Polygon and their Classification:
1. Polygon - Definition
2. Classification of Polygons
3. Properties of Polygons
4. Naming polygons
5. Generalizations of polygons
6. Tessellation

1. Polygon – Definition:
The word "polygon" comes from Late Latin polygōnum (a noun), from Greek polygōnon / polugōnon, meaning "many-angled".

2. Classification of Polygons:
• 2a. Polygons Based on Number of sides
• 2b. Polygon Based on Convexity and types of non-convexity
• 2c. Polygon Based on Nature of Symmetry
• 2d. Miscellaneous

Shown below are graphical representations of the different kinds of Polygons that can be classified based on a  set (or combinations thereof) of criteria. These can be classified as: Simple; Convex and Non-convex; Cyclic; Equilateral and Equiangular; Regular etc. The criteria for classifications are highlighted below.



2a. Polygons Based on Number of sides:
Polygons are primarily classified by the number of sides. 
For Eg. Trigon, Tetragon, Pentagon, Hexagon, Octogon etc. There is an order in naming polygons and this is outlined in the section Naming Polygons.

2b. Polygon Based on Convexity and types of non-convexity:
Polygons may be characterised by their convexity or type of non-convexity:

(1) Convex: Any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. Equivalently, all its interior angles are less than 180°.

(2) Non-convex: A line may be found which meets its boundary more than twice. In other words, it contains at least one interior angle with a measure larger than 180°.

(3) Simple: The boundary of the polygon does not cross itself. All convex polygons are simple.

(4) Concave: Non-convex and simple.

(5) Star Shaped: The whole interior is visible from a single point, without crossing any edge. The polygon must be simple, and may be convex or concave.

(6) Self-intersecting: The boundary of the polygon crosses itself.

(7) Star Polygon: A polygon which self-intersects in a regular way.

2c. Polygon Based on Nature of Symmetry:

(1) Equiangular: All its corner angles are equal.

(2) Cyclic: All corners lie on a single circle.

(3) Isogonal or vertex-transitive: All corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.

(4) Equilateral: All edges are of the same length. (A polygon with 5 or more sides can be equilateral without being convex)

(5) Isotoxal or edge-transitive: All sides lie within the same symmetry orbit. The polygon is also equilateral.

(6) Regular: A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.

2d. Miscellaneous: 

(1) Rectilinear: A polygon whose sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees.

(2) Monotone: With respect to a given line L, if every line orthogonal to L intersects the polygon not more than twice.

3. Properties of Polygons:

These are based on Euclidean geometry assumptions.

 • 3a. Angles

 • 3b. Self-intersecting polygons

 • 3c. Degrees of freedom

 • 3d. Product of diagonals of a regular polygon

3a. Angles:

Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides. Each corner has several angles. The two most important ones are:

(1) Interior Angle:
The sum of the interior angles of a simple n-gon is (n − 2) radians  or  (n − 2)180 degrees. This is because any simple n-gon can be considered to be made up of (n − 2) triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is radians or  (180 - ) degrees. 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.

(2) Exterior Angle: 
Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way round the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. 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°, e.g. 720° for a pentagram and 0° for an angular "eight", where d is the density or starriness of the polygon.

The exterior angle is the supplementary angle to the interior angle. From this the sum of the interior angles can be easily confirmed, even if some interior angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number  of the orientation of the sides, where at every vertex the contribution is between −½ and ½ winding.)

3b. Self-intersecting polygons:

The area of  a self intersecting polygon can be defined in two different ways, each of which gives a different answer:

- Using the above methods for simple polygons, we discover that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region. For example the central convex pentagon in the centre of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure.

- Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon, or to the area of a simple polygon having the same outline as the self-intersecting one (or, in the case of the cross-quadrilateral, the two simple triangles).

3c. Degrees of freedom:

An n-gon has 2n degree of freedom, including 2 for position, 1 for rotational orientation, and 1 for over-all size, so 2n − 4 for shape. In the case of a line of symmetry the latter reduces to n − 2.

Let k ≥ 2. For an nk-gon with k-fold rotational symmetry (Ck), there are 2n − 2 degrees of freedom for the shape. With additional mirror-image symmetry (Dk) there are n − 1 degrees of freedom.

3d. Product of diagonals of a regular polygon:

For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n.

4. Naming polygons:

Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon.  The triangle and quadrilateral or quadrangle, and nonagon are exceptions. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usuallyn-gon. This is useful if the number of sides is used in a formula.

Some special polygons also have their own names; for example the regular star pentagon is also known as the pentagram.

Below are listed names of polygons:

4a. Constructing higher names:

To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows:

A complete covering of a plane using a limited number of different shapes. Usually the shapes are polygons (as in the Dirichlet tesselation). The plane can be tessellated with rectangles, or hexagons, or triangles (for example, using Delaunay triangles).

In a regular tessellation all the shapes are regular polygons (i.e. with all sides equal and all angles equal) of the same shape and size, and there are only three possible regular tessellations, using squares, equilateral triangles, or regular hexagons.

Other semi-regular tessellations use two or more regular polygonal shapes, for example, squares and octagons. Many tessellations are periodic, i.e. the pattern repeats at regular intervals. A non-periodic tessellation, using two basic shapes, was invented by Sir Roger Penrose and is usually referred to as Penrose tiling.

Websites:
Impossible world: Articles: The principles of artistic illusions
Tessellation -- from Wolfram MathWorld