Polygon and their Classification:
1. Polygon – Definition
2. Classification of Polygons
3. Properties of Polygons
4. Naming polygons
5. Generalizations of polygons
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:
• 4a. Constructing higher names
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:
|1||In the Euclidean plane, degenerates to a closed curve with a single vertex point on it.|
|Digon||2||In the Euclidean plane, degenerates to a closed curve with two vertex points on it.|
|3||The simplest polygon which can exist in the Euclidean plane.|
|4||The simplest polygon which can cross itself.|
|Pentagon||5||The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle.|
|Hexagon||6||avoid "sexagon" = Latin [sex-] + Greek
|Heptagon||7||avoid "septagon" = Latin [sept-] + Greek|
|9||"Nonagon" is commonly used but mixes Latin [novem = 9] with Greek. Some modern authors prefer "enneagon".|
|Hendecagon||11||Avoid "undecagon" = Latin [un-] + Greek|
|Dodecagon||12||Avoid "duodecagon" = Latin [duo-] + Greek|
|Icosagon||20||Oak Knoll Press|
|Hectogon||100||"Hectogon" is the Greek name, "centagon" is a Latin-Greek hybrid; neither is widely attested.|
|Chiliagon||1000||The measure of each angle in a regular chiliagon is 179.64°.|
|Myriagon||10000||The internal angle of a regular myriagon is 179.964°.|
|Megagon||1,000,000||The internal angle of a regular megagon is 179.99964 degrees.|
|Apeirogon||A degenerate polygon of infinitely many sides.|
• 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:
The "kai" is not always used. Opinions differ on exactly when it should, or need not, be used.
5. Generalizations of polygons:
In a broad sense, a polygon is an unbounded (without ends) sequence or circuit of alternating segments (sides) and angles (corners). An ordinary polygon is unbounded because the sequence closes back in itself in a loop or circuit, while an infinite polygon is unbounded because it goes on forever so you can never reach any bounding end point. The modern mathematical understanding is to describe such a structural sequence in terms of an “abstract" polygon which is a partially ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.
A geometric polygon is understood to be a "realization" of the associated abstract polygon; this involves some "mapping" of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. So when we talk about "polygons" we must be careful to explain what kind we are talking about.
A digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle. Add another arc of a different great circle and you have a digon. Tile the sphere with digons and you have apolyhedron called a hosohedron. Take just one great circle instead, run it all the way round, and add just one "corner" point, and you have a monogon or henagon—although many authorities do not regard this as a proper polygon.
Other realizations of these polygons are possible on other surfaces, but in the Euclidean (flat) plane, their bodies cannot be sensibly realized and we think of them as degenerate.
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.
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Tessellation -- from Wolfram MathWorld