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A regular decagon has all sides of equal length and each internal angle will always be equal to 144°. [1] Its Schläfli symbol is {10} [2] and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.
Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well.
Specifically, the n-th decagonal numbers counts the dots in a pattern of n nested decagons, all sharing a common corner, where the ith decagon in the pattern has sides made of i dots spaced one unit apart from each other. The n-th decagonal number is given by the following formula
The regular dodecagon is the Petrie polygon for many higher-dimensional polytopes, seen as orthogonal projections in Coxeter planes. Examples in 4 dimensions are the 24-cell, snub 24-cell, 6-6 duoprism, 6-6 duopyramid. In 6 dimensions 6-cube, 6-orthoplex, 2 21, 1 22. It is also the Petrie polygon for the grand 120-cell and great stellated 120-cell.
12 decagons: 90 60 I h: Truncated icosahedron: 5.6.6: 12 pentagons 20 hexagons 90 60 I h: Rhombicosidodecahedron: 3.4.5.4: 20 triangles 30 squares 12 pentagons 120 60 I h: Truncated icosidodecahedron: 4.6.10: 30 squares 20 hexagons 12 decagons 180 120 I h: Snub dodecahedron: 3.3.3.3.5: 80 triangles 12 pentagons 150 60 I
The {5/2} star pentagon is also known as a pentalpha or pentangle, and historically has been considered by many magical and religious cults to have occult significance. The {7/2} and {7/3} star polygons ( heptagrams ) also have occult significance, particularly in the Kabbalah and in Wicca .
12 decagons x 6 100 irregular hexagons x 4 Petrie polygons: 1 pentagon x 2 1 octagon x 3 2 octagons x 4 2 dodecagons x 4 ... Many other 4-polytopes are possible, but ...
This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 3 6; 3 6; 3 4.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling.