a mathematical view of our world 1 st ed. parks, musser, trimpe, maurer, and maurer

A Mathematical View A Mathematical View of Our Worldof Our World

11stst ed. ed.

Parks, Musser, Trimpe, Parks, Musser, Trimpe, Maurer, and MaurerMaurer, and Maurer

Chapter 2Chapter 2

Shapes in Our LivesShapes in Our Lives

Section 2.1Section 2.1

TilingsTilings• GoalsGoals

• Study polygonsStudy polygons• Vertex anglesVertex angles• Regular tilingsRegular tilings• Semiregular tilingsSemiregular tilings• Miscellaneous tilingsMiscellaneous tilings

• Study the Pythagorean theoremStudy the Pythagorean theorem

2.1 Initial Problem2.1 Initial Problem• A portion of a ceramic A portion of a ceramic

tile wall composed of tile wall composed of two differently shaped two differently shaped tiles is shown. Why do tiles is shown. Why do these two types of tiles these two types of tiles fit together without gaps fit together without gaps or overlaps?or overlaps?• The solution will be given The solution will be given

at the end of the section.at the end of the section.

TilingsTilings

• Geometric patterns of tiles have been Geometric patterns of tiles have been used for thousands of years all around used for thousands of years all around the world. the world.

• TilingsTilings, also called , also called tessellationstessellations, , usually involve geometric shapes usually involve geometric shapes called polygons.called polygons.

PolygonsPolygons

• A A polygonpolygon is a plane figure consisting is a plane figure consisting of line segments that can be traced of line segments that can be traced so that the starting and ending points so that the starting and ending points are the same and the path never are the same and the path never crosses itself.crosses itself.

Question:Question:

Choose the figure below that is NOT Choose the figure below that is NOT a polygon.a polygon.

a.a. c.c.

b. b. d. all are polygons d. all are polygons

Polygons, cont’dPolygons, cont’d

• The line segments forming a polygon The line segments forming a polygon are called its are called its sidessides. .

• The endpoints of the sides are called The endpoints of the sides are called its its verticesvertices. .

• The singular of vertices is The singular of vertices is vertexvertex..

Polygons, cont’dPolygons, cont’d• A polygon with A polygon with nn sides and sides and nn vertices is vertices is

called an called an nn-gon-gon. . • For small values of For small values of nn, more familiar names are , more familiar names are

used.used.

Polygonal RegionsPolygonal Regions• A A polygonal regionpolygonal region is a polygon together with is a polygon together with

the portion of the plan enclosed by the the portion of the plan enclosed by the polygon.polygon.

Polygonal Regions, cont’dPolygonal Regions, cont’d

• 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 An example of a tiling, made up of rectangles, is

shown below.shown below.


• Polygonal regions form a tiling if:Polygonal regions form a tiling if:

• The entire plane is covered without The entire plane is covered without gaps.gaps.

• No two polygonal regions overlap.No two polygonal regions overlap.


• Examples of tilings with polygonal Examples of tilings with polygonal regions are shown below.regions are shown below.

Vertex AnglesVertex Angles• A tiling of triangles illustrates the fact that the A tiling of triangles illustrates the fact that the

sum of the measures of the angles in a sum of the measures of the angles in a triangle is 180triangle is 180°°..

Vertex Angles, cont’dVertex Angles, cont’d

• The angles in a polygon are called its The angles in a polygon are called its vertex anglesvertex angles..• The symbol The symbol indicates an angle. indicates an angle.

• Line segments that join nonadjacent Line segments that join nonadjacent vertices in a polygon are called vertices in a polygon are called diagonalsdiagonals of the polygon. of the polygon.

Example 1Example 1

• The vertex angles in the pentagon are called The vertex angles in the pentagon are called V, V, W, W, X, X, Y, and Y, and Z.Z.

• Two diagonals shown are WZ and WY. Two diagonals shown are WZ and WY.

Vertex Angles, cont’dVertex Angles, cont’d• Any polygon can be divided, using Any polygon can be divided, using

diagonals, into triangles.diagonals, into triangles.• A polygon with A polygon with nn sides can be divided into sides can be divided into

n n – 2 triangles.– 2 triangles.


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

2 180n

Example 2Example 2• Find the sum of measures of the Find the sum of measures of the

vertex angles of a hexagon. vertex angles of a hexagon. • Solution: Solution:

• A hexagon has 6 sides, so A hexagon has 6 sides, so nn = 6. = 6.

