pentagonal

Pentagonal is an adjective relating to five angles or five-sided figures. It derives from the Greek pente meaning five and gōnía meaning angle, and it is used across mathematics, science, and culture to describe objects, patterns, or symmetries that involve five.

In geometry, a pentagon is a polygon with five sides. The sum of the interior angles of any pentagon is 540 degrees, and a regular pentagon has all sides and angles equal, with each interior angle measuring 108 degrees. The ratio of a pentagon’s diagonal to its side in a regular pentagon is the golden ratio, approximately 1.618. A regular pentagon can be combined to form a pentagram, a five-pointed star shape. Regular pentagons exhibit fivefold rotational symmetry and five lines of reflection symmetry, though fivefold symmetry cannot tile the plane periodically.

Pentagonal figures also appear in number theory, where pentagonal numbers form a sequence given by P(n) = n(3n − 1)/2 for integers n. These numbers index certain figurate patterns and appear in partition-related identities studied by Euler.

In tiling and symmetry, fivefold (pentagonal) symmetry is a common motif in nonperiodic structures, such as quasicrystals and Penrose tilings, which exhibit order without repeating patterns in the ordinary sense.

In applied contexts, pentagonal descriptors appear in coordination chemistry and crystallography to describe seven-coordinate environments known as pentagonal bipyramidal geometries, as well as in architecture and art, where pentagonal shapes or symbols, including the five-pointed star, are used for decorative or symbolic purposes.

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