3Section 2.1 Tilings Goals Study polygons Study the Pythagorean theorem Vertex anglesRegular tilingsSemiregular tilingsMiscellaneous tilingsStudy the Pythagorean theorem
42.1 Initial ProblemA 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.
5TilingsGeometric patterns of tiles have been used for thousands of years all around the world.Tilings, also called tessellations, usually involve geometric shapes called polygons.
6PolygonsA 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.
7Question: Choose the figure below that is NOT a polygon. a. c. b. d. all are polygons
8Polygons, cont’dThe line segments forming a polygon are called its sides.The endpoints of the sides are called its vertices.The singular of vertices is vertex.
9Polygons, cont’dA polygon with n sides and n vertices is called an n-gon.For small values of n, more familiar names are used.
10Polygonal RegionsA polygonal region is a polygon together with the portion of the plan enclosed by the polygon.
11Polygonal Regions, cont’d A tiling is a special collection of polygonal regions.An example of a tiling, made up of rectangles, is shown below.
12Polygonal Regions, cont’d Polygonal regions form a tiling if:The entire plane is covered without gaps.No two polygonal regions overlap.
13Polygonal Regions, cont’d Examples of tilings with polygonal regions are shown below.
14Vertex AnglesA tiling of triangles illustrates the fact that the sum of the measures of the angles in a triangle is 180°.
15Vertex Angles, cont’dThe 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.
16Example 1The vertex angles in the pentagon are called V, W, X, Y, and Z.Two diagonals shown are WZ and WY.
17Vertex Angles, cont’dAny polygon can be divided, using diagonals, into triangles.A polygon with n sides can be divided into n – 2 triangles.
18Vertex Angles, cont’dThe sum of the measures of the vertex angles in a polygon with n sides is equal to:
19Example 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:
20Regular 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.
25Regular TilingsA 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.
26Regular Tilings, cont’d Examples of edge-to-edge regular tilings.
27Regular Tilings, cont’d Example of a regular tiling that is not edge-to-edge.
28Regular 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.
29Regular 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.
30Example 4 Equilateral Triangles (Regular 3-gons) In a tiling of equilateral triangles, there are 6(60°) = 360° at each vertex point.
31Example 5 Squares (Regular 4-gons) In a tiling of squares, there are 4(90°) = 360° at each vertex point.
32Question:Will a regular pentagon tile the plane?a. yesb. no
33Example 6 Regular hexagons (Regular 6-gons) In a tiling of regular hexagons, there are 3(120°) = 360° at each vertex point.
34Regular 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 pointVertex angles measuring more than 120°Angle measures at each vertex point that add to 360°
35Regular 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.
36Vertex FiguresA 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.
37Vertex Figures, cont’dVertex figures for the three regular tilings are shown below.
38Semiregular 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.
39Example 7Verify that the tiling shown is a semiregular tiling.
40Example 7, cont’d Solution: The tiling is made of 3 regular polygons. Every vertex figure is the same shape and size.
41Example 8Verify that the tiling shown is not a semiregular tiling.
42Example 8, cont’d Solution: The tiling is made of 3 regular polygons. Every vertex figure is not the same shape and size.
44Miscellaneous 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.
45Miscellaneous Tilings, cont’d Any triangle will tile the plane.An example is given below:
46Miscellaneous Tilings, cont’d Any quadrilateral (4-gon) will tile the plane.An example is given below:
47Miscellaneous Tilings, cont’d Some irregular pentagons (5-gons) will tile the plane.An example is given below:
48Miscellaneous Tilings, cont’d Some irregular hexagons (6-gons) will tile the plane.An example is given below:
49Miscellaneous 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.
51Pythagorean TheoremIn 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.
52Example 9Find the length x in the figure.Solution: Use the theorem.
53Pythagorean Theorem Converse Ifthen the triangle is a right triangle.
54Example 10Show 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:
552.1 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.
57SymmetryWe 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 symmetryRotation symmetryTranslation symmetry
58Strip PatternsAn example of a strip pattern, also called a one-dimensional pattern, is shown below.
59Strip Patterns, cont’dThis 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.
60Strip Patterns, cont’dThis strip pattern has horizontal reflection symmetry because the pattern looks the same when it is reflected across a horizontal line.
61Strip Patterns, cont’dThis 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°.
62Strip Patterns, cont’dThis 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.
63Example 1 Describe the symmetries of the pattern. Solution: This pattern has translation symmetry only.
