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A Mathematical View of Our World 1 st 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 GoalsGoals Study polygonsStudy polygons Vertex anglesVertex angles Regular tilingsRegular tilings Semiregular tilingsSemiregular tilings Miscellaneous tilingsMiscellaneous tilings Study the Pythagorean theoremStudy 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?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.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.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.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.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 line segments forming a polygon are called its sides. The endpoints of the sides are called its vertices.The endpoints of the sides are called its vertices. The singular of vertices is vertex.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.A polygon with n sides and n vertices is called an n-gon. For small values of n, more familiar names are used.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.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.A tiling is a special collection of polygonal regions. An example of a tiling, made up of rectangles, is shown below.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:Polygonal regions form a tiling if: The entire plane is covered without gaps.The entire plane is covered without gaps. No two polygonal regions overlap.No two polygonal regions overlap.

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Polygonal Regions, contd Examples of tilings with polygonal regions are shown below.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°.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 angles in a polygon are called its vertex angles. The symbol indicates an angle.The symbol indicates an angle. Line segments that join nonadjacent vertices in a polygon are called diagonals of the polygon.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.The vertex angles in the pentagon are called V, W, X, Y, and Z. Two diagonals shown are WZ and WY.Two diagonals shown are WZ and WY.

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Vertex Angles, contd Any polygon can be divided, using diagonals, into triangles.Any polygon can be divided, using diagonals, into triangles. A polygon with n sides can be divided into n – 2 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: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.Find the sum of measures of the vertex angles of a hexagon. Solution:Solution: A hexagon has 6 sides, so n = 6.A hexagon has 6 sides, so n = 6. The sum of the measures of the angles is found to be:The sum of the measures of the angles is found to be:

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Regular Polygons Regular polygons are polygons in which:Regular polygons are polygons in which: All sides have the same length.All sides have the same length. All vertex angles have the same measure.All vertex angles have the same measure. Polygons that are not regular are called irregular polygons.Polygons that are not regular are called irregular polygons.

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Regular Polygons, contd

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

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Example 3 Find the measure of any vertex angle in a regular hexagon.Find the measure of any vertex angle in a regular hexagon. Solution:Solution: A hexagon has 6 sides, so n = 6.A hexagon has 6 sides, so n = 6. Each vertex angle in the regular hexagon has the measure:Each vertex angle in the regular hexagon has the measure:

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Vertex Angles, contd

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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.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 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.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.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.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.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.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?What regular polygons will form tilings of the plane? Whether or not a tiling is formed depends on the measure of the vertex angles.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.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)Equilateral Triangles (Regular 3-gons) In a tiling of equilateral triangles, there are 6(60°) = 360° at each vertex point.In a tiling of equilateral triangles, there are 6(60°) = 360° at each vertex point.

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Example 5 SquaresSquares (Regular 4-gons) In a tiling of squares, there are 4(90°) = 360° at each vertex point.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 hexagonsRegular hexagons (Regular 6-gons) In a tiling of regular hexagons, there are 3(120°) = 360° at each vertex point.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?Do any regular polygons, besides n = 3, 4, and 6, tile the plane? Note: Every regular tiling with n > 6 must have:Note: Every regular tiling with n > 6 must have: At least three vertex angles at each pointAt least three vertex angles at each point Vertex angles measuring more than 120°Vertex angles measuring more than 120° Angle measures at each vertex point that add to 360°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.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.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.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.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.Vertex figures for the three regular tilings are shown below.

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Semiregular Tilings Semiregular tilingsSemiregular tilings Are edge-to-edge tilings.Are edge-to-edge tilings. Use two or more regular polygonal regions.Use two or more regular polygonal regions. Vertex figures are the same shape and size no matter where in the tiling they are drawn.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.Verify that the tiling shown is a semiregular tiling.

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Example 7, contd Solution:Solution: The tiling is made of 3 regular polygons.The tiling is made of 3 regular polygons. Every vertex figure is the same shape and size.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.Verify that the tiling shown is not a semiregular tiling.

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Example 8, contd Solution:Solution: The tiling is made of 3 regular polygons.The tiling is made of 3 regular polygons. Every vertex figure is not the same shape and size.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 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.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.Any triangle will tile the plane. An example is given below:An example is given below:

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

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

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Miscellaneous Tilings, contd Some irregular hexagons (6-gons) will tile the plane.Some irregular hexagons (6-gons) will tile the plane. An example is given below: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 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.A polygonal region that is not convex is called concave.

