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Contemporary Mathematics

13.1 Math and Art

Contemporary Mathematics13.1 Math and Art

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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
A close-up view of a sunflower.
Figure 13.2 Sunflower seeds appear in a pattern that involves the Fibonacci sequence. (credit: “Sunflower Surprise” by frankieleon/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Identify and describe the golden ratio.
  2. Identify and describe the Fibonacci sequence and its application to nature.
  3. Apply the golden ratio and the Fibonacci sequence relationship.
  4. Identify and compute golden rectangles.

Art is the expression or application of human creative skill and imagination, typically in a visual form such as painting or sculpture, producing works to be appreciated primarily for their beauty or emotional power.

Oxford Dictionary

Art, like other disciplines, is an area that combines talent and experience with education. While not everyone considers themself skilled at creating art, there are mathematical relationships commonly found in artistic masterpieces that drive what is considered attractive to the eye. Nature is full of examples of these mathematical relationships.

Enroll in a cake decorating class and, when you learn how to create flowers out of icing, you will likely be directed as to the number of petals to use. Depending on the desired size of a rose flower, the recommendation for the number of petals to use is commonly 5, 8, or 13 petals. If learning to draw portraits, you may be surprised to learn that eyes are approximately halfway between the top of a person’s head and their chin. Studying architecture, we find examples of buildings that contain golden rectangles and ratios that add to the beautifying of the design. The Parthenon (Figure 13.3), which was built around 400 BC, as well as modern-day structures such the Washington Monument are two examples containing these relationships. These seemingly unrelated examples and many more highlight mathematical relationships that we associate with beauty in artistic form.

The ancient citadel Acropolis of Athens from the side perspective.
Figure 13.3 The Parthenon in Greece demonstrates the golden ratio. (credit: “Parthénon” by Julien Maury/Flickr, Public Domain Mark 1.0)

Golden Ratio

The golden ratio, also known as the golden proportion, is a ratio aspect that can be found in beauty from nature to human anatomy as well as in golden rectangles that are commonly found in building structures. The golden ratio is expressed in nature from plants to creatures such as the starfish, honeybees, seashells, and more. It is commonly noted by the Greek letter ϕ (pronounced “fee”). ϕ=1+52ϕ=1+52, which has a decimal value approximately equal to 1.618.

Consider Figure 13.3: Note how the building is balanced in dimension and has a natural shape. The overall structure does not appear as if it is too wide or too tall in comparison to the other dimensions.

A scanned image of Vitruvian Man sketch.
Figure 13.4 Vitruvian Man by Leonardo da Vinci (credit: "Vitruvian Man" by Leonardo da Vinci/Wikimedia Commons, Public Domain)

The golden ratio has been used by artists through the years and can be found in art dating back to 3000 BC. Leonardo da Vinci is considered one of the artists who mastered the mathematics of the golden ratio, which is prevalent in his artwork such as Virtuvian Man (Figure 13.4). This famous masterpiece highlights the golden ratio in the proportions of an ideal body shape.

The golden ratio is approximated in several physical measurements of the human body and parts exhibiting the golden ratio are simply called golden. The ratio of a person’s height to the length from their belly button to the floor is ϕ or approximately 1.618. The bones in our fingers (excluding the thumb), are golden as they form a ratio that approximates ϕ. The human face also includes several ratios and those faces that are considered attractive commonly exhibit golden ratios.

Example 13.1

Using Golden Ratio and a Person’s Height

If a person’s height is 5 ft 6 in, what is the approximate length from their belly button to the floor rounded to the nearest inch, assuming the ratio is golden?

Your Turn 13.1

1.
If a person’s height is 6 ft 2 in, what is the approximate length from their belly button to the floor rounded to the nearest inch if the ratio is golden?

Fibonacci Sequence and Application to Nature

A close-up view of a rose.
Figure 13.5 Rose petals appear in a Fibonacci spiral. (credit: “rilke4” by monchoohcnom/Flickr, Public Domain Mark 1.0)

The Fibonacci sequence can be found occurring naturally in a wide array of elements in our environment from the number of petals on a rose flower to the spirals on a pine cone to the spines on a head of lettuce and more. The Fibonacci sequence can be found in artistic renderings of nature to develop aesthetically pleasing and realistic artistic creations such as in sculptures, paintings, landscape, building design, and more. It is the sequence of numbers beginning with 1, 1, and each subsequent term is the sum of the previous two terms in the sequence (1, 1, 2, 3, 5, 8, 13, …).