64Question:Describe the symmetries of the strip pattern, assuming it continues to the left and right indefinitelya. horizontal reflection, vertical reflection, translationb. vertical reflection, translationc. translationd. vertical reflection
65Two-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.
66Two-Dimensional Patterns, cont’d The pattern also hashorizontal and vertical translation symmetries.180° rotation symmetry.
67Two-Dimensional Patterns, cont’d This pattern has120° rotation symmetry.240° rotation symmetry.
68Rigid MotionsAny 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.
69ReflectionA 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’.
70Example 2Find the image of the triangle under reflection about the line l.
71Example 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.
72Vectors 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:ParallelHave the same lengthPoint in the same direction.
73Vectors, cont’dA 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.
74Translation 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.
75Example 3Find the image of the triangle under a translation determined by the vector v.
76Example 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.
77RotationA rotation involves turning a figure around a point O, clockwise or counterclockwise, through an angle less than 360°.
78Rotation, 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.
79Rotation, cont’dA rotation of a point X about the center O determined by a directed angle AOB is illustrated in the figure below.
80Example 4Find the image of the triangle under the given rotation.
81Example 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.
82Example 4, cont’d Solution cont’d: Repeat this process for each vertex.Connect the three image points to form the new triangle.
83Glide ReflectionA 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.
84Example 5A strip pattern of footprints can be created using a glide reflection.
85Crystallographic Classification The rigid motions can be used to classify strip patterns.
86Classification, cont’d There are only seven basic one-dimensional repeated patterns.
87Example 6Use the crystallographic system to describe the strip pattern.Solution: The classification is pmm2.
88Example 7Use the crystallographic system to describe the strip pattern.Solution: The classification is p111.
89Question:Use the crystallographic classification system to describe the pattern.a. p112b. pmm2c. p1m1d. p111
90Escher PatternsMaurits Escher was an artist who used rigid motions in his work.You can view some examples of Escher’s work in your textbook.
91Escher 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.
92Escher Patterns, cont’d The figure has been decorated and repeated.Notice that the pattern has vertical and horizontal translation symmetry and vertical reflection symmetry.
93Section 2.3 Fibonacci Numbers and the Golden Mean GoalsStudy the Fibonacci SequenceRecursive sequencesFibonacci number occurrences in natureGeometric recursionThe golden ratio
942.3 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.
95Sequences 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.
97Fibonacci SequenceThe 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.
98Fibonacci 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.
99Fibonacci 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, …
100RecursionRecursion, 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.
101Question: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 = 67b. A4 = 26 d. A4 = 30
102Fibonacci 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.
103Example 1Suppose 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?
104Example 1, cont’dSolution: The number of branches each month in the first year is given in the table and drawn in the figure below.
105Fibonacci 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.
106Fibonacci Numbers In Nature, cont’d The number of petals on a flower are often Fibonacci numbers.
107Fibonacci 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.
108Fibonacci Numbers In Nature, cont’d Example: The branch at right has a phyllotactic ratio of 3/8.Both 3 and 8 are Fibonacci numbers.
109Fibonacci 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.
110Geometric RecursionIn 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.
111Example 2Beginning 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?
112Example 2, cont’d Solution: The first seven rectangles in the sequence are shown below.
113Example 2, cont’d Solution cont’d: Notice that the dimensions of each rectangle are consecutive Fibonacci numbers.
114The Golden RatioConsider the ratios of pairs of consecutive Fibonacci numbers.Some of the ratios are calculated in the table shown on the following slide.
116The 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
117The 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”.
118The Golden Ratio, cont’d The golden ratio has an exact value ofThe golden ratio has been used in mathematics, art, and architecture for more than 2000 years.
119Golden RectanglesA 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.
120Golden Rectangles, cont’d The rectangle enclosing the diagram of the Parthenon is an example of a golden rectangle.
121Creating a Golden Rectangle Start with a square, WXYZ, that measures one unit on each side.Label the midpoint of side WX as point M.
122Creating a Golden Rectangle, cont’d Draw an arc centered at M with radius MY.Label the point P as shown.
123Creating 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.
1242.3 Initial Problem Solution How can you find the exact decimal equivalent of this number?
125Initial 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 isThe recursion rule is
126Initial Problem Solution, cont’d We find:The first term isThe second term is
127Initial Problem Solution, cont’d The third term isThe fourth term is
128Initial Problem Solution, cont’d The fractions in this sequence are2, 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.