and f 1 , f 2 (conjugate vectors), such that the ellipse can be represented by the parametric equation
Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih 1, and 2 cyclic group symmetries: Z 5, and Z 1. These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order. [10] Full symmetry of the regular form is r10 and no symmetry is labeled a1.
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
Though a good set of parameters permits identification of every point in the object space, it may be that, for a given parametrization, different parameter values can refer to the same point. Such mappings are surjective but not injective. An example is the pair of cylindrical polar coordinates (ρ, φ, z) and (ρ, φ + 2π, z).
A subdivision rule takes a tiling of the plane by polygons and turns it into a new tiling by subdividing each polygon into smaller polygons. It is finite if there are only finitely many ways that every polygon can subdivide. Each way of subdividing a tile is called a tile type. Each tile type is represented by a label (usually a letter).