• The sum of the measures of the angles The sum of the measures of the angles is found to be:is found to be:

2 180 6 2 180 4 180 720n

Regular PolygonsRegular Polygons

• Regular polygonsRegular polygons are polygons in are polygons in which:which:• All sides have the same length.All sides have the same length.

• All vertex angles have the same All vertex angles have the same measure.measure.

• Polygons that are not regular are Polygons that are not regular are called called irregular polygonsirregular polygons..

Regular Polygons, cont’dRegular Polygons, cont’d

Regular Polygons, cont’dRegular Polygons, cont’d

• A regular A regular nn-gon has -gon has nn angles. angles.• All vertex angles have the same All vertex angles have the same

measure.measure.

• The measure of each vertex angle must The measure of each vertex angle must bebe 2 180n

n

Example 3Example 3• Find the measure of any vertex angle Find the measure of any vertex angle

in a regular hexagon.in a regular hexagon.• Solution:Solution:

• A hexagon has 6 sides, so A hexagon has 6 sides, so nn = 6. = 6.

• Each vertex angle in the regular Each vertex angle in the regular hexagon has the measure:hexagon has the measure:

2 180 6 2 180 4 180 720120

6 6 6

n

n

Regular TilingsRegular Tilings• A A regular tilingregular tiling is a tiling composed of is a tiling composed of

regular polygonal regions in which all the regular polygonal regions in which all the polygons are the same shape and size.polygons are the same shape and size.• Tilings can be edge-to-edge, meaning the Tilings can be edge-to-edge, meaning the

polygonal regions have entire sides in polygonal regions have entire sides in common.common.

• Tilings can be Tilings can be notnot edge-to-edge, meaning the edge-to-edge, meaning the polygonal regions do not have entire sides in polygonal regions do not have entire sides in common.common.

Regular Tilings, cont’dRegular Tilings, cont’d

• Examples of edge-to-edge regular tilings. Examples of edge-to-edge regular tilings.

Regular Tilings, cont’dRegular Tilings, cont’d• Example of a regular tiling that is not edge-to-Example of a regular tiling that is not edge-to-

edge. edge.


• Only regular edge-to-edge tilings are Only regular edge-to-edge tilings are generally called generally called regular tilingsregular tilings. .

• In every such tiling the vertex angles In every such tiling the vertex angles of the tiles meet at a point.of the tiles meet at a point.


• What regular polygons will form tilings What regular polygons will form tilings of the plane?of the plane?• Whether or not a tiling is formed Whether or not a tiling is formed

depends on the measure of the vertex depends on the measure of the vertex angles.angles.

• The vertex angles that meet at a point The vertex angles that meet at a point must add up to exactly 360must add up to exactly 360° so that no ° so that no gap is left and no overlap occurs.gap is left and no overlap occurs.

Example 4Example 4

• Equilateral Triangles Equilateral Triangles (Regular 3-gons)(Regular 3-gons)

• In a tiling of In a tiling of equilateral equilateral triangles, there are triangles, there are 6(606(60°) = 360° at °) = 360° at each vertex point.each vertex point.

Example 5Example 5

• Squares Squares

(Regular 4-gons)(Regular 4-gons)

• In a tiling of In a tiling of squares, there squares, there are 4(90are 4(90°) = 360° °) = 360° at each vertex at each vertex point.point.

Question:Question:

Will a regular pentagon tile the plane? Will a regular pentagon tile the plane?

a. yesa. yes

b. nob. no

Example 6Example 6

• Regular hexagonsRegular hexagons

(Regular 6-gons)(Regular 6-gons)

• In a tiling of regular In a tiling of regular hexagons, there hexagons, there are 3(120are 3(120°) = 360° °) = 360° at each vertex at each vertex point.point.


• Do any regular polygons, besides Do any regular polygons, besides nn = 3, 4, = 3, 4, and 6, tile the plane?and 6, tile the plane?

• Note: Every regular tiling with Note: Every regular tiling with nn > 6 must > 6 must have:have:• At least three vertex angles at each pointAt least three vertex angles at each point

• Vertex angles measuring more than 120Vertex angles measuring more than 120°°

• Angle measures at each vertex point that add Angle measures at each vertex point that add to 360°to 360°


• In a previous question, you determined In a previous question, you determined that a regular pentagon does not tile the that a regular pentagon does not tile the plane.plane.

