Download Week 7 Notes - Arvind Borde

Arvind Borde
MATH 1: Week 7, Flat Geometry and Tilings
A polygon is a flat two dimensional shape
bounded by straight edges.
NOTES:
Examples:
But not
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NOTES:
(1) One of these polygons is different from
the other two. Which do you think?
Polygons such as the first and last on the
previous page are called convex. The middle
one is called concave or non-convex.
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NOTES:
If a straight line enters a convex polygon,
it can never re-enter after it leaves. For a
concave polygon, reentry is possible:
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A subclass of convex polygons are ones that
are regular. For such polygons all internal
angles are the same size and all edges are the
same length.
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(2) Which of these is regular?
We’re going to study the interior angles and
exterior angles of regular polygons.
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The total exterior angle is the sum of all
the exterior angles and is easy to calculate.
One way is to walk around the polygon:
By looking at the total rotation during the
walk, we see that the total exterior angle of a
convex polygon is 360◦ .
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For a regular polygon all the exterior angles
are equal. Knowing the total we can calculate
each individual exterior angle.
(3) If the polygon has 6 sides, what will each
exterior angle be?
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This leads to the formula:
NOTES:
Each exterior angle of a regular polygon of n
sides is 360◦ /n.
What is the exterior angle of a regular
polygon with
(4) 4 sides?
(5) 10 sides?
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We can now work out what each interior angle
must be. Look at one vertex in the diagram
on page 6:
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(6) What is the interior angle in terms of the
exterior angle?
What are the interior angles of a regular
polygon with
(7) 9 sides?
(8) 3 sides?
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Int. and Ext. Angles of Regular Polygons
Edges
3
4
5
6
7
8
9
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Name
Ext.
What about a polygon with
(9) 12 edges?
(10) 3, 751 edges?
(11) n edges?
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Int.
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As you notice the internal angle gets larger as
the number of sides increases. Visually this
means the polygons “open out” and get more
circular:
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NOTES:
Here’s what a 20-gon and a 40-gon look like:
Tilings
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A tiling of the plane is a set of shapes that
can completely cover it.
We will focus initially on tiling by regular
polygons, starting with tilings by single tiles:
only squares, or only equilateral triangles, for
example.
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NOTES:
The key requirement for a tiling to work is
that the tiles fit together without gaps. The
next page shows how a tiling by squares fits
together.
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NOTES:
(12) At every vertex, the interior angles fit
together to exactly total what?
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This is the key criterion for tilings by polygons
(regular or not): If a set of polygons meet
at a vertex, the interior angles at that vertex
must total 360◦ .
NOTES:
If we are looking at tilings by a single regular
polygon, then this is the same as saying that
the interior angle must be a factor of 360◦ .
Go back to the table on page 13, and check
off which interior angles obey this, and
how may times they divide into 360◦ . (For
example, 90◦ divides into 360◦ 4 times.)
This means that the only tilings with single
regular polygons that are possible are with
triangles, squares and hexagons.
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NOTES:
(13) Looking at the table, what tilings
can you come up with that use two regular
polygons?
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