Download Polygons

§8.1 Polygons
The student will learn:
the definition of polygons,
The terms associated
with polygons, and
how to tessellate a
surface.
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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.
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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.
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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.
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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
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Theorem
The sum of the interior angles of a polygon of n
sides is 180 (n – 2).
Proof left for homework.
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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.
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Theorem
The sum of the exterior angles of a polygon is
360.
Proof left for homework.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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.
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Tessellations with
Nonregular
Polygons
Triangles
Any triangle may be used to form a monohedral
tessellation.
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Quadrilaterals
Any quadrilateral may be used to create a
monohedral tessellation. Bricks, ceiling tiles and
all others.
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Pentagons
A regular pentagon does not tessellate, but there
are some nonregular that do.
Cairo tiling
Some others
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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.
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Tessellation Rocks
Tasmania, Under,
Down Under
Assignment: §8.1