Polygon

• 1. Introduction

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  A pentagram scratched onto a clay krater, painted by Aristophanes and found at Caere, sits today in the Capitoline Museum. It is one of the oldest traces of a polygon in human hands, dating to as early as the 7th century B.C. A polygon is a plane figure built from line segments joined into a closed polygonal chain. The segments are its edges, the meeting points its vertices. That definition sounds simple, almost too simple to fill a documentary. Yet from this skeleton of straight lines comes a sprawling family of shapes. Some refuse to lie flat. Some cross over themselves into stars. Some carry a million sides. How do mathematicians tell one polygon from another, and what counts as a polygon at all? Why does a shape with two sides exist on a sphere but not on a page? And how did a figure of pure geometry end up rendering the faces on a computer screen and tiling the cells of a beehive?

• 2. What Counts As Closed

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  A simple polygon never intersects itself. The only intersections allowed are the shared endpoints of consecutive segments in the chain. That restraint matters, because once a polygon obeys it, it becomes the boundary of a region of the plane, a solid polygon whose interior is its body, also called a polygonal region or polygonal area. Let the chain cross over itself instead, and the result is a star polygon or some other self-intersecting figure. The word complex is sometimes used in contrast to simple, but that usage risks confusion with the genuine complex polygon of the complex Hilbert plane. A polygon need not even stay in two dimensions. Some sources count a closed polygonal chain in Euclidean space as a skew polygon, even when the chain does not lie in a single plane. Across all these cases, a polygon remains a 2-dimensional example of the more general polytope, which can be defined in any number of dimensions.

• 3. Sorting The Family

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  Convexity is the first dividing line. A polygon is convex when any line drawn through it, not tangent to an edge or corner, meets its boundary exactly twice, and every interior angle stays below 180 degrees. A non-convex polygon allows a line that crosses its boundary more than twice. A concave polygon is non-convex but still simple, with at least one interior angle greater than 180 degrees. A star-shaped polygon hides nothing: its whole interior is visible from at least one point without crossing an edge, and every convex polygon qualifies. Symmetry adds another layer of names. Equiangular polygons have equal corner angles, equilateral ones have equal edges, and a regular polygon is both at once. Cyclic polygons place every corner on a single circle, the circumcircle, while tangential ones touch an inscribed circle along every side. Isogonal polygons are vertex-transitive and isotoxal ones edge-transitive, and a polygon is regular if and only if it is both. A non-convex regular polygon earns a name of its own: the regular star polygon.

• 4. The Arithmetic Of Corners

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  The sum of the interior angles of a simple n-gon is (n minus 2) times 180 degrees, because any such shape can be cut into n minus 2 triangles, each contributing 180 degrees. The exterior angle is the supplementary partner of the interior angle, the amount you turn at each corner as you trace the outline. Walk all the way around a convex n-gon and you make one full turn, so those exterior angles sum to 360 degrees. The picture grows stranger with self-intersecting shapes. Trace any n-gon and the total rotation can be any integer multiple of 360 degrees, giving 720 degrees for a pentagram and 0 degrees for an angular eight, or antiparallelogram. That multiple is the density or turning number. The interior angles of regular star polygons were first studied by Poinsot, in the same paper where he described the four regular star polyhedra.

• 5. Measuring The Inside

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  The shoelace formula, also called the surveyor's formula, computes the signed area of a simple polygon from its ordered vertices. The sign depends on whether those vertices run counterclockwise or clockwise, but the area is correct in absolute value either way. Lopshits described another route in 1963, finding the area from the side lengths and the exterior angles alone. When a polygon's vertices land on an equally spaced grid, Pick's theorem offers something simpler still: the interior grid points plus half the boundary points, minus 1. Side lengths usually fail to fix an area, but a simple cyclic polygon is the exception, and of all n-gons with given side lengths the cyclic one is largest. The Bolyai-Gerwien theorem promises that any two simple polygons of equal area can be cut into pieces that reassemble from one into the other. Self-intersecting figures resist a single answer. Treat regions by density and the central pentagon of a pentagram counts twice; treat a cross-quadrilateral the same way and its opposite-signed densities can cancel to a total area of zero.

• 6. Naming The Many Angled

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  The word descends from the Greek for many-angled, polús meaning many and gōnía meaning corner or angle, with a suggestion that gónu, knee, may lie behind the gon. Greek-derived prefixes build the standard names, from pentagon to dodecagon, though triangle, quadrilateral and nonagon break the pattern, and nonagon itself mixes Latin novem with Greek while enneagon stays pure. Beyond the 10-sided decagon and 12-sided dodecagon, mathematicians usually switch to numerical notation like 17-gon and 257-gon. Geometry decides some names through construction. The heptagon is the simplest polygon whose regular form cannot be drawn with compass and straightedge, though a neusis construction succeeds. The 11-sided hendecagon defeats even compass, straightedge and angle trisector together, yet neusis still works. The 23-sided icositrigon is the simplest whose regular form neusis cannot build, and the 25-sided icosipentagon sits at the edge of the unknown, where it is not known whether neusis can construct it or not. Far out at the top of the family, the chiliagon of 1000 sides served philosophers including René Descartes, Immanuel Kant and David Hume, while the million-sided megagon illustrates a concept too large to picture, converging visibly toward a circle.

• 7. Stars, Spheres, And Beyond

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  A spherical polygon traces arcs of great circles across the surface of a sphere, and it permits the digon, a two-sided, two-cornered shape impossible on a flat plane. Such polygons matter in cartography and in Wythoff's construction of the uniform polyhedra. The skew polygon zigzags through three or more dimensions instead of lying flat, and the Petrie polygons of the regular polytopes are its best-known examples. Stretch a polygon to infinity and it becomes an apeirogon, an endless sequence of sides and angles that never closes because it extends forever in both directions. A polygon with holes carries one outer boundary and one or more interior boundaries within it. The abstract polygon strips the shape down to an algebraic partially ordered set of sides, vertices and their connections, with every geometric polygon a realization of one. Raise the dimension once more and the flat polygonal faces bound a polyhedron, the three-dimensional cousin, with polytopes waiting in four dimensions and higher. The first systematic study of non-convex polygons fell to Thomas Bradwardine in the 14th century, and in 1952 Geoffrey Colin Shephard carried polygons into the complex plane.

• 8. From Basalt To The Screen

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  Cooling lava can pack itself into tight columns of basalt whose tops form regular hexagons, visible at the Giant's Causeway in Northern Ireland and at the Devil's Postpile in California. The same six-sided cell builds the wax honeycomb of bees, where the sides and base of each cell are polygons in their own right. Crystals show polygons too, as the flat facets whose angles depend on the mineral that formed them. Inside a computer, the polygon turns into a primitive for modelling and rendering, stored in a database of vertex coordinates, connectivity, color, shading, texture and materials. Surfaces become a tessellation called a polygon mesh, and a square mesh with n plus 1 points per side holds n squared squares, or twice as many triangles. The imaging system calls up the polygons a scene needs, renders them in correct perspective, and places these flat shapes into a three-dimensional orientation for the display. One small question recurs throughout this work and throughout computational geometry: whether a given point lies inside a simple polygon, the test known as point in polygon.