• Since 3(120Since 3(120°) = 360°, no polygon with °) = 360°, no polygon with vertex angles larger than 120° [i.e. vertex angles larger than 120° [i.e. nn > 6] > 6] can form a regular tiling.can form a regular tiling.

• Conclusion: The only regular tilings are Conclusion: The only regular tilings are those for those for nn = 3, = 3, nn = 4, and = 4, and nn = 6. = 6.

Vertex FiguresVertex Figures

• A A vertex figurevertex figure of a tiling is the of a tiling is the polygon formed when line segments polygon formed when line segments join consecutive midpoints of the join consecutive midpoints of the sides of the polygons sharing that sides of the polygons sharing that vertex point.vertex point.

Vertex Figures, cont’dVertex Figures, cont’d

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

Semiregular TilingsSemiregular Tilings

• Semiregular tilingsSemiregular tilings• Are edge-to-edge tilings.Are edge-to-edge tilings.

• Use two or more regular polygonal Use two or more regular polygonal regions.regions.

• Vertex figures are the same shape and Vertex figures are the same shape and size no matter where in the tiling they size no matter where in the tiling they are drawn.are drawn.

Example 7Example 7

• Verify that the Verify that the tiling shown is a tiling shown is a semiregular tiling.semiregular tiling.

Example 7, cont’dExample 7, cont’d

• Solution:Solution:• The tiling is made of The tiling is made of

3 regular polygons.3 regular polygons.

• Every vertex figure Every vertex figure is the same shape is the same shape and size.and size.

Example 8Example 8

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


• Solution:Solution:• The tiling The tiling isis made of made of

3 regular polygons.3 regular polygons.

• Every vertex figure Every vertex figure is is notnot the same the same shape and size.shape and size.

Semiregular TilingsSemiregular Tilings

Miscellaneous TilingsMiscellaneous Tilings

• Tilings can also be made of other Tilings can also be made of other types of shapes.types of shapes.

• Tilings consisting of irregular polygons Tilings consisting of irregular polygons that are all the same size and shape that are all the same size and shape will be considered.will be considered.

Miscellaneous Tilings, cont’dMiscellaneous Tilings, cont’d

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


• 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:


• SomeSome irregular pentagons (5-gons) will tile the irregular pentagons (5-gons) will tile the plane.plane.

• An example is given below: An example is given below:


• SomeSome irregular hexagons (6-gons) will tile the irregular hexagons (6-gons) will tile the plane.plane.

• An example is given below: An example is given below:


• A polygonal region is A polygonal region is convexconvex if, for any two if, for any two points in the region, the line segment having the points in the region, the line segment having the two points as endpoints also lies in the region.two points as endpoints also lies in the region.

• A polygonal region that is not convex is called A polygonal region that is not convex is called concaveconcave..

Pythagorean TheoremPythagorean Theorem

• In a right triangle, the In a right triangle, the sum of the areas of sum of the areas of the squares on the the squares on the sides of the triangle is sides of the triangle is equal to the area of equal to the area of the square on the the square on the hypotenuse.hypotenuse.

•

2 2 2a b c

Example 9Example 9

• Find the length Find the length xx in in the figure. the figure.

• Solution: Use the Solution: Use the theorem.theorem.•

•

2 2 21 1 2y 2 2 21 1 2 3x y

Pythagorean Theorem ConversePythagorean Theorem Converse

• If If

then the then the triangle is a triangle is a right triangle.right triangle.

2 2 2a b c

Example 10Example 10• Show that any triangle with sides of length Show that any triangle with sides of length

3, 4 and 5 is a right triangle.3, 4 and 5 is a right triangle.• Solution: The longest side must be the Solution: The longest side must be the

hypotenuse. Let hypotenuse. Let aa = 3, = 3, bb = 4, and = 4, and cc = 5. = 5. We find: We find: 2 2 23 4 5 ?

9 16 25 ?

25 25

2.1 Initial Problem Solution2.1 Initial Problem Solution• The tiling consists of The tiling consists of

squares and regular squares and regular octagons.octagons.

• The vertex angle The vertex angle measures add up to 90measures add up to 90° ° + 2(135°) = 360°.+ 2(135°) = 360°.

