Download A Mathematical View of Our World

A Mathematical View
of Our World
1st ed.
Parks, Musser, Trimpe,
Maurer, and Maurer
Chapter 2
Shapes in Our Lives
Section 2.1
Tilings
• Goals
• Study polygons
• Vertex angles
• Regular tilings
• Semiregular tilings
• Miscellaneous tilings
• Study the Pythagorean theorem
2.1 Initial Problem
• A portion of a ceramic
tile wall composed of
two differently shaped
tiles is shown. Why do
these two types of tiles
fit together without gaps
or overlaps?
• The solution will be given
at the end of the section.
Tilings
• Geometric patterns of tiles have been
used for thousands of years all around
the world.
• Tilings, also called tessellations,
usually involve geometric shapes
called polygons.
Polygons
• A polygon is a plane figure consisting
of line segments that can be traced
so that the starting and ending points
are the same and the path never
crosses itself.
Question:
Choose the figure below that is NOT
a polygon.
a.
c.
b.
d. all are polygons
Polygons, cont’d
• The line segments forming a polygon
are called its sides.
• The endpoints of the sides are called
its vertices.
• The singular of vertices is vertex.
Polygons, cont’d
• A polygon with n sides and n vertices is
called an n-gon.
• For small values of n, more familiar names are
used.
Polygonal Regions
• A polygonal region is a polygon together with
the portion of the plan enclosed by the
polygon.
Polygonal Regions, cont’d
• A tiling is a special collection of polygonal regions.
• An example of a tiling, made up of rectangles, is
shown below.
Polygonal Regions, cont’d
• Polygonal regions form a tiling if:
• The entire plane is covered without
gaps.
• No two polygonal regions overlap.
Polygonal Regions, cont’d
• Examples of tilings with polygonal
regions are shown below.
Vertex Angles
• A tiling of triangles illustrates the fact that the
sum of the measures of the angles in a
triangle is 180°.
Vertex Angles, cont’d
• The angles in a polygon are called its
vertex angles.
• The symbol  indicates an angle.
• Line segments that join nonadjacent
vertices in a polygon are called
diagonals of the polygon.
Example 1
• The vertex angles in the pentagon are called
 V, W, X, Y, and Z.
• Two diagonals shown are WZ and WY.
Vertex Angles, cont’d
• Any polygon can be divided, using
diagonals, into triangles.
• A polygon with n sides can be divided into
n – 2 triangles.
Vertex Angles, cont’d
• The sum of the measures of the
vertex angles in a polygon with n
sides is equal to:
 n  2180
Example 2
• Find the sum of measures of the
vertex angles of a hexagon.
• Solution:
• A hexagon has 6 sides, so n = 6.
• The sum of the measures of the angles
is found to be:
 n  2180
  6  2180   4180  720
Regular Polygons
• Regular polygons are polygons in
which:
• All sides have the same length.
• All vertex angles have the same
measure.
• Polygons that are not regular are
called irregular polygons.
Regular Polygons, cont’d
Regular Polygons, cont’d
• A regular n-gon has n angles.
• All vertex angles have the same
measure.
• The measure of each vertex angle must
be
 n  2 180
n
Example 3
• Find the measure of any vertex angle
in a regular hexagon.
• Solution:
• A hexagon has 6 sides, so n = 6.
• Each vertex angle in the regular
hexagon has the measure:
 n  2 180
n
6  2 180


6
4 180


6
720

 120
6
Vertex Angles, cont’d
Regular Tilings
• A regular tiling is a tiling composed of
regular polygonal regions in which all the
polygons are the same shape and size.
• Tilings can be edge-to-edge, meaning the
polygonal regions have entire sides in
common.
• Tilings can be not edge-to-edge, meaning the
polygonal regions do not have entire sides in
common.
Regular Tilings, cont’d
• Examples of edge-to-edge regular tilings.
Regular Tilings, cont’d
• Example of a regular tiling that is not edge-toedge.
Regular Tilings, cont’d
• Only regular edge-to-edge tilings are
generally called regular tilings.
• In every such tiling the vertex angles
of the tiles meet at a point.
Regular Tilings, cont’d
• What regular polygons will form tilings
of the plane?
• Whether or not a tiling is formed
depends on the measure of the vertex
angles.
• The vertex angles that meet at a point
must add up to exactly 360° so that no
gap is left and no overlap occurs.
