§8.1 Polygons

The student will learn:

the definition of polygons,

1

The terms associated with polygons, and how to tessellate a surface.

Polygon, Convexity

2

Definitions

A polygon is a closed plane figure formed by three or more line segments.

Definitions

A polygon is a closed plane figure formed by three or more line segments.

A convex polygon is a polygon in which all of the interior angles are less than 180.

A concave polygon has at least one interior angles that is greater than 180.

More Terms

3

Definitions

An side is one of the segments forming the polygon.

An interior angle is formed by the intersection of two adjacent sides.

A exterior angle is formed by one of the sides and an adjacent side extended.

More Terms

4

Definitions

An equilateral polygon is one in which all sides are equal.

An equiangular polygon is one in which all interior angles are equal.

A regular polygon is both equilateral and equiangular.

An diagonal is a segment joining two non-adjacent vertices.

And Even More Terms

5

Polygons are named according to their number of sides.

# Sides Name

5 pentagon

6 hexagon

7 heptagon

8 octagon

9 nonagon

10 decagon

12 dodecagon

TheoremThe sum of the interior angles of a polygon of n sides is 180 (n – 2).

Proof left for homework.

6

TheoremIn a regular polygon of n sides each interior angle has a measure of:

7

Proof left for homework.

180(n 2)

n

TheoremThe exterior angle of a regular polygon of n sides has a measure of:

8

360

n

Proof left for homework.

TheoremThe sum of the exterior angles of a polygon is 360.

9

Proof left for homework.

Tessellations with Regular Polygons

Tessellation

Definition

A tiling or tessellation of the plane is a collection of regions T 1, T 2, . . . , T n, called tiles such that

1. no two tiles have any interior points in common, and

2. the collection of tiles completely covers the plane.

11

Terms

12

Definitions

A tiling that uses only one shape is called a monohedral tessellation.

A tiling that uses congruent regular polyhedron is called a regular tessellation.

You should be able to prove (One of my favorite final questions.) that there are only three regular tessellations.

Semiregular Tessellations

13

A semiregular tessellation is a tessellation that is made up of regular polyhedron only and have the same combination of polyhedron at each vertex.

There are eight semiregular tessellations.

There is a way to classify tessellations by choosing a vertex and listing in sequence the number of sides of each regular polygon surrounding that vertex.

Examples

14

This is a 4.8.8 tessellation.

This is a 3.6.3.6 tessellation.

The three regular and eight semiregular tessellations are called Archimedean tilings.

Uniform Tessellations

15

The Archimedean tilings are also called 1-uniform tilings because all the vertices in a tiling are identical.

A 2-uniform tiling has two different types of vertices and a 3-uniform has three.

Examples

16

2-uniform tiling 3,6,3,6 or 3,4,4,6

3-uniform tiling 4,4,4,4 or 3,3,4,3,4 or 3,3,3,4,4

Examples

17

2-uniform tiling 3,6,3,6 or 3,4,4,6

3-uniform tiling 4,4,4,4 or 3,3,4,3,4 or 3,3,3,4,4

Note

There are 20 different 2-uniform tessellations.

18

There are 61 different 3-uniform tessellations.

The number of different 4-uniform tessellations is still an unsolved problem.

Tessellations with Nonregular

Polygons

TrianglesAny triangle may be used to form a monohedral tessellation.

20

QuadrilateralsAny quadrilateral may be used to create a monohedral tessellation. Bricks, ceiling tiles and all others.

21

PentagonsA regular pentagon does not tessellate, but there are some nonregular that do.

22

Cairo tilingSome others

Other TessellationsThe literature on tessellations is very extensive. Indeed geometric translations, reflections and rotations may be used along with other geometric techniques.

It is a fun and rewarding topic.

23

Tessellation Rocks Tasmania, Under,

Down Under

Assignment: §8.1