• This is an example of This is an example of one of the eight possible one of the eight possible semiregular tilings. semiregular tilings.


Symmetry, Rigid Motions, Symmetry, Rigid Motions,

and Escher Patternsand 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

SymmetrySymmetry• We say a figure has We say a figure has symmetrysymmetry if it can be if it can be

moved in such a way that the resulting moved in such a way that the resulting figure looks identical to the original figure.figure looks identical to the original figure.

• Types of symmetry that will be studied Types of symmetry that will be studied here are:here are:• Reflection symmetryReflection symmetry

• Rotation symmetryRotation symmetry

• Translation symmetryTranslation symmetry

Strip PatternsStrip Patterns• An example of a An example of a strip patternstrip pattern, also , also

called a called a one-dimensional patternone-dimensional pattern, is , is shown below.shown below.

Strip Patterns, cont’dStrip Patterns, cont’d• This strip pattern has This strip pattern has vertical reflection vertical reflection

symmetrysymmetry because the pattern looks the because the pattern looks the same when it is reflected across a vertical same when it is reflected across a vertical line.line.• The dashed line is called a The dashed line is called a line of symmetryline of symmetry..

Strip Patterns, cont’dStrip Patterns, cont’d• This strip pattern has This strip pattern has horizontal horizontal

reflection symmetryreflection symmetry because the because the pattern looks the same when it is pattern looks the same when it is reflected across a horizontal line.reflected across a horizontal line.

Strip Patterns, cont’dStrip Patterns, cont’d• This strip pattern has This strip pattern has rotation symmetryrotation symmetry because because

the pattern looks the same when it is rotated 180the pattern looks the same when it is rotated 180° ° about a given point.about a given point.• The point around which the pattern is turned is called the The point around which the pattern is turned is called the

center of rotationcenter of rotation..

• Note that the degree of rotation must be less than 360Note that the degree of rotation must be less than 360°.°.

Strip Patterns, cont’dStrip Patterns, cont’d• This strip pattern has This strip pattern has translation symmetrytranslation symmetry

because the pattern looks the same when because the pattern looks the same when it is translated a certain amount to the it is translated a certain amount to the right. right. • The pattern is understood to extend The pattern is understood to extend

indefinitely to the left and right. indefinitely to the left and right.

Example 1Example 1• Describe the symmetries of the pattern.Describe the symmetries of the pattern.

• Solution: This pattern has translation Solution: This pattern has translation symmetry only.symmetry only.

Question:Question:

Describe the symmetries of the strip Describe the symmetries of the strip pattern, assuming it continues to the left pattern, assuming it continues to the left and right indefinitely and right indefinitely

a. horizontal reflection, vertical reflection, a. horizontal reflection, vertical reflection, translationtranslation

b. vertical reflection, translationb. vertical reflection, translationc. translationc. translationd. vertical reflectiond. vertical reflection

Two-Dimensional PatternsTwo-Dimensional Patterns• Two-dimensional Two-dimensional

patterns that fill the patterns that fill the plane can also have plane can also have symmetries.symmetries.

• The pattern shown The pattern shown here has horizontal here has horizontal and vertical reflection and vertical reflection symmetries.symmetries.• Some lines of Some lines of

symmetry have been symmetry have been drawn in. drawn in.

Two-Dimensional Patterns, cont’dTwo-Dimensional Patterns, cont’d

• The pattern also The pattern also has has • horizontal and horizontal and

vertical translation vertical translation symmetries.symmetries.

• 180180° rotation ° rotation ssymmetry. ymmetry.

Two-Dimensional Patterns, cont’dTwo-Dimensional Patterns, cont’d

• This pattern has This pattern has • 120120° rotation ° rotation

ssymmetry. ymmetry.

• 240240° rotation ° rotation ssymmetry. ymmetry.

Rigid MotionsRigid Motions• Any combination of translations, Any combination of translations,

reflections across lines, and/or rotations reflections across lines, and/or rotations around a point is called a around a point is called a rigid motionrigid motion, or , or an an isometryisometry. .

• Rigid motions may change the location of Rigid motions may change the location of the figure in the plane.the figure in the plane.

• Rigid motions do not change the size or Rigid motions do not change the size or shape of the figure.shape of the figure.

ReflectionReflection• A A reflectionreflection with with

respect to line respect to line ll is is defined as follows, defined as follows, with with AA’ being the ’ being the imageimage of point of point AA under under the reflection.the reflection.• If If AA is a point on the is a point on the

line line ll, , AA = = AA’.’.