Example 4
• Equilateral Triangles
(Regular 3-gons)
• In a tiling of
equilateral
triangles, there are
6(60°) = 360° at
each vertex point.
Example 5
• Squares
(Regular 4-gons)
• In a tiling of
squares, there
are 4(90°) = 360°
at each vertex
point.
Question:
Will a regular pentagon tile the plane?
a. yes
b. no
Example 6
• Regular hexagons
(Regular 6-gons)
• In a tiling of regular
hexagons, there
are 3(120°) = 360°
at each vertex
point.
Regular Tilings, cont’d
• Do any regular polygons, besides n = 3, 4,
and 6, tile the plane?
• Note: Every regular tiling with n > 6 must
have:
• At least three vertex angles at each point
• Vertex angles measuring more than 120°
• Angle measures at each vertex point that add
to 360°
Regular Tilings, cont’d
• In a previous question, you determined that
a regular pentagon does not tile the plane.
• Since 3(120°) = 360°, no polygon with
vertex angles larger than 120° [i.e. n > 6]
can form a regular tiling.
• Conclusion: The only regular tilings are
those for n = 3, n = 4, and n = 6.
Vertex Figures
• A vertex figure of a tiling is the
polygon formed when line segments
join consecutive midpoints of the
sides of the polygons sharing that
vertex point.
Vertex Figures, cont’d
• Vertex figures for the three regular tilings are
shown below.
Semiregular Tilings
• Semiregular tilings
• Are edge-to-edge tilings.
• Use two or more regular polygonal
regions.
• Vertex figures are the same shape and
size no matter where in the tiling they
are drawn.
Example 7
• Verify that the
tiling shown is a
semiregular tiling.
Example 7, cont’d
• Solution:
• The tiling is made of
3 regular polygons.
• Every vertex figure
is the same shape
and size.
Example 8
• Verify that the
tiling shown is
not a semiregular
tiling.
Example 8, cont’d
• Solution:
• The tiling is made of
3 regular polygons.
• Every vertex figure
is not the same
shape and size.
Semiregular Tilings
Miscellaneous Tilings
• Tilings can also be made of other
types of shapes.
• Tilings consisting of irregular polygons
that are all the same size and shape
will be considered.
Miscellaneous Tilings, cont’d
• Any triangle will tile the plane.
• An example is given below:
Miscellaneous Tilings, cont’d
• Any quadrilateral (4-gon) will tile the plane.
• An example is given below:
Miscellaneous Tilings, cont’d
• Some irregular pentagons (5-gons) will tile the
plane.
• An example is given below:
Miscellaneous Tilings, cont’d
• Some irregular hexagons (6-gons) will tile the
plane.
• An example is given below:
Miscellaneous Tilings, cont’d
• A polygonal region is convex if, for any two
points in the region, the line segment having the
two points as endpoints also lies in the region.
• A polygonal region that is not convex is called
concave.
Miscellaneous Tilings, cont’d
Pythagorean Theorem
• In a right triangle, the
sum of the areas of
the squares on the
sides of the triangle is
equal to the area of
the square on the
hypotenuse.
•
a b  c
2
2
2
Example 9
• Find the length x in
the figure.
• Solution: Use the
theorem.
•
y  1 1  2
•
x  1  y  1 2  3
2
2
2
2
2
2
Pythagorean Theorem Converse
• If a  b  c
then the
triangle is a
right triangle.
2
2
2
Example 10
• Show that any triangle with sides of length
3, 4 and 5 is a right triangle.
• Solution: The longest side must be the
hypotenuse. Let a = 3, b = 4, and c = 5.
We find: 2
2
2
3 4 5 ?
9  16  25 ?
25  25
2.1 Initial Problem Solution
• The tiling consists of
squares and regular
octagons.
• The vertex angle
measures add up to 90°
+ 2(135°) = 360°.
• This is an example of
one of the eight possible
semiregular tilings.
Section 2.2
Symmetry, Rigid Motions,
and Escher Patterns
• Goals
• Study symmetries
• One-dimensional patterns
• Two-dimensional patterns
• Study rigid motions
• Study Escher patterns
Symmetry
• We say a figure has symmetry if it can be
moved in such a way that the resulting
figure looks identical to the original figure.