The petal counts on some flowers are represented in the Fibonacci sequence. A daisy is sometimes associated with plucking petals to answer the question “They love me, they love me not.” Interestingly, a daisy found growing wild typically contains 13, 21, or 34 petals and it is noted that these numbers are part of the Fibonacci sequence. The number of petals aligns with the spirals in the flower family.

Example 13.2

Applying the Fibonacci Sequence to Rose Petals

Suppose you were creating a rose out of icing, assuming a Fibonacci sequence in the petals, how many petals would be in the row following a row containing 13 petals?

Your Turn 13.2

1.
If a circular row on a pinecone contains 21 scales and models the Fibonacci sequence, approximately how many scales would be found on the next circular row?

Golden Ratio and the Fibonacci Sequence Relationship

Mathematicians for years have explored patterns and applications to the world around us and continue to do so today. One such pattern can be found in ratios of two adjacent terms of the Fibonacci sequence.

Recall that the Fibonacci sequence = 1, 1, 3, 5, 8, 13,… with 5 and 8 being one example of adjacent terms. When computing the ratio of the larger number to the preceding number such as 8/5 or 13/8, it is fascinating to find the golden ratio emerge. As larger numbers from the Fibonacci sequence are utilized in the ratio, the value more closely approaches ϕ, the golden ratio.

Example 13.3

Finding Golden Ratio in Adjacent Fibonacci Terms

The 24th Fibonacci number is 46,368 and the 25th is 75,025. Show that the ratio of the 25th and 24th Fibonacci numbers is approximately ϕ. Round your answer to the nearest thousandth.

Your Turn 13.3

1.
The 23rd Fibonacci number is 28,657 and the 24th is 46,368. Show that the ratio of the 24th and 23rd Fibonacci numbers is approximately \mathit{ϕ}. Round your answer to the nearest thousandth.
A distant view of the Pyramids of Giza.
Figure 13.6 The pyramids of Giza in Egypt (credit: “Giza Pyramids” by Vincent Brown/Flickr, CC BY 2.0)

Golden Rectangles

Turning our attention to man-made elements, the golden ratio can be found in architecture and artwork dating back to the ancient pyramids in Egypt (Figure 13.6) to modern-day buildings such as the UN headquarters. The ancient Greeks used golden rectangles—any rectangles where the ratio of the length to the width is the golden ratio—to create aesthetically pleasing as well as solid structures, with examples of the golden rectangle often being used multiple times in the same building such as the Parthenon, which is shown in Figure 13.3. Golden rectangles can be found in twentieth-century buildings as well, such as the Washington Monument.

Looking at another man-made element, artists paintings often contain golden rectangles. Well-known paintings such as Leonardo da Vinci’s The Last Supper and the Vitruvian Man contain multiple golden rectangles as do many of da Vinci’s masterpieces.

Whether framing a painting or designing a building, the golden rectangle has been widely utilized by artists and are considered to be the most visually pleasing rectangles.

Example 13.4

Finding Golden Rectangle in Frames

A frame has dimensions of 8 in by 6 in. Calculate the ratio of the sides rounded to the nearest thousandth and determine if the size approximates a golden rectangle.

Your Turn 13.4

1.
A frame has dimensions of 10 in by 8 in. Calculate the ratio of the sides rounded to the nearest thousandth and determine if the size approximates a golden rectangle.

People in Mathematics

M.C. Escher

A photo of Maurits Cornelis Escher.
Figure 13.7 M.C. Escher (credit: "M.C. Escher" by Hans Peters (ANEFO)/Dutch National Archives, CC0 1.0 Public Domain)

Mauritis Cornelis Escher was a Dutch-born world-famous graphic artist and his work can be found in murals, stamps, wallpaper designs, illustrations in books, and even carpets. Over his lifetime, M.C. Escher created hundreds of lithographs and wood engravings as well as more than 2,000 sketches.