• If If AA is not on line is not on line ll, then , then ll is the perpendicular is the perpendicular bisector of line bisector of line AAAA’.’.

Example 2Example 2

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

Example 2, cont’dExample 2, cont’d• Solution: Solution:

• Find the image of each vertex point of the triangle, Find the image of each vertex point of the triangle, using a protractor.using a protractor.

• AA and and AA’ are equal distances from ’ are equal distances from ll..

• Connect the image points to form the new triangle.Connect the image points to form the new triangle.

VectorsVectors• A A vectorvector is a directed line segment. 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 The other endpoint, labeled with an arrow, is the ending point.ending point.

• Two vectors are Two vectors are equivalentequivalent if they are: if they are:• ParallelParallel

• Have the same lengthHave the same length

• Point in the same direction. Point in the same direction.

Vectors, cont’dVectors, cont’d• A vector A vector vv is has a length and a direction, as is has a length and a direction, as

shown below.shown below.• A translation can be defined by moving every A translation can be defined by moving every

point of a figure the distance and direction point of a figure the distance and direction indicated by a vector. indicated by a vector.

TranslationTranslation• A A translationtranslation is defined as follows. is defined as follows.

• A vector A vector vv assigns to every point assigns to every point AA an image an image point point AA’.’.

• The directed line segment between The directed line segment between AA and and AA’ ’ is equivalent to is equivalent to vv..

Example 3Example 3

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


• Find the image of each vertex point by drawing the Find the image of each vertex point by drawing the three vectors.three vectors.

• Connect the image points to form the new triangle.Connect the image points to form the new triangle.

RotationRotation

• A A rotationrotation involves turning a figure involves turning a figure around a point around a point OO, clockwise or , clockwise or counterclockwise, through an angle counterclockwise, through an angle less than 360less than 360°.°.

Rotation, cont’dRotation, cont’d• The point The point OO is called the center of rotation. is called the center of rotation. • The directed angle indicates the amount and The directed angle indicates the amount and

direction of the rotation.direction of the rotation.

• A positive angle indicates a A positive angle indicates a counterclockwise rotation.counterclockwise rotation.

• A negative angle indicates a clockwise A negative angle indicates a clockwise rotation.rotation.

• A point and its image are the same distance A point and its image are the same distance from from OO..

Rotation, cont’dRotation, cont’d• A rotation of a point A rotation of a point XX about the center about the center O O

determined by a directed angle determined by a directed angle AOBAOB is is illustrated in the figure below.illustrated in the figure below.

Example 4Example 4

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

Example 4, cont’dExample 4, cont’d• Solution:Solution:

• Create a 50Create a 50° angle ° angle with initial side with initial side OAOA. .

• Mark Mark AA’ on the ’ on the terminal side, terminal side, recalling that recalling that AA and and AA’ are the same ’ are the same distance from distance from OO..


• Solution cont’d:Solution cont’d:• Repeat this process Repeat this process

for each vertex.for each vertex.

• Connect the three Connect the three image points to form image points to form the new triangle.the new triangle.

Glide ReflectionGlide Reflection• A A glide reflectionglide reflection is the result of a is the result of a

reflection followed by a translation.reflection followed by a translation.• The line of reflection must not be perpendicular to the The line of reflection must not be perpendicular to the

translation vector.translation vector.

• The line of reflection is usually parallel to the The line of reflection is usually parallel to the translation vector.translation vector.

Example 5Example 5• A strip pattern of footprints can be A strip pattern of footprints can be

created using a glide reflection.created using a glide reflection.

Crystallographic ClassificationCrystallographic Classification

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

Classification, cont’dClassification, cont’d

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

Example 6Example 6

• Use the crystallographic system to Use the crystallographic system to describe the strip pattern.describe the strip pattern.

• Solution: The classification is pmm2.Solution: The classification is pmm2.

Example 7Example 7

• Use the crystallographic system to Use the crystallographic system to describe the strip pattern.describe the strip pattern.

• Solution: The classification is p111.Solution: The classification is p111.

