Pentagram
| Regular pentagram | |
|---|---|
A regular pentagram
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| Type | Regular star polygon |
| Edges and vertices | 5 |
| Schläfli symbol | {5/2} |
| Coxeter diagram |      |
| Symmetry group | Dihedral (D5) |
| Internal angle (degrees) | 36° |
| Dual polygon | self |
| Properties | star, cyclic, equilateral, isogonal, isotoxal |
A pentagram (sometimes known as a pentalpha or pentangle or a star pentagon) is the shape of a five-pointed star drawn with five straight strokes.
The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), from πέντε (pente), "five" + γραμμή (grammē), "line". The word "pentacle" is sometimes used synonymously with "pentagram" The word pentalpha is a learned modern (17th-century) revival of a post-classical Greek name of the shape.
The pentagram is the simplest regular star polygon. The pentagram contains ten points (the five points of the star, and the five vertices of the inner pentagon) and fifteen line segments. It is represented by the Schläfli symbol {5/2}. Like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10.
The pentagram can be constructed by connecting alternate vertices of a pentagon; see details of the construction. It can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect.
The golden ratio, φ = (1 + √5) / 2 ≈ 1.618, satisfying
plays an important role in regular pentagons and pentagrams. Each intersection of edges sections the edges in the golden ratio: the ratio of the length of the edge to the longer segment is φ, as is the length of the longer segment to the shorter. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ. As the four-color illustration shows:
The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon.
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