Escher’s work is characterized with the infusion of geometric designs that obey most of the mathematical rules. If you study his work closely, you can see where he breaks a mathematical relationship to create famous illusions such as soldiers marching around the top of a square turret where the soldiers appear to be always going uphill but are contained on a single set of stairs in a square. Look closely and the golden ratio as well as golden rectangles abound in Escher’s work.

Like many famous people, M.C. Escher did not find success in his early school years. Before finding success, Escher failed his final school exam and quit a short stint in architecture. Finding a graphic arts teacher who recognized Escher’s talent, Escher completed art school and enjoyed traveling through Italy, where he found much of his inspiration for his work.

Check Your Understanding

1.
What is the value of the golden ratio to the nearest thousandth?
2.
What are the first 10 terms of the Fibonacci sequence?
3.
What is a golden rectangle?

Section 13.1 Exercises

1.
A person’s height is 5 ft 2 in. What is the approximate length from their belly button to the floor rounded to the nearest inch?
2.
A person’s height is 6 ft 3 in. What is the approximate length from their belly button to the floor rounded to the nearest inch?
3.
A person’s length from their belly button to the floor is 3 ft 11 in. What is the person’s approximate height rounded to the nearest inch?
4.
A person’s length from their belly button to the floor is 58 in. What is the person’s approximate height rounded to the nearest inch?
5.
The spikes on a pineapple mirror the Fibonacci sequence. If a row on a pineapple contains five spikes, approximately how many spikes would be found on the next larger row of spikes?
6.
The leaves on a plant mirror the Fibonacci sequence. If a set of leaves on the plant contains 5 leaves, how many leaves would be found on the previous smaller set of leaves?
7.
The spines on a head of lettuce mirror the Fibonacci sequence. If a head of lettuce contains 13 spines, approximately how many spines would be found on the next inside layer?
8.
The seeds on a sunflower mirror the Fibonacci sequence. If a circular layer on the sunflower contains 55 seeds, approximately how many seeds would be found on the next larger circular layer?
9.
The segments on a palm frond mirror the Fibonacci sequence. If a palm frond contains 89 segments, approximately how many segments would be found on the next larger palm frond?
10.
The 19th term of the Fibonacci sequence is 4,181 and the 20th term is 6,765. What is the 21st term of the sequence?
11.
The 23rd term of the Fibonacci sequence is 28,657 and the 24th term is 46,368. What is the 22nd term of the sequence?
12.
The 18th term of the Fibonacci sequence is 2,584 and the 20th term is 6,765. What is the 19th term of the sequence?
13.
The 25th term of the Fibonacci sequence is 75,025 and the 20th term is 6,765. What is the 24th term of the sequence?
14.
The 10th Fibonacci number is 55 and the 11th is 89. Show that the ratio of the 11th and 10th Fibonacci numbers is approximately \mathit{ϕ}. Round your answer to the nearest thousandth.
15.
The 23rd Fibonacci number is 28,657 and the 24th is 46,368. Show that the ratio of the 24th and 23rd Fibonacci numbers is approximately \mathit{ϕ}. Round your answer to the nearest ten-thousandth.
16.
The 22nd Fibonacci number is 17,711 and the 21st is 10,946. Show that the ratio of the 22nd and 21st Fibonacci numbers is approximately \mathit{ϕ}. Round your answer to the nearest ten-thousandth.
17.
The 16th term of the Fibonacci sequence is 987. Use the approximate value of \mathit{ϕ} of 1.618 to estimate the 15th term. Round your answer to the nearest whole number.
18.
The 26th term of the Fibonacci sequence is 121,393. Use the approximate value of \mathit{ϕ} of 1.618 to estimate the 25th term. Round your answer to the nearest whole number.
19.
A frame has dimensions of 20 in by 24 in. Calculate the ratio of the sides rounded to the nearest tenth and determine if the size approximates a golden rectangle.
20.
A fence has dimensions of 75 in by 45 in. Calculate the ratio of the sides rounded to the nearest tenth and determine if the size approximates a golden rectangle.
21.
A frame has a length of 50 in. Calculate the width rounded to the nearest inch if the frame is to be a golden rectangle.
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