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Polygon: the story on HearLore | HearLore
Polygon
The word polygon derives from the Greek adjective polys meaning much or many and gonia meaning corner or angle, though some scholars suggest the root may actually be gony meaning knee. This linguistic journey began in ancient times when the regular polygons were known to the ancient Greeks, with the pentagram, a non-convex regular polygon, appearing as early as the 7th century B.C. on a krater by Aristophanes found at Caere and now in the Capitoline Museum. Two pentagrams are visible near the center of the image, marking one of the earliest known artistic representations of these geometric forms. The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century, expanding the understanding beyond simple shapes to include complex intersections. In 1952, Geoffrey Colin Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons that exist in two real and two imaginary dimensions.
The Shape Of Things
A polygon is a plane figure made up of line segments connected to form a closed polygonal chain, where the segments are called edges or sides and the points where two edges meet are the vertices or corners. An n-gon is a polygon with n sides, for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself, meaning the only allowed intersections among the line segments are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a solid polygon, and the interior of a solid polygon is its body, also known as a polygonal region or polygonal area. A polygonal chain may cross over itself, creating star polygons and other self-intersecting polygons, and some sources also consider closed polygonal chains in Euclidean space to be a type of polygon, known as a skew polygon, even when the chain does not lie in a single plane.
Convexity And Concavity
Polygons may be characterized by their convexity or type of non-convexity, where a convex polygon is one where any line drawn through the polygon and not tangent to an edge or corner meets its boundary exactly twice. As a consequence, all its interior angles are less than 180 degrees, and equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. This condition is true for polygons in any geometry, not just Euclidean. A non-convex polygon is one where a line may be found which meets its boundary more than twice, or equivalently, there exists a line segment between two boundary points that passes outside the polygon. A concave polygon is non-convex and simple, meaning there is at least one interior angle greater than 180 degrees. A star-shaped polygon is one where the whole interior is visible from at least one point, without crossing any edge, and the polygon must be simple, and may be convex or concave. All convex polygons are star-shaped, but a polygon cannot be both a star and star-shaped.
Common questions
What is the origin of the word polygon?
The word polygon derives from the Greek adjective polys meaning much or many and gonia meaning corner or angle, though some scholars suggest the root may actually be gony meaning knee.
When did the ancient Greeks first use the pentagram as a polygon?
The pentagram, a non-convex regular polygon, appeared as early as the 7th century B.C. on a krater by Aristophanes found at Caere and now in the Capitoline Museum.
Who made the first known systematic study of non-convex polygons?
Thomas Bradwardine made the first known systematic study of non-convex polygons in the 14th century, expanding the understanding beyond simple shapes to include complex intersections.
What is the sum of the interior angles of a simple n-gon?
The sum of the interior angles of a simple n-gon is 180 times n minus 2 degrees because any simple n-gon can be considered to be made up of n minus 2 triangles, each of which has an angle sum of 180 degrees.
Where can regular hexagons be found in nature?
Regular hexagons occur when the cooling of lava forms areas of tightly packed columns of basalt, which may be seen at the Giant's Causeway in Northern Ireland, or at the Devil's Postpile in California.
Which philosophers used the chiliagon to explore the limits of human imagination?
Philosophers including René Descartes, Immanuel Kant, and David Hume used the chiliagon, a polygon with 1000 sides, as an example in discussions to explore the limits of human imagination and the nature of mathematical concepts.
A polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral, meaning all corner angles are equal and all edges are of the same length. A non-convex regular polygon is called a regular star polygon, and the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra. For a regular n-gon, a p-gon with central density q, each interior angle is radians or degrees. The regular star pentagon is also known as a pentagram or pentacle, and the simplest polygon such that the regular form is not constructible with compass and straightedge is the heptagon. The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and angle trisector is the hendecagon, and the simplest polygon such that the regular form cannot be constructed with neusis is the icosagon.
Area And Angle
Any polygon has as many corners as it has sides, and the sum of the interior angles of a simple n-gon is radians or degrees, because any simple n-gon can be considered to be made up of n-2 triangles, each of which has an angle sum of radians or 180 degrees. The exterior angle is the supplementary angle to the interior angle, and tracing around a convex n-gon, the angle turned at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360 degrees. Tracing around an n-gon in general, the sum of the exterior angles, the total amount one rotates at the vertices, can be any integer multiple d of 360 degrees, for example 720 degrees for a pentagram and 0 degrees for an angular eight or antiparallelogram, where d is the density or turning number of the polygon. The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by the formula A equals one-half times r times p, and this radius is also termed its apothem and is often represented as a.
The Infinite And The Complex
A polygon with holes is an area-connected or multiply-connected planar polygon with one external boundary and one or more interior boundaries, and an apeirogon is an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions. A skew apeirogon is an infinite sequence of sides and angles that do not lie in a flat plane, and a complex polygon is a configuration analogous to an ordinary polygon, which exists in the complex plane of two real and two imaginary dimensions. An abstract polygon is an algebraic partially ordered set representing the various elements, sides, vertices, and their connectivity, and a real geometric polygon is said to be a realization of the associated abstract polygon. A polyhedron is a three-dimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions, and the corresponding shapes in four or higher dimensions are called polytopes. A spherical polygon is a circuit of arcs of great circles, sides and vertices on the surface of a sphere, and it allows the digon, a polygon having only two sides and two corners, which is impossible in a flat plane.
Nature And Computation
Polygons appear in rock formations, most commonly as the flat facets of crystals, where the angles between the sides depend on the type of mineral from which the crystal is made. Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of basalt, which may be seen at the Giant's Causeway in Northern Ireland, or at the Devil's Postpile in California. In biology, the surface of the wax honeycomb made by bees is an array of hexagons, and the sides and base of each cell are also polygons. In computer graphics, a polygon is a primitive used in modelling and rendering, and they are defined in a database, containing arrays of vertices, the coordinates of the geometrical vertices, as well as other attributes of the polygon, such as color, shading and texture, connectivity information, and materials. Any surface is modelled as a tessellation called polygon mesh, and if a square mesh has n points per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square.
The Philosophical Polygon
The chiliagon, a polygon with 1000 sides, has been used by philosophers including René Descartes, Immanuel Kant, and David Hume as an example in discussions, and the megagon, a polygon with 1,000,000 sides, has been used as an illustration of a well-defined concept that cannot be visualized. The megagon is also used as an illustration of the convergence of regular polygons to a circle, and as with René Descartes's example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualized. Philosophers have used the chiliagon to explore the limits of human imagination and the nature of mathematical concepts, and the megagon serves as a bridge between the finite and the infinite, demonstrating how a polygon with a million sides becomes indistinguishable from a circle to the human eye. The 257-gon and the 65537-gon are constructible polygons, and the 17-gon is the simplest polygon such that the regular form is constructible with compass and straightedge, marking a significant milestone in the history of geometric construction.