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§8.1 Polygons
The student will learn:
the definition of polygons,
The terms associated
with polygons, and
how to tessellate a
surface.
1
Polygon, Convexity
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.
2
More Terms
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.
3
More Terms
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 nonadjacent vertices.
4
And Even More Terms
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
5
Theorem
The sum of the interior angles of a polygon of n
sides is 180 (n – 2).
Proof left for homework.
6
Theorem
In a regular polygon of n sides each interior
angle has a measure of:
180(n  2)
n
Proof left for homework.
7
Theorem
The exterior angle of a regular polygon of n
sides has a measure of:
360
n
Proof left for homework.
8
Theorem
The sum of the exterior angles of a polygon is
360.
Proof left for homework.
9
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
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.
12
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.
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.
13
Examples
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.
14
Uniform Tessellations
The Archimedean tilings are also called 1uniform 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.
15
Examples
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
16
Examples
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
17
Note
There are 20 different 2-uniform tessellations.
There are 61 different 3-uniform tessellations.
The number of different 4-uniform tessellations
is still an unsolved problem.
18
Tessellations with
Nonregular
Polygons
Triangles
Any triangle may be used to form a monohedral
tessellation.
20
Quadrilaterals
Any quadrilateral may be used to create a
monohedral tessellation. Bricks, ceiling tiles and
all others.
21
Pentagons
A regular pentagon does not tessellate, but there
are some nonregular that do.
Cairo tiling
Some others
22
Other Tessellations
The 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
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