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Miscellaneous Tilings, contd

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

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Pythagorean Theorem Converse IfIf 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.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: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 tiling consists of squares and regular octagons. The vertex angle measures add up to 90° + 2(135°) = 360°.The vertex angle measures add up to 90° + 2(135°) = 360°. This is an example of one of the eight possible semiregular tilings.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 GoalsGoals Study symmetriesStudy symmetries One-dimensional patternsOne-dimensional patterns Two-dimensional patternsTwo-dimensional patterns Study rigid motionsStudy rigid motions Study Escher patternsStudy 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.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:Types of symmetry that will be studied here are: Reflection symmetryReflection symmetry Rotation symmetryRotation symmetry Translation symmetryTranslation symmetry

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Strip Patterns An example of a strip pattern, also called a one-dimensional pattern, is shown below.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.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.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.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.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.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°.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.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.The pattern is understood to extend indefinitely to the left and right.

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Example 1 Describe the symmetries of the pattern.Describe the symmetries of the pattern. Solution: This pattern has translation symmetry only.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.Two-dimensional patterns that fill the plane can also have symmetries. The pattern shown here has horizontal and vertical reflection symmetries.The pattern shown here has horizontal and vertical reflection symmetries. Some lines of symmetry have been drawn in.Some lines of symmetry have been drawn in.

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Two-Dimensional Patterns, contd The pattern also hasThe pattern also has horizontal and vertical translation symmetries.horizontal and vertical translation symmetries. 180° rotation symmetry.180° rotation symmetry.

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Two-Dimensional Patterns, contd This pattern hasThis pattern has 120° rotation symmetry.120° rotation symmetry. 240° 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.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 may change the location of the figure in the plane. Rigid motions do not change the size or shape of the figure.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.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 a point on the line l, A = A. If A is not on line l, then l is the perpendicular bisector of line AA.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.Find the image of the triangle under reflection about the line l.

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Example 2, contd Solution:Solution: Find the image of each vertex point of the triangle, using a protractor.Find the image of each vertex point of the triangle, using a protractor. A and A are equal distances from l.A and A are equal distances from l. Connect the image points to form the new triangle.Connect the image points to form the new triangle.

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Vectors A vector is a directed line segment.A vector is a directed line segment. One endpoint is the beginning point.One endpoint is the beginning point. The other endpoint, labeled with an arrow, is the ending point.The other endpoint, labeled with an arrow, is the ending point. Two vectors are equivalent if they are:Two vectors are equivalent if they are: ParallelParallel Have the same lengthHave the same length Point in the same direction.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 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.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 translation is defined as follows. A vector v assigns to every point A an image point A.A vector v assigns to every point A an image point A. The directed line segment between A and A is equivalent to v.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.Find the image of the triangle under a translation determined by the vector v.

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Example 3, contd Solution:Solution: Find the image of each vertex point by drawing the three vectors.Find the image of each vertex point by drawing the three vectors. Connect the image points to form the new triangle.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°.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 point O is called the center of rotation. The directed angle indicates the amount and direction of the rotation.The directed angle indicates the amount and direction of the rotation. A positive angle indicates a counterclockwise rotation.A positive angle indicates a counterclockwise rotation. A negative angle indicates a clockwise rotation.A negative angle indicates a clockwise rotation. A point and its image are the same distance from O.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.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.Find the image of the triangle under the given rotation.

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Example 4, contd Solution:Solution: Create a 50° angle with initial side OA.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.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:Solution contd: Repeat this process for each vertex.Repeat this process for each vertex. Connect the three image points to form the new triangle.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.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 must not be perpendicular to the translation vector. The line of reflection is usually parallel 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.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.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.There are only seven basic one- dimensional repeated patterns.

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Example 6 Use the crystallographic system to describe the strip pattern.Use the crystallographic system to describe the strip pattern. Solution: The classification is pmm2.Solution: The classification is pmm2.

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Example 7 Use the crystallographic system to describe the strip pattern.Use the crystallographic system to describe the strip pattern. Solution: The classification is p111.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.Maurits Escher was an artist who used rigid motions in his work. You can view some examples of Eschers work in your textbook.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.An example of the process used to create Escher-type patterns is shown next. Begin with a square.Begin with a square. Cut a piece from the upper left and translate it to the right.Cut a piece from the upper left and translate it to the right. Reflect the left side to the right side.Reflect the left side to the right side.