Question:Question:

Use the crystallographic classification Use the crystallographic classification system to describe the pattern.system to describe the pattern.

a. p112a. p112b. pmm2b. pmm2c. p1m1c. p1m1d. p111d. p111

Escher PatternsEscher Patterns

• Maurits Escher was an artist who used Maurits Escher was an artist who used rigid motions in his work.rigid motions in his work.

• You can view some examples of You can view some examples of Escher’s work in your textbook. Escher’s work in your textbook.

Escher Patterns, cont’dEscher Patterns, cont’d• An example of the process used to create An example of the process used to create

Escher-type patterns is shown next. 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 Cut a piece from the upper left and translate it to the right.right.

• Reflect the left side to the right side. Reflect the left side to the right side.

Escher Patterns, cont’dEscher Patterns, cont’d

• The figure has been The figure has been decorated and decorated and repeated.repeated.

• Notice that the Notice that the pattern has vertical pattern has vertical and horizontal and horizontal translation symmetry translation symmetry and vertical and vertical reflection symmetry. reflection symmetry.


Fibonacci Numbers and the Fibonacci Numbers and the

Golden MeanGolden 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

2.3 Initial Problem2.3 Initial Problem• This expression is called a continued This expression is called a continued

fraction.fraction.

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

SequencesSequences

• A A sequencesequence is an ordered collection of is an ordered collection of numbers.numbers.

• A sequence can be written in the form A sequence can be written in the form a a11, a, a22, a, a33, …, a, …, ann, …, …

• The symbol aThe symbol a11 represents the first number in the represents the first number in the

sequence.sequence.

• The symbol aThe symbol ann represents the represents the nnth number in the th number in the

sequence.sequence.

Question:Question:

Given the sequence: 1, 3, 5, 7, 9, 11, Given the sequence: 1, 3, 5, 7, 9, 11, 13, 15, … , find the values of the 13, 15, … , find the values of the numbers Anumbers A11, A, A33, and A, and A99..

a. Aa. A11 = 1, A = 1, A3 3 = 5,= 5, AA9 9 = 15= 15

b. Ab. A11 = 1, A = 1, A3 3 = 3,= 3, AA9 9 = 17= 17

c. Ac. A11 = 1, A = 1, A3 3 = 5,= 5, AA9 9 = 17= 17

d. Ad. A11 = 1, A = 1, A3 3 = 5,= 5, AA9 9 = 16= 16

Fibonacci SequenceFibonacci Sequence• The famous Fibonacci sequence is the The famous Fibonacci sequence is the

result of a question posed by Leonardo de result of a question posed by Leonardo de Fibonacci, a mathematician during the Fibonacci, a mathematician during the Middle Ages.Middle Ages.• If you begin with one pair of rabbits on the first If you begin with one pair of rabbits on the first

day of the year, how many pairs of rabbits will day of the year, how many pairs of rabbits will you have on the first day of the next year?you have on the first day of the next year?

• It is assumed that each pair of rabbits produces a It is assumed that each pair of rabbits produces a new pair every month and each new pair begins to new pair every month and each new pair begins to produce two months after birth. produce two months after birth.

Fibonacci Sequence, cont’dFibonacci Sequence, cont’d

• 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, The sequence that appears three times in the table, 1, 1,

2, 3, 5, 8, 13, 21, … is called the 2, 3, 5, 8, 13, 21, … is called the Fibonacci sequenceFibonacci sequence. .

Fibonacci Sequence, cont’dFibonacci Sequence, cont’d• The The Fibonacci sequenceFibonacci sequence is the sequence is the sequence

of numbers 1, 1, 2, 3, 5, 8, 13, 21, …of numbers 1, 1, 2, 3, 5, 8, 13, 21, …• The Fibonacci sequence is found many The Fibonacci sequence is found many

places in nature.places in nature.

• Any number in the sequence is called a Any number in the sequence is called a Fibonacci number.Fibonacci number.

• The sequence is usually written The sequence is usually written f f11, f, f22, f, f33, …, f, …, fnn, …, …

RecursionRecursion

• RecursionRecursion, in a sequence, indicates that , in a sequence, indicates that each number in the sequence is found each number in the sequence is found using previous numbers in the sequence.using previous numbers in the sequence.

• Some sequences, such as the Fibonacci Some sequences, such as the Fibonacci sequence, are generated by a sequence, are generated by a recursion recursion rulerule along with along with starting valuesstarting values for the first for the first two, or more, numbers in the sequence.two, or more, numbers in the sequence.