• Types of symmetry that will be studied
here are:
• Reflection symmetry
• Rotation symmetry
• Translation symmetry
Strip Patterns
• An example of a strip pattern, also
called a one-dimensional pattern, is
shown below.
Strip Patterns, cont’d
• This strip pattern has vertical reflection
symmetry because the pattern looks the
same when it is reflected across a vertical
line.
• The dashed line is called a line of symmetry.
Strip Patterns, cont’d
• This strip pattern has horizontal
reflection symmetry because the
pattern looks the same when it is
reflected across a horizontal line.
Strip Patterns, cont’d
• This strip pattern has rotation symmetry because
the pattern looks the same when it is rotated 180°
about a given point.
• The point around which the pattern is turned is called the
center of rotation.
• Note that the degree of rotation must be less than 360°.
Strip Patterns, cont’d
• This strip pattern has translation symmetry
because the pattern looks the same when
it is translated a certain amount to the
right.
• The pattern is understood to extend
indefinitely to the left and right.
Example 1
• Describe the symmetries of the pattern.
• Solution: This pattern has translation
symmetry only.
Question:
Describe the symmetries of the strip
pattern, assuming it continues to the left
and right indefinitely
a. horizontal reflection, vertical reflection,
translation
b. vertical reflection, translation
c. translation
d. vertical reflection
Two-Dimensional Patterns
• Two-dimensional
patterns that fill the
plane can also have
symmetries.
• The pattern shown
here has horizontal
and vertical reflection
symmetries.
• Some lines of
symmetry have been
drawn in.
Two-Dimensional Patterns, cont’d
• The pattern also
has
• horizontal and
vertical translation
symmetries.
• 180° rotation
symmetry.
Two-Dimensional Patterns, cont’d
• This pattern has
• 120° rotation
symmetry.
• 240° rotation
symmetry.
Rigid Motions
• Any combination of translations,
reflections across lines, and/or rotations
around a point is called a rigid motion, or
an isometry.
• Rigid motions may change the location of
the figure in the plane.
• Rigid motions do not change the size or
shape of the figure.
Reflection
• A reflection with
respect to line l is
defined as follows,
with A’ being the
image of point A under
the reflection.
• If A is a point on the
line l, A = A’.
• If A is not on line l, then
l is the perpendicular
bisector of line AA’.
Example 2
• Find the image of
the triangle under
reflection about the
line l.
Example 2, cont’d
• Solution:
• Find the image of each vertex point of the triangle,
using a protractor.
• A and A’ are equal distances from l.
• Connect the image points to form the new triangle.
Vectors
• A vector is a directed line segment.
• One endpoint is the beginning point.
• The other endpoint, labeled with an arrow, is the
ending point.
• Two vectors are equivalent if they are:
• Parallel
• Have the same length
• Point in the same direction.
Vectors, cont’d
• A vector v is has a length and a direction, as
shown below.
• A translation can be defined by moving every
point of a figure the distance and direction
indicated by a vector.
Translation
• A translation is defined as follows.
• A vector v assigns to every point A an image
point A’.
• The directed line segment between A and A’
is equivalent to v.
Example 3
• Find the image of
the triangle under a
translation
determined by the
vector v.
Example 3, cont’d
• Solution:
• Find the image of each vertex point by drawing the
three vectors.
• Connect the image points to form the new triangle.
Rotation
• A rotation involves turning a figure
around a point O, clockwise or
counterclockwise, through an angle
less than 360°.
Rotation, cont’d
• The point O is called the center of rotation.
• The directed angle indicates the amount and
direction of the rotation.
• A positive angle indicates a
counterclockwise rotation.
• A negative angle indicates a clockwise
rotation.
• A point and its image are the same distance
from O.
Rotation, cont’d
• A rotation of a point X about the center O
determined by a directed angle AOB is
illustrated in the figure below.
Example 4
• Find the image
of the triangle
under the given
rotation.
Example 4, cont’d
• Solution:
• Create a 50° angle
with initial side OA.
• Mark A’ on the
terminal side,
recalling that A and
A’ are the same
distance from O.
Example 4, cont’d
• Solution cont’d:
• Repeat this process
for each vertex.
• Connect the three
image points to form
the new triangle.
Glide Reflection
• A glide reflection is the result of a
reflection followed by a translation.
• The line of reflection must not be perpendicular to the
translation vector.