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Escher Patterns, contd The figure has been decorated and repeated.The figure has been decorated and repeated. Notice that the pattern has vertical and horizontal translation symmetry and vertical reflection symmetry.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 GoalsGoals Study the Fibonacci SequenceStudy the Fibonacci Sequence Recursive sequencesRecursive sequences Fibonacci number occurrences in natureFibonacci number occurrences in nature Geometric recursionGeometric recursion The golden ratioThe golden ratio

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2.3 Initial Problem This expression is called a continued fraction.This expression is called a continued fraction. How can you find the exact decimal equivalent of this number?How can you find the exact decimal equivalent of this number? The solution will be given at the end of the section.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 is an ordered collection of numbers. A sequence can be written in the form a 1, a 2, a 3, …, a n, …A sequence can be written in the form a 1, a 2, a 3, …, a n, … The symbol a 1 represents the first number in the sequence.The symbol a 1 represents the first number in the sequence. The symbol a n represents the nth number in the sequence.The symbol a n 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 A 1, A 3, and A 9. a. A 1 = 1, A 3 = 5, A 9 = 15 b. A 1 = 1, A 3 = 3, A 9 = 17 c. A 1 = 1, A 3 = 5, A 9 = 17 d. A 1 = 1, A 3 = 5, A 9 = 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.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?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.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 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.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 the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, … The Fibonacci sequence is found many places in nature.The Fibonacci sequence is found many places in nature. Any number in the sequence is called a Fibonacci number.Any number in the sequence is called a Fibonacci number. The sequence is usually written f 1, f 2, f 3, …, f n, …The sequence is usually written f 1, f 2, f 3, …, f n, …

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Recursion Recursion, in a sequence, indicates that each number in the sequence is found using previous numbers in the sequence.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.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 A n =4A n-1 – A n-2, with starting values of A 1 = 2, A 2 =7. What is the fourth term in the sequence? a. A 4 = 45c. A 4 = 67 b. A 4 = 26d. A 4 = 30

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Fibonacci Sequence, contd For the Fibonacci sequence, the starting values are f 1 = 1 and f 2 = 1.For the Fibonacci sequence, the starting values are f 1 = 1 and f 2 = 1. The recursion rule for the Fibonacci sequence is:The recursion rule for the Fibonacci sequence is: Example: Find the third number in the sequence using the formula.Example: Find the third number in the sequence using the formula. Let n = 3.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 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?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.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 Fibonacci numbers are found many places in the natural world, including: The number of flower petals.The number of flower petals. The branching behavior of plants.The branching behavior of plants. The growth patterns of sunflowers and pinecones.The growth patterns of sunflowers and pinecones. It is believed that the spiral nature of plant growth accounts for this phenomenon.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.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.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.The numbers in the phyllotactic ratio are usually Fibonacci numbers.

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Fibonacci Numbers In Nature, contd Example: The branch at right has a phyllotactic ratio of 3/8.Example: The branch at right has a phyllotactic ratio of 3/8. Both 3 and 8 are Fibonacci numbers.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.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 numbers of spirals in each set are usually a pair of adjacent Fibonacci numbers. The most common number of spirals is 34 and 55.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.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.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.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?

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Example 2, contd Solution:Solution: a)The first seven rectangles in the sequence are shown below.

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Example 2, contd Solution contd:Solution contd: b) b) Notice that the dimensions of each rectangle are consecutive Fibonacci numbers.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.Consider the ratios of pairs of consecutive Fibonacci numbers. Some of the ratios are calculated in the table shown on the following slide.Some of the ratios are calculated in the table shown on the following slide.

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The Golden Ratio, contd

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The ratios of pairs of consecutive Fibonacci numbers are also represented in the graph below.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 The ratios approach the dashed line which represents a number around

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The Golden Ratio, contd The irrational number, approximately 1.618, is called the golden ratio.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.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 is represented by the Greek letter φ, which is pronounced fe or fi.

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The Golden Ratio, contd The golden ratio has an exact value ofThe golden ratio has an exact value of The golden ratio has been used in mathematics, art, and architecture for more than 2000 years.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.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 are used in architecture, art, and packaging.

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Golden Rectangles, contd The rectangle enclosing the diagram of the Parthenon is an example of a golden rectangle.The rectangle enclosing the diagram of the Parthenon is an example of a golden rectangle.

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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.

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Creating a Golden Rectangle, contd 3)Draw an arc centered at M with radius MY. 4)Label the point P as shown.

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Creating a Golden Rectangle, contd 5)Draw a line perpendicular to WP. 6)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?How can you find the exact decimal equivalent of this number?

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Initial Problem Solution, contd We can find the value of the continued fraction by using a recursion rule that generates a sequence of fractions.We can find the value of the continued fraction by using a recursion rule that generates a sequence of fractions. The first term isThe first term is The recursion rule isThe recursion rule is

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Initial Problem Solution, contd We find:We find: The first term isThe first term is The second term isThe second term is

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Initial Problem Solution, contd The third term isThe third term is The fourth term isThe fourth term is

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Initial Problem Solution, contd The fractions in this sequence areThe 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.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.The numbers in this sequence of fractions get closer and closer to φ.

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