Question:Question:

A recursive sequence uses the rule AA recursive sequence uses the rule An n

=4A=4An-1 n-1 –– AAn-2, n-2, with starting values of Awith starting values of A1 1 = = 2, A2, A22 =7. =7.

What is the fourth term in the What is the fourth term in the sequence?sequence?

a. Aa. A44 = 45 = 45 c. Ac. A44 = 67 = 67

b. Ab. A44 = 26 = 26 d. Ad. A44 = 30 = 30

Fibonacci Sequence, cont’dFibonacci Sequence, cont’d• For the Fibonacci sequence, the starting values For the Fibonacci sequence, the starting values

are are ff11 = 1 and = 1 and ff22 = 1. = 1.

• The recursion rule for the Fibonacci sequence The recursion rule for the Fibonacci sequence is: is:

• Example: Find the third number in the Example: Find the third number in the sequence using the formula.sequence using the formula.• Let Let nn = 3. = 3.

1 2n n nf f f

3 3 1 3 2 2 1 1 1 2f f f f f

Example 1Example 1• Suppose a tree starts from one shoot that Suppose a tree starts from one shoot that

grows for two months and then sprouts a grows for two months and then sprouts a second branch. If each established second branch. If each established branch begins to spout a new branch branch begins to spout a new branch after one month’s growth, and if every after one month’s growth, and if every new branch begins to sprout its own first new branch begins to sprout its own first new branch after two month’s growth, new branch after two month’s growth, how many branches does the tree have how many branches does the tree have at the end of the year?at the end of the year?

Example 1, cont’dExample 1, cont’d• Solution: The number of branches each month Solution: The number of branches each month

in the first year is given in the table and drawn in the first year is given in the table and drawn in the figure below. in the figure below.

Fibonacci Numbers In NatureFibonacci Numbers In Nature

• The Fibonacci numbers are found many The Fibonacci numbers are found many places in the natural world, including: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 The growth patterns of sunflowers and pinecones.pinecones.

• It is believed that the spiral nature of plant It is believed that the spiral nature of plant growth accounts for this phenomenon.growth accounts for this phenomenon.

Fibonacci Numbers In Nature, cont’dFibonacci Numbers In Nature, cont’d

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

Fibonacci Numbers In Nature, cont’dFibonacci Numbers In Nature, cont’d• Plants grow in a spiral pattern. The ratio of the Plants grow in a spiral pattern. The ratio of the

number of spirals to the number of branches is number of spirals to the number of branches is called the called the phyllotactic ratiophyllotactic ratio..

• The numbers in the phyllotactic ratio are usually The numbers in the phyllotactic ratio are usually Fibonacci numbers.Fibonacci numbers.


• Example: The Example: The branch at right has branch at right has a phyllotactic ratio a phyllotactic ratio of 3/8. of 3/8.

• Both 3 and 8 are Both 3 and 8 are Fibonacci numbers.Fibonacci numbers.


• Mature sunflowers have one set of spirals Mature sunflowers have one set of spirals going clockwise and another set going going clockwise and another set going counterclockwise.counterclockwise.

• The numbers of spirals in each set are The numbers of spirals in each set are usually a pair of adjacent Fibonacci usually a pair of adjacent Fibonacci numbers.numbers.• The most common number of spirals is 34 The most common number of spirals is 34

and 55. and 55.

Geometric RecursionGeometric Recursion

• In addition to being used to generate In addition to being used to generate a sequence, the recursion process a sequence, the recursion process can also be used to create shapes.can also be used to create shapes.

• The process of building a figure step-The process of building a figure step-by-step by repeating a rule is called by-step by repeating a rule is called geometric recursiongeometric recursion..

Example 2Example 2• Beginning with a 1-by-1 square, Beginning with a 1-by-1 square,

form a sequence of rectangles by form a sequence of rectangles by adding a square to the bottom, then adding a square to the bottom, then to the right, then to the bottom, then to the right, then to the bottom, then to the right, and so on.to the right, and so on.

a)a) Draw the resulting rectangles.Draw the resulting rectangles.

b)b) What are the dimensions of the What are the dimensions of the rectangles?rectangles?


a)a) The first seven rectangles in the sequence The first seven rectangles in the sequence are shown below.are shown below.