• The line of reflection is usually parallel to the
translation vector.
Example 5
• A strip pattern of footprints can be
created using a glide reflection.
Crystallographic Classification
• The rigid motions can be used to classify
strip patterns.
Classification, cont’d
• There are
only seven
basic onedimensional
repeated
patterns.
Example 6
• Use the crystallographic system to
describe the strip pattern.
• Solution: The classification is pmm2.
Example 7
• Use the crystallographic system to
describe the strip pattern.
• Solution: The classification is p111.
Question:
Use the crystallographic classification
system to describe the pattern.
a. p112
b. pmm2
c. p1m1
d. p111
Escher Patterns
•
Maurits Escher was an artist who used
rigid motions in his work.
•
You can view some examples of
Escher’s work in your textbook.
Escher Patterns, cont’d
•
An example of the process used to create
Escher-type patterns is shown next.
•
Begin with a square.
•
Cut a piece from the upper left and translate it to the
right.
•
Reflect the left side to the right side.
Escher Patterns, cont’d
•
The figure has been
decorated and
repeated.
•
Notice that the
pattern has vertical
and horizontal
translation symmetry
and vertical
reflection symmetry.
Section 2.3
Fibonacci Numbers and the
Golden Mean
• Goals
• Study the Fibonacci Sequence
• Recursive sequences
• Fibonacci number occurrences in nature
• Geometric recursion
• The golden ratio
2.3 Initial Problem
• This expression is called a continued
fraction.
• How can you find the exact decimal
equivalent of this number?
• The solution will be given at the end of the section.
Sequences
• A sequence is an ordered collection of
numbers.
• A sequence can be written in the form
a1, a2, a3, …, an, …
• The symbol a1 represents the first number in the
sequence.
• The symbol an represents the nth number in the
sequence.
Question:
Given the sequence: 1, 3, 5, 7, 9,
11, 13, 15, … , find the values of the
numbers A1, A3, and A9.
a. A1 = 1, A3 = 5, A9 = 15
b. A1 = 1, A3 = 3, A9 = 17
c. A1 = 1, A3 = 5, A9 = 17
d. A1 = 1, A3 = 5, A9 = 16
Fibonacci Sequence
• The famous Fibonacci sequence is the
result of a question posed by Leonardo de
Fibonacci, a mathematician during the
Middle Ages.
• If you begin with one pair of rabbits on the first
day of the year, how many pairs of rabbits will
you have on the first day of the next year?
• It is assumed that each pair of rabbits produces a
new pair every month and each new pair begins to
produce two months after birth.
Fibonacci Sequence, cont’d
• The solution to this question is shown in the
table below.
• The sequence that appears three times in the
table, 1, 1, 2, 3, 5, 8, 13, 21, … is called the
Fibonacci sequence.
Fibonacci Sequence, cont’d
• The Fibonacci sequence is the sequence
of numbers 1, 1, 2, 3, 5, 8, 13, 21, …
• The Fibonacci sequence is found many
places in nature.
• Any number in the sequence is called a
Fibonacci number.
• The sequence is usually written
f1, f2, f3, …, fn, …
Recursion
• Recursion, in a sequence, indicates that
each number in the sequence is found
using previous numbers in the sequence.
• Some sequences, such as the Fibonacci
sequence, are generated by a recursion
rule along with starting values for the first
two, or more, numbers in the sequence.
Question:
A recursive sequence uses the rule
An =4An-1 – An-2, with starting values
of A1 = 2, A2 =7.
What is the fourth term in the
sequence?
a. A4 = 45
b. A4 = 26
c. A4 = 67
d. A4 = 30
Fibonacci Sequence, cont’d
• For the Fibonacci sequence, the starting values
are f1 = 1 and f2 = 1.
• The recursion rule for the Fibonacci sequence
is:
f n  f n1  f n2
• Example: Find the third number in the
sequence using the formula.
• Let n = 3.
f3  f31  f32  f 2  f1  1  1  2
Example 1
• Suppose a tree starts from one shoot that
grows for two months and then sprouts a
second branch. If each established
branch begins to spout a new branch
after one month’s growth, and if every
new branch begins to sprout its own first
new branch after two month’s growth,
how many branches does the tree have
at the end of the year?
Example 1, cont’d
• Solution: The number of branches each month
in the first year is given in the table and drawn
in the figure below.
