Triangles, squares, pentagons, hexagons: polygons are in honeycombs, road signs and screw nuts.

Who discovered

The simplest regular polygons were constructed with a ruler and compass by Euclid (c. 300 BC), in book IV of The Elements: triangle, square, pentagon, hexagon and the polygon of 15 sides. For more than two thousand years, no one has managed to construct another regular polygon with a prime number of sides. In 1796, at the 18 age, Carl Friedrich Gauss constructed the heptadecagon, the regular polygon with 17 sides.

Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.
Euclid, Renaissance representation traditionally associated with Euclid or Archimedes.Rafael Sanzio · Wikimedia Commons · Public domain
Carl Friedrich Gauss.
Carl Friedrich Gauss.Christian Albrecht Jensen (1792–1870) · Wikimedia Commons · Public domain

The story

Gauss’s discovery was so remarkable that, according to him, it made him choose mathematics over philology. He showed which regular polygons could be constructed with straightedge and compass; in 1837, Pierre Wantzel completed the proof that the others are impossible. It is said that Gauss wanted a heptadecagon on his tomb, but the sculptor refused, saying it would look like a circle.

Fragment of the Elements known as P.Oxy. I 29, found in Oxirrinco. The text and figure record the ancient transmission of Euclidean geometry.
Fragment of the Elements known as P.Oxy. I 29, found in Oxirrinco. The text and figure record the ancient transmission of Euclidean geometry.Text by Euclid; unidentified early copyist · Wikimedia Commons · Public domain.

The names of polygons come from the Greek: polys, “many”, and gonia, “angle”. Pentagon has five angles; hexagon, six; octagon, eight.

In everyday life

  • Honeycombs. Bees build regular hexagons. Cop of Alexandria already admired this choice in the 4th century, and in 1999 Thomas Hales proved that the hexagon divides the plane into cells of the same area with the smallest possible perimeter.
  • STOP sign. The regular red octagon is recognized even from behind or from afar.
  • Nuts and screws. The hex heads are hexagons: the wrench fits in several positions.
  • Floors and sidewalks. Hexagonal, square and triangular tiles cover the floor without leaving gaps.

Sources: Gauss (MacTutor) · Elements, proposition IV.11 · Wantzel (MacTutor).

Theory and constructions

DefiniçãoPolygon

Given a sequence of points on a plane $(A_1, A_2, \ldots, A_n)$, with $n \geq 3$, all distinct and with three consecutive non-collinear points (also considering consecutive $A_{n-1}, A_n, A_1$ and $A_n, A_1, A_2$), the meeting of segments is called polygon

$$A_1A_2 \cup A_2A_3 \cup \cdots \cup A_{n-1}A_n \cup A_nA_1.$$

The points $A_i$ are the vertices, the segments are the sides and the angles $\hat A_1 = A_n\hat{A_1}A_2$, …, $\hat A_n = A_{n-1}\hat{A_n}A_1$ are the angles of the polygon. A polygon with $n$ vertices has $n$ sides and $n$ angles. The sum of the sides is the perimeter.

DefiniçãoSimple, convex and concave

  • A polygon is simple if two non-consecutive sides never intersect. Otherwise, it is complex (and may be interlaced, like a star).
  • A simple polygon is convex if the straight line of any two consecutive vertices leaves all other $n - 2$ vertices in the same half-plane. If it is not convex, it is concave.

Interior and exterior. Given a simple polygon and a point outside it, draw a ray originating at the point that cuts the polygon without passing through vertices. If the number of cuts is odd, the point is internal; if it is par, it is external. The interior of a convex polygon is a convex region; that of a concave, concave. The polygon with its interior forms the polygonal surface.

Drag the point $P$ and rotate the ray: the number of cuts changes, but its parity always tells you whether $P$ is inside.

ResumoPolygon names

$n$ name $n$ name
3 triangle (trilateral) 9 eneagon
4 quadrilateral (quadangle) 10 decagon
5 Pentagon 11 undecagon
6 hexagon 12 dodecagon
7 heptagon 15 pentadecagon
8 octagon 20 icosagon

In general, a polygon with $n$ sides is a $n$-side.

DefiniçãoRegular polygon

A polygon that has congruent sides is equilateral; if it has congruent angles, it is equiangular. A convex polygon is regular if it is equilateral e equiangular. The regular triangle is equilateral; the regular quadrilateral is the square. A rhombus is equilateral without being regular; a rectangle is equiangular without being regular.