Example 2, cont’dExample 2, cont’d• Solution cont’d: Solution cont’d:

b)b)

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

The Golden RatioThe Golden Ratio

• Consider the ratios of pairs of Consider the ratios of pairs of consecutive Fibonacci numbers. consecutive Fibonacci numbers.

• Some of the ratios are calculated in Some of the ratios are calculated in the table shown on the following the table shown on the following slide.slide.

The Golden Ratio, cont’dThe Golden Ratio, cont’d

The Golden Ratio, cont’dThe Golden Ratio, cont’d• The ratios of pairs of consecutive Fibonacci The ratios of pairs of consecutive Fibonacci

numbers are also represented in the graph numbers are also represented in the graph below.below.

• The ratios approach the dashed line which The ratios approach the dashed line which represents a number around 1.618.represents a number around 1.618.

The Golden Ratio, cont’dThe Golden Ratio, cont’d• The irrational number, approximately The irrational number, approximately

1.618, is called the 1.618, is called the golden ratiogolden ratio..• Other names for the golden ratio include Other names for the golden ratio include

the the golden sectiongolden section, the , the golden meangolden mean, and , and the the divine proportiondivine proportion..

• The golden ratio is represented by the The golden ratio is represented by the Greek letter Greek letter φφ, which is pronounced “fe” , which is pronounced “fe” or “fi”.or “fi”.

The Golden Ratio, cont’dThe Golden Ratio, cont’d

• The golden ratio has an exact value ofThe golden ratio has an exact value of

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

1 5

2

Golden RectanglesGolden Rectangles

• A A golden rectanglegolden rectangle has a ratio of the has a ratio of the longer side to the shorter side that is the longer side to the shorter side that is the golden ratio.golden ratio.

• Golden rectangles are used in Golden rectangles are used in architecture, art, and packaging.architecture, art, and packaging.

Golden Rectangles, cont’dGolden Rectangles, cont’d• The rectangle enclosing the diagram of the The rectangle enclosing the diagram of the

Parthenon is an example of a golden Parthenon is an example of a golden rectangle.rectangle.

Creating a Golden RectangleCreating a Golden Rectangle

1)1) Start with a Start with a square, square, WXYZWXYZ, , that measures that measures one unit on each one unit on each side.side.

2)2) Label the Label the midpoint of side midpoint of side WXWX as point as point MM..

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

3)3) Draw an arc Draw an arc centered at centered at MM with radius with radius MYMY..

4)4) Label the point Label the point PP as shown. as shown.

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

5)5) Draw a line Draw a line perpendicular to perpendicular to WP.WP.

6)6) Extend ZY to meet Extend ZY to meet this line, labeling this line, labeling point Q as shown. point Q as shown. The completed The completed rectangle is shown.rectangle is shown.

2.3 Initial Problem Solution2.3 Initial Problem Solution• How can you find the exact decimal How can you find the exact decimal

equivalent of this number?equivalent of this number?

Initial Problem Solution, cont’dInitial Problem Solution, cont’d• We can find the value of the We can find the value of the

continued fraction by using a continued fraction by using a recursion rule that generates a recursion rule that generates a sequence of fractions.sequence of fractions.

• The first term isThe first term is

• The recursion rule is The recursion rule is

1 1 1a

1

11n

n

aa

Initial Problem Solution, cont’dInitial Problem Solution, cont’d

• We find:We find:• The first term isThe first term is

• The second term is The second term is

21

1 1 31 1

2 2a

a

1 1 1 2a

Initial Problem Solution, cont’dInitial Problem Solution, cont’d

• The third term isThe third term is

• The fourth term is The fourth term is

32

1 1 51 1

3 32

aa

43

1 1 81 1

5 53

aa

Initial Problem Solution, cont’dInitial Problem Solution, cont’d• The fractions in this sequence are The fractions in this sequence are

2, 3/2, 5/3, 8/5, … 2, 3/2, 5/3, 8/5, … • This is recognized to be the same as This is recognized to be the same as

the ratios of consecutive pairs of the ratios of consecutive pairs of Fibonacci numbers.Fibonacci numbers.

• The numbers in this sequence of The numbers in this sequence of fractions get closer and closer to fractions get closer and closer to φφ..

a mathematical view of our world 1 st ed. parks, musser, trimpe, maurer, and maurer

Documents

n sides

n vertices

contd polygonal regions

maurer slide

irregular polygons

sum of measures

pythagorean theorem

tiling of triangles