Polygon
The word polygon derives from the Greek adjective polús meaning much or many and gōnía meaning corner or angle. Some scholars suggest that gónu meaning knee may be the origin of gon instead. The term entered Late Latin as polygōnum before becoming standard in mathematical vocabulary. Individual polygons are named by combining a Greek-derived numerical prefix with the suffix -gon. A triangle is technically a 3-gon while a quadrilateral serves as a 4-gon. Exceptions exist for side counts easily expressed verbally like twenty-sided icosagons or thirty-sided triacontagons. Mathematicians generally use numerical notation beyond decagons and dodecagons such as 17-gon or 257-gon. Philosophers including René Descartes and Immanuel Kant used the chiliagon as an example in discussions about well-defined concepts that cannot be visualized. The million-sided megagon illustrates how regular polygons converge to a circle.
A simple polygon does not intersect itself except at shared endpoints of consecutive segments. Any line drawn through a convex polygon meets its boundary exactly twice if it is not tangent to an edge or corner. All interior angles of a convex polygon measure less than 180 degrees. Non-convex polygons allow lines to meet their boundaries more than twice. Concave polygons are non-convex yet remain simple with at least one interior angle exceeding 180 degrees. Star-shaped polygons permit visibility from at least one point without crossing any edge. Self-intersecting boundaries create star polygons where regions may have density factors multiplying area calculations. A pentagram exemplifies a self-intersecting regular polygon distinct from star-shaped forms. Spherical polygons exist on curved surfaces allowing digons with only two sides impossible in flat planes. Skew polygons zigzag through three or more dimensions rather than lying within a single plane.
The sum of interior angles for any simple n-gon equals n minus 2 multiplied by 180 degrees. This occurs because any simple n-gon can be divided into triangles each containing 180 degrees. The exterior angle represents the supplementary angle to the interior angle measured during traversal around the shape. Tracing all the way around a convex n-gon produces exactly 360 degrees total rotation. For concave simple polygons external angles turning opposite directions subtract from the total turned amount. The shoelace formula calculates signed area using vertex coordinates ordered counterclockwise or clockwise. Lopshits described an alternative method in 1963 computing area from side lengths and exterior angles. Pick's theorem applies when vertices lie on equally spaced grid points counting interior and boundary intersections. The isoperimetric inequality holds for every polygon relating perimeter p and area A. Regular polygons maximize area among all shapes sharing identical side lengths or perimeters. Self-intersecting regions may assign density factors multiplying specific areas while other definitions treat enclosed point sets differently.
Regular polygons were known to ancient Greeks with pentagrams appearing as early as the 7th century B.C. on kraters found at Caere. Thomas Bradwardine conducted the first systematic study of non-convex polygons in general during the 14th century. Geoffrey Colin Shephard generalized polygons to complex planes in 1952 pairing real dimensions with imaginary ones. Spherical polygons play important roles in cartography and Wythoff's construction of uniform polyhedra. Abstract polygons represent algebraic partially ordered sets connecting sides vertices and their connectivity structures. Polytopes extend these concepts into four or higher dimensions beyond three-dimensional polyhedra. Mathematical Plums edited by R. Honsberger published Chakerian's work on distorted views of geometry in 1979. Coxeter's Regular Polytopes third edition appeared through Dover Publications in 1973 after initial Methuen release in 1948.
Polygons appear in rock formations most commonly as flat facets of crystals where angles depend on mineral types. Cooling lava forms tightly packed columns of basalt creating regular hexagons visible at Giant's Causeway in Northern Ireland. Similar formations occur at Devil's Postpile in California where geological processes produce polygonal patterns. The surface of wax honeycombs made by bees consists entirely of hexagonal arrays forming cellular structures. Each cell side and base represents a polygon within biological systems. These natural occurrences demonstrate how geometric principles manifest without human design intent. Mineral composition determines specific angular relationships between adjacent crystal faces across different environments.
A polygon serves as a primitive used in modeling and rendering within computer graphics databases. Vertices contain coordinates alongside attributes like color shading and texture information stored in connectivity arrays. Any surface becomes modeled as tessellation called polygon mesh containing squared squares or triangles depending on vertex counts. Square meshes with n points per side generate n squared squares or 2n squared triangles since two triangles fill each square. Imaging systems transfer structure from database to active memory before sending processed data to display screens. Polygons remain two-dimensional yet placed in three-dimensional orientation through system calculations during scene creation. Point-in-polygon tests determine whether given locations lie inside simple polygons defined by line segment sequences. Computational geometry algorithms compare capabilities speed and numerical robustness for boolean operations on polygon shapes.
Common questions
What is the origin of the word polygon?
The word polygon derives from the Greek adjective polús meaning much or many and gōnía meaning corner or angle. Some scholars suggest that gónu meaning knee may be the origin of gon instead.
How are interior angles calculated for any simple n-gon?
The sum of interior angles for any simple n-gon equals n minus 2 multiplied by 180 degrees. This occurs because any simple n-gon can be divided into triangles each containing 180 degrees.
When did regular polygons appear in ancient history?
Regular polygons were known to ancient Greeks with pentagrams appearing as early as the 7th century B.C. on kraters found at Caere. Thomas Bradwardine conducted the first systematic study of non-convex polygons in general during the 14th century.
Where do natural polygon formations occur in geology?
Cooling lava forms tightly packed columns of basalt creating regular hexagons visible at Giant's Causeway in Northern Ireland. Similar formations occur at Devil's Postpile in California where geological processes produce polygonal patterns.
How are polygons used in computer graphics databases?
A polygon serves as a primitive used in modeling and rendering within computer graphics databases. Vertices contain coordinates alongside attributes like color shading and texture information stored in connectivity arrays.