Fibonacci Numbers In Nature
• The Fibonacci numbers are found many
places in the natural world, including:
• The number of flower petals.
• The branching behavior of plants.
• The growth patterns of sunflowers and
pinecones.
• It is believed that the spiral nature of plant
growth accounts for this phenomenon.
Fibonacci Numbers In Nature, cont’d
• The number of petals on a flower are
often Fibonacci numbers.
Fibonacci Numbers In Nature, cont’d
• Plants grow in a spiral pattern. The ratio of the
number of spirals to the number of branches is
called the phyllotactic ratio.
• The numbers in the phyllotactic ratio are usually
Fibonacci numbers.
Fibonacci Numbers In Nature, cont’d
• Example: The
branch at right has
a phyllotactic ratio
of 3/8.
• Both 3 and 8 are
Fibonacci numbers.
Fibonacci Numbers In Nature, cont’d
• Mature sunflowers have one set of spirals
going clockwise and another set going
counterclockwise.
• The numbers of spirals in each set are
usually a pair of adjacent Fibonacci
numbers.
• The most common number of spirals is 34
and 55.
Geometric Recursion
• In addition to being used to generate
a sequence, the recursion process
can also be used to create shapes.
• The process of building a figure stepby-step by repeating a rule is called
geometric recursion.
Example 2
• Beginning with a 1-by-1 square,
form a sequence of rectangles by
adding a square to the bottom, then
to the right, then to the bottom, then
to the right, and so on.
a) Draw the resulting rectangles.
b) What are the dimensions of the
rectangles?
Example 2, cont’d
•
Solution:
a) The first seven rectangles in the sequence
are shown below.
Example 2, cont’d
•
Solution cont’d:
b)
•
Notice that the dimensions of each rectangle
are consecutive Fibonacci numbers.
The Golden Ratio
• Consider the ratios of pairs of
consecutive Fibonacci numbers.
• Some of the ratios are calculated in
the table shown on the following
slide.
The Golden Ratio, cont’d
The Golden Ratio, cont’d
• The ratios of pairs of consecutive Fibonacci
numbers are also represented in the graph
below.
• The ratios approach the dashed line which
represents a number around 1.618.
The Golden Ratio, cont’d
• The irrational number, approximately
1.618, is called the golden ratio.
• Other names for the golden ratio include
the golden section, the golden mean, and
the divine proportion.
• The golden ratio is represented by the
Greek letter φ, which is pronounced “fe”
or “fi”.
The Golden Ratio, cont’d
• The golden ratio has an exact value of
1 5

2
• The golden ratio has been used in
mathematics, art, and architecture for
more than 2000 years.
Golden Rectangles
• A golden rectangle has a ratio of the
longer side to the shorter side that is the
golden ratio.
• Golden rectangles are used in
architecture, art, and packaging.
Golden Rectangles, cont’d
• The rectangle enclosing the diagram of the
Parthenon is an example of a golden
rectangle.
Creating a Golden Rectangle
1) Start with a
square, WXYZ,
that measures
one unit on each
side.
2) Label the
midpoint of side
WX as point M.
Creating a Golden Rectangle, cont’d
3) Draw an arc
centered at M
with radius MY.
4) Label the point
P as shown.
Creating a Golden Rectangle, cont’d
5) Draw a line
perpendicular to
WP.
6) Extend ZY to meet
this line, labeling
point Q as shown.
The completed
rectangle is shown.
2.3 Initial Problem Solution
• How can you find the exact decimal
equivalent of this number?
Initial Problem Solution, cont’d
• We can find the value of the
continued fraction by using a
recursion rule that generates a
sequence of fractions.
• The first term is
a1  1  1
1
• The recursion rule is an  1 
an 1
Initial Problem Solution, cont’d
• We find:
• The first term is
a1  1  1  2
• The second term is
1
1 3
a2  1   1  
a1
2 2
Initial Problem Solution, cont’d
• The third term is
1
1 5
a3  1   1 

3
a2
3
2
• The fourth term is
1
1 8
a4  1   1 

5
a3
5
3
Initial Problem Solution, cont’d
• The fractions in this sequence are
2, 3/2, 5/3, 8/5, …
• This is recognized to be the same as
the ratios of consecutive pairs of
Fibonacci numbers.
• The numbers in this sequence of
fractions get closer and closer to φ.