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**A Mathematical View of Our World**

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

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Chapter 2 Shapes in Our Lives

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**Section 2.1 Tilings Goals Study polygons Study the Pythagorean theorem**

Vertex angles Regular tilings Semiregular tilings Miscellaneous tilings Study the Pythagorean theorem

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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, 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.

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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.

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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, cont’d**

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, cont’d**

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

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**Polygonal Regions, cont’d**

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, 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.

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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, cont’d 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, cont’d 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, cont’d**

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**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

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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, cont’d

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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, cont’d**

Examples of edge-to-edge regular tilings.

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**Regular Tilings, cont’d**

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

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**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.

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**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.

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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, 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°

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**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.

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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, cont’d 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, cont’d 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, cont’d 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, cont’d**

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

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**Miscellaneous Tilings, cont’d**

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

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**Miscellaneous Tilings, cont’d**

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

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**Miscellaneous Tilings, cont’d**

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

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**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.

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**Miscellaneous Tilings, cont’d**

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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 2.2 Symmetry, 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, 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.

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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.

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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°.

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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.

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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, cont’d**

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

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**Two-Dimensional Patterns, cont’d**

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, 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.

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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, 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.

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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, cont’d 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, 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.

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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.

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Example 4 Find the image of the triangle under the given rotation.

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**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.

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**Example 4, cont’d Solution cont’d:**

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, cont’d**

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 Escher’s work in your textbook.

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**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.

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**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.

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**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

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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, 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.

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**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, …

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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, cont’d**

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.

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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?

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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.

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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, cont’d**

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

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**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.

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**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.

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**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.

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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, cont’d Solution:**

The first seven rectangles in the sequence are shown below.

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**Example 2, cont’d Solution cont’d:**

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, cont’d**

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**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

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**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”.

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**The Golden Ratio, cont’d**

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.

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**Golden Rectangles, cont’d**

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

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**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, cont’d**

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

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**Creating a Golden Rectangle, cont’d**

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

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**2.3 Initial Problem Solution**

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

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**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 The recursion rule is

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**Initial Problem Solution, cont’d**

We find: The first term is The second term is

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**Initial Problem Solution, cont’d**

The third term is The fourth term is

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**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.

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