A Mathematical View of Our World

1
A Mathematical View of Our World

• 1st ed.
• Parks, Musser, Trimpe, Maurer, and Maurer

2
Chapter 2

• Shapes in Our Lives

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Section 2.1Tilings

• Goals
• Study polygons
• Vertex angles
• Regular tilings
• Semiregular tilings
• Miscellaneous tilings
• Study the Pythagorean theorem

4
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.

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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.

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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.

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Question

• Choose the figure below that is NOT a polygon.
• a. c.
• b. d. all are polygons

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Polygons, contd

• 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.

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Polygons, contd

• A polygon with n sides and n vertices is called
  an n-gon.
• For small values of n, more familiar names are
  used.

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Polygonal Regions

• A polygonal region is a polygon together with the
  portion of the plan enclosed by the polygon.

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Polygonal Regions, contd

• A tiling is a special collection of polygonal
  regions.
• An example of a tiling, made up of rectangles, is
  shown below.

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Polygonal Regions, contd

• Polygonal regions form a tiling if
• The entire plane is covered without gaps.
• No two polygonal regions overlap.

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Polygonal Regions, contd

• Examples of tilings with polygonal regions are
  shown below.

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Vertex Angles

• A tiling of triangles illustrates the fact that
  the sum of the measures of the angles in a
  triangle is 180.

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Vertex Angles, contd

• 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.

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Example 1

• The vertex angles in the pentagon are called ? V,
  ?W, ?X, ?Y, and ?Z.
• Two diagonals shown are WZ and WY.

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Vertex Angles, contd

• Any polygon can be divided, using diagonals, into
  triangles.
• A polygon with n sides can be divided into n 2
  triangles.

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Vertex Angles, contd

• The sum of the measures of the vertex angles in a
  polygon with n sides is equal to

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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

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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.

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Regular Polygons, contd
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Regular Polygons, contd

• A regular n-gon has n angles.
• All vertex angles have the same measure.
• The measure of each vertex angle must be

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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

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Vertex Angles, contd
25
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.

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Regular Tilings, contd

• Examples of edge-to-edge regular tilings.

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Regular Tilings, contd

• Example of a regular tiling that is not
  edge-to-edge.

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Regular Tilings, contd

• 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.

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Regular Tilings, contd

• 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.

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Example 4

• Equilateral Triangles (Regular 3-gons)
• In a tiling of equilateral triangles, there are
  6(60) 360 at each vertex point.

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Example 5

• Squares
• (Regular 4-gons)
• In a tiling of squares, there are 4(90) 360
  at each vertex point.

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Question

• Will a regular pentagon tile the plane?
• a. yes
• b. no

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Example 6

• Regular hexagons
• (Regular 6-gons)
• In a tiling of regular hexagons, there are
  3(120) 360 at each vertex point.

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Regular Tilings, contd

• Do any regular polygons, besides n 3, 4, and 6,
  tile the plane?
• Note Every regular tiling with n gt 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

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Regular Tilings, contd

• 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 gt 6 can form a
  regular tiling.
• Conclusion The only regular tilings are those
  for n 3, n 4, and n 6.

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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.

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Vertex Figures, contd

• Vertex figures for the three regular tilings are
  shown below.

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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.

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Example 7

• Verify that the tiling shown is a semiregular
  tiling.

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Example 7, contd

• Solution
• The tiling is made of 3 regular polygons.
• Every vertex figure is the same shape and size.

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Example 8

• Verify that the tiling shown is not a semiregular
  tiling.

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Example 8, contd

• Solution
• The tiling is made of 3 regular polygons.
• Every vertex figure is not the same shape and
  size.

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Semiregular Tilings
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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.

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Miscellaneous Tilings, contd

• Any triangle will tile the plane.
• An example is given below

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Miscellaneous Tilings, contd

• Any quadrilateral (4-gon) will tile the plane.
• An example is given below

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Miscellaneous Tilings, contd

• Some irregular pentagons (5-gons) will tile the
  plane.
• An example is given below

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Miscellaneous Tilings, contd

• Some irregular hexagons (6-gons) will tile the
  plane.
• An example is given below

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Miscellaneous Tilings, contd

• 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.

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Miscellaneous Tilings, contd
51
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.

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Example 9

• Find the length x in the figure.
• Solution Use the theorem.

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Pythagorean Theorem Converse

• If
• then the triangle is a right triangle.

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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

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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.

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Section 2.2Symmetry, Rigid Motions, and Escher
Patterns

• Goals
• Study symmetries
• One-dimensional patterns
• Two-dimensional patterns
• Study rigid motions
• Study Escher patterns

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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

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Strip Patterns

• An example of a strip pattern, also called a
  one-dimensional pattern, is shown below.

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Strip Patterns, contd

• 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.

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Strip Patterns, contd

• This strip pattern has horizontal reflection
  symmetry because the pattern looks the same when
  it is reflected across a horizontal line.

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Strip Patterns, contd

• 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.

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Strip Patterns, contd

• 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.

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Example 1

• Describe the symmetries of the pattern.
• Solution This pattern has translation symmetry
  only.

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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

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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.

