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In geometry, a pentagon (from Greek πέντε (pente) ' five ' and γωνία (gonia) ' angle ' [1]) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.
The Order of the Eastern Star, an organization (established 1850) associated with Freemasonry, uses a pentagram as its symbol, with the five isosceles triangles of the points colored blue, yellow, white, green, and red. In most Grand Chapters the pentagram is used pointing down, but in a few, it is pointing up.
The parametric angle 𝛼 (in degrees or radians) can be chosen to match internal angles of neighboring polygons in a tessellation pattern. In his 1619 work Harmonices Mundi , among periodic tilings, Johannes Kepler includes nonperiodic tilings, like that with three regular pentagons and one regular star pentagon fitting around certain vertices ...
Angles are A = 140°, B = 60°, C = 160°, D = 80°, E = 100°. [13] [14] In 2016 it could be shown by Bernhard Klaassen that every discrete rotational symmetry type can be represented by a monohedral pentagonal tiling from the same class of pentagons. [15] Examples for 5-fold and 7-fold symmetry are shown below.
In 1809, Louis Poinsot rediscovered Kepler's figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two more regular stars, the great icosahedron and great dodecahedron. Some people call these two the Poinsot polyhedra. Poinsot did not know if he had discovered all the ...
The concept of the position angle is inherited from nautical navigation on the oceans, where the optimum compass course is the course from a known position s to a target position t with minimum effort. Setting aside the influence of winds and ocean currents, the optimum course is the course of smallest distance between the two positions on the ...
Pentagramma mirificum (Latin for "miraculous pentagram") is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles.
The heptagram is known among neopagans as the Elven Star or Fairy Star. It is treated as a sacred symbol in various modern pagan and witchcraft traditions. Blue Star Wicca also uses the symbol, where it is referred to as a septegram. The second heptagram is a symbol of magical power in some pagan spiritualities.