Nature of Mathematics: Fibonacci Numbers and the Golden Ratio – Study Notes

Patterns in Nature

• Patterns exist when a number, shape, or color occurs repeatedly; they are found in plants, animals, humans, and the universe.
• Examples of natural regularities include sunrise/sunset and the difference in starting months of spring between the Northern and Southern hemispheres.
• Patterns help organize ideas and information, aiding understanding of ourselves, life, and the world.
• Number patterns are often linked to mathematics.

Fibonacci Numbers and the Golden Ratio (overview)

• Fibonacci sequence and the Golden Ratio offer a bridge between mathematics and nature.
• Fibonacci sequence is named after Leonardo Pisano (Fibonacci), an Italian mathematician (1170–1250).
• Historical context:
  • Liber Abaci (Book of Calculation, 1202) introduced Hindu-Arabic numerals to Europe; zero originated in India.
  • Fibonacci’s rabbit puzzle (1202) popularized the sequence.
• Learning outcomes (summary): identify patterns in nature, explain the importance of mathematics, discuss the nature and representation of mathematics, and appreciate mathematics as a human endeavor.

The Rabbit Puzzle and Fibonacci Numbers

• Problem setup (Fibonacci’s Rabbit Puzzle): start with one male–female rabbit pair; rabbits mature in one month; after maturation, each month the pair produces a new pair (male and female) which also mature in one month.
• Assumptions: no deaths; reproduction begins after one month; each mature pair produces one new pair per month.
• Result: the total number of rabbit pairs after each month forms the Fibonacci sequence.
• Key values and examples:
  • Month indices: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
  • Young + adult totals (Total column): 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233
  • Therefore, the number of pairs at the start of the 13th month is 233.
• Definition (Fibonacci sequence):
  • Fn are Fibonacci numbers; initial values and recursion:
  • F1 = 1, F2 = 1,
    Fn = F{n-1} + F_{n-2} ext{ for } n \ge 3.
• Table-based illustration and examples:
  • F7 = 13, F8 = 21, F14 = 377.
• Notes on indexing and variation:
  • Some sources include F0 = 0 (which still satisfies the recursion with F1 = 1, F2 = 1).
  • The recursion can be extended to negative indices with appropriate sign patterns.
• Observations:
  • The Fibonacci numbers describe the growth pattern of rabbit pairs after the first two months.
  • The sequence is defined by the sum of the two previous terms.

Properties of the Fibonacci Sequence

• Recursion relation: $F<em>n = F</em>{n-1} + F<em>{n-2},\quad F</em>1 = 1,\quad F_2 = 1.$
• First several terms: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, \dots$
• The sequence can be used to illustrate additive growth and simple recursive definitions.

The Golden Ratio (φ) and the Fibonacci Connection

• The golden ratio, denoted by $\,\phi\,$ (phi), is defined such that the ratio of a larger segment to a smaller segment equals the ratio of the whole segment to the larger segment.
• Mathematical definition (quadratic relation):
  • If a and b are positive with a > b and a/b = (a+b)/a, then $\phi$ satisfies $\phi^2 = \phi + 1.$
  • Solving gives $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618…$
• Relationship with Fibonacci numbers:
  • The ratio of consecutive Fibonacci numbers tends to $\phi$ as n grows:
  • $\lim<em>{n\to\infty} \frac{F</em>{n+1}}{F_n} = \phi.$
• Examples:
  • $\frac{F<em>6}{F</em>5} = \frac{8}{5} = 1.6.$
  • Larger example: $\frac{F<em>{12}}{F</em>{11}} = \frac{144}{89} \approx 1.61798…$
• Implication: the long-run behavior of the Fibonacci sequence encodes the golden ratio.

Fibonacci and Golden Ratio in Nature

• Plant petals (common Fibonacci-related petal counts):
  • Calla lily: 1 petal
  • Trillium: 3 petals
  • Buttercup, wild rose, hibiscus: 5 petals
  • Cosmos: 8 petals
  • Corn marigold, cineraria, ragwort: 13 petals
  • Asters: 21 petals
  • Daisy petals: can be 13, 21, 34, 55, or 89
• Sunflower spirals:
  • Counterclockwise and clockwise spirals typically correspond to consecutive Fibonacci numbers, e.g., 21 CCW and 34 CW, or 34 CCW and 55 CW.
  • This pattern holds across many sunflowers.
• Pineapples:
  • Spirals formed by hexagonal nubs: 8 spirals left, 13 spirals right, and 21 parallel rows of nubs.
• Other natural occurrences:
  • Growth patterns in plant and vegetable branches, vegetable spirals, seeds (pine nuts) spiraling right/left with counts 8 and 13, respectively.
  • Spiral patterns observed in broccoli, snail shells, human bone, dolphins, hurricane shapes, and galaxy structures.
• Important takeaway: many natural patterns exhibit numbers from the Fibonacci sequence and related Fibonacci spirals, suggesting an underlying structural harmony, not mere coincidence.

The Golden Rectangle and the Golden Spiral

• The golden rectangle: a rectangle whose side lengths are in the Golden Ratio, i.e., the ratio of length to width is $\phi \approx 1.618…$
• Construction steps for a golden rectangle (as given in Activity 1.4):
  1. Construct a unit square.
  2. Draw a line from the midpoint M of one side to the opposite corner O; denote this segment as $P = MO$.
  3. With center M and radius P, draw an arc from O to the extension of the side past M; where the arc intersects the extension is the corner point Q of the golden rectangle.
  4. Complete the golden rectangle and verify the ratio length/width approximately equals $\phi\,.$
• The golden spiral:
  • A logarithmic spiral whose radius changes by a factor of the golden ratio upon each quarter turn (90°, or $\frac{\pi}{2}$ radians).
  • The Parthenon façade is often cited as fitting a golden spiral pattern, aligning with the golden rectangle/spiral concept.
  • The spiral center is the accumulation point of the golden rectangles; diagonals mark this center in illustrative figures.

Golden Ratio in Humans and Art

• The golden ratio is often associated with perceived beauty; the term “divine proportion” is used.
• Human facial and limb proportions cited as approximately following golden ratio-guided measurements:
  • (a) Center of pupil : bottom of teeth : bottom of chin
  • (b) Outer and inner edge of eye : center of nose
  • (c) Outer edges of lips : upper ridges of lips
  • (d) Width of center tooth : width of second tooth
  • (e) Width of eye : width of iris
• Forearm to hand proportion also yields a value near the golden ratio, with many human body proportions approximating $\phi$ in different measurements.

Activity Highlights and Practical Applications

• Activity 1.1: Map of Mathematics
  • Watch video on how mathematics is summarized in a single map, showing connections between pure and applied mathematics.
  • Task: write 4–5 sentences on the discussion board answering: What is math? Where is math? Who uses math?
  • Submission: face-to-face with instructor, rubric evaluates keywords, clarity, and relevance.
• Activity 1.2: Science Documentary
  • Watch video on how humankind explored mathematics across centuries; discuss whether math is an invention or a discovery.
  • Rubric focuses on content, description/grammar, and relevance.
• Activity 1.3: Fibonacci Numbers Video
  • Short video (10–15 minutes) about Fibonacci numbers in nature to reinforce concepts.
• Activity 1.4: The Golden Rectangle
  • Construct a golden rectangle using the steps above; verify the ratio numerically.
• Activity 1.5: Face Analysis for Module 1
  • Face-beauty analysis activity: for women, measure length/width of face; for men, measure shoulder circumference and waist size; compute ratios; use an online calculator to analyze facial features.
  • Instructions emphasize consistency in measurement units (same metric system) for accuracy.
  • Submission