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Two-Dimensional Patterns, contd

• The pattern also has
• horizontal and vertical translation symmetries.
• 180 rotation symmetry.

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Two-Dimensional Patterns, contd

• This pattern has
• 120 rotation symmetry.
• 240 rotation symmetry.

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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.

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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.

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Example 2

• Find the image of the triangle under reflection
  about the line l.

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Example 2, contd

• 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.

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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.

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Vectors, contd

• 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.

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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.

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Example 3

• Find the image of the triangle under a
  translation determined by the vector v.

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Example 3, contd

• Solution
• Find the image of each vertex point by drawing
  the three vectors.
• Connect the image points to form the new triangle.

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Rotation

• A rotation involves turning a figure around a
  point O, clockwise or counterclockwise, through
  an angle less than 360.

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Rotation, contd

• 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.

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Rotation, contd

• A rotation of a point X about the center O
  determined by a directed angle ?AOB is
  illustrated in the figure below.

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Example 4

• Find the image of the triangle under the given
  rotation.

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Example 4, contd

• 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.

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Example 4, contd

• Solution contd
• Repeat this process for each vertex.
• Connect the three image points to form the new
  triangle.

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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.

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Example 5

• A strip pattern of footprints can be created
  using a glide reflection.

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Crystallographic Classification

• The rigid motions can be used to classify strip
  patterns.

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Classification, contd

• There are only seven basic one-dimensional
  repeated patterns.

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Example 6

• Use the crystallographic system to describe the
  strip pattern.
• Solution The classification is pmm2.

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Example 7

• Use the crystallographic system to describe the
  strip pattern.
• Solution The classification is p111.

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Question

• Use the crystallographic classification system
  to describe the pattern.
• a. p112
• b. pmm2
• c. p1m1
• d. p111

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Escher Patterns

• Maurits Escher was an artist who used rigid
  motions in his work.
• You can view some examples of Eschers work in
  your textbook.

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Escher Patterns, contd

• 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.

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Escher Patterns, contd

• The figure has been decorated and repeated.
• Notice that the pattern has vertical and
  horizontal translation symmetry and vertical
  reflection symmetry.

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Section 2.3Fibonacci Numbers and the Golden Mean

• Goals
• Study the Fibonacci Sequence
• Recursive sequences
• Fibonacci number occurrences in nature
• Geometric recursion
• The golden ratio

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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.

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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.

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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

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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.

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Fibonacci Sequence, contd

• 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.

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Fibonacci Sequence, contd

• 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,

100
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.

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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 c. A4 67
• b. A4 26 d. A4 30

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Fibonacci Sequence, contd

• For the Fibonacci sequence, the starting values
  are f1 1 and f2 1.
• The recursion rule for the Fibonacci sequence is
  
• Example Find the third number in the sequence
  using the formula.
• Let n 3.

103
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 months growth, and if every new
  branch begins to sprout its own first new branch
  after two months growth, how many branches does
  the tree have at the end of the year?

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Example 1, contd

• Solution The number of branches each month in
  the first year is given in the table and drawn in
  the figure below.

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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.

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Fibonacci Numbers In Nature, contd

• The number of petals on a flower are often
  Fibonacci numbers.

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Fibonacci Numbers In Nature, contd

• 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.

108
Fibonacci Numbers In Nature, contd

• Example The branch at right has a phyllotactic
  ratio of 3/8.
• Both 3 and 8 are Fibonacci numbers.

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Fibonacci Numbers In Nature, contd

• 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.

110
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 step-by-step by
  repeating a rule is called geometric recursion.

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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.
• Draw the resulting rectangles.
• What are the dimensions of the rectangles?

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Example 2, contd

• Solution
• The first seven rectangles in the sequence are
  shown below.

113
Example 2, contd

• Solution contd

• Notice that the dimensions of each rectangle are
  consecutive Fibonacci numbers.

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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.

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The Golden Ratio, contd
116
The Golden Ratio, contd

• 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.

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The Golden Ratio, contd

• 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 f, which is pronounced fe or fi.

118
The Golden Ratio, contd

• The golden ratio has an exact value of
• The golden ratio has been used in mathematics,
  art, and architecture for more than 2000 years.

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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.

120
Golden Rectangles, contd

• The rectangle enclosing the diagram of the
  Parthenon is an example of a golden rectangle.

121
Creating a Golden Rectangle

• Start with a square, WXYZ, that measures one unit
  on each side.
• Label the midpoint of side WX as point M.

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Creating a Golden Rectangle, contd

1. Draw an arc centered at M with radius MY.
2. Label the point P as shown.

123
Creating a Golden Rectangle, contd

1. Draw a line perpendicular to WP.
2. Extend ZY to meet this line, labeling point Q as
   shown. The completed rectangle is shown.

124
2.3 Initial Problem Solution

• How can you find the exact decimal equivalent of
  this number?

125
Initial Problem Solution, contd

• We can find the value of the continued fraction
  by using a recursion rule that generates a
  sequence of fractions.
• The first term is
• The recursion rule is

126
Initial Problem Solution, contd

• We find
• The first term is
• The second term is

127
Initial Problem Solution, contd

• The third term is
• The fourth term is

128
Initial Problem Solution, contd

• 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 f.