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First, to prove a pentagon cannot form a regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 3 1 ⁄ 3 (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps ...
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
{3,4} Defect 120° {3,5} Defect 60° {3,6} Defect 0° {4,3} Defect 90° {4,4} Defect 0° {5,3} Defect 36° {6,3} Defect 0° A vertex needs at least 3 faces, and an angle defect. A 0° angle defect will fill the Euclidean plane with a regular tiling. By Descartes' theorem, the number of vertices is 720°/defect.
If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side. [1]: pp. 261–264
12 Miscellaneous surfaces. ... a quasi-helical shape characterized by multiple tendril perversions; ... 5: 4: 2: ∞ 6: 3 convex 6-polytopes: 0: 1 pentacombs: 0: 0: 5:
If two angles of a triangle have measures equal to the measures of two angles of another triangle, then the triangles are similar. Corresponding sides of similar polygons are in proportion, and corresponding angles of similar polygons have the same measure. Two congruent shapes are similar, with a scale factor of 1. However, some school ...
[12] [6] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. [13] Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly ...
It is straightforward to construct with ruler and compasses. It is the angle of the equilateral triangle or is 1 / 6 turn. 1 Babylonian unit = 60° = π /3 rad ≈ 1.047197551 rad. hexacontade: 60: 6° The hexacontade is a unit used by Eratosthenes. It equals 6°, so a whole turn was divided into 60 hexacontades. pechus: 144 to 180: 2 ...