A star polygon is concave.
Star polygons are characterized by their star shape. A polygon , meanwhile, is a figure composed of a certain number of sides, which are non-aligned and straight segments. Depending on their characteristics, there are multiple classifications of polygons.
Concave polygons are those that have at least one interior angle that measures more than 180° or pi radians. A star polygon, therefore, is concave since it has one or more interior angles of more than 180° or pi radians. Other characteristics of concave polygons and star polygons are that they also have one or more exterior diagonals and have two or more vertices that, when joined by a segment, cut at least one side of the figure .
Characteristics of a star polygon
A star polygon is not only concave, but can also be part of regular polygons when its interior angles and sides are equal. Through certain "unions" made through new segments that link the vertices, a star polygon can be created from a regular polygon (like a pentagon, for example).
Regular star polygons can also be simple. This occurs when its vertices are found, alternatively, on a pair of concentric circles with central angles that are equal.
Star polygons can be classified as regular.
Your creation
One way to construct star polygons is by overlapping and rotating other polygons. Thus it is possible to develop numerous star-shaped polygons, such as the famous Star of David , which is a symbol of the Jewish religion .
By dividing a circle into n parts and joining them successively it is possible to obtain a regular convex polygon; If the unions between the vertices are made two by two, three by three, etc., a concave and star polygon is obtained. In other words, to construct a star polygon you can start from a regular convex polygon and join its vertices in a continuous sequence, maintaining the interval between one and the other, so that the following conditions are met:
- The number of vertices of the original polygon ( N ) over the space between one and the other ( M ) must form an irreducible fraction , that is, its denominator and numerator do not have factors in common, so the fraction cannot be simplified.
- The star polygon formed by joining the vertices of a regular convex polygon must be the same regardless of the direction in which the segments are drawn. In other words, N/M and N/(NM) must represent the same polygon.
Other considerations about star polygons
Some concepts related to the star polygon are the following: genus , the number of sides (or chords) it has, which must coincide with its number of vertices, which is why its name is the same as that of convex polygons (with a type 6 is called a star hexagon , for example); step , the number of parts into which the circle is divided, and the value that includes the sides of the polygon; species , a property with an ordinal name that refers to the step, such that if the unions are two by two it is called second species , and so on.
Of the best-known polygons, it is known that the triangle and the square do not have a star; the pentagon , the octagon, the decagon and the dodecagon, on the other hand, have one each, of the first, second, second and fifth or fourth species, respectively; the heptagon and the enneagon have two each, of the first and second kind; The one with eleven sides, finally, has four, ranging from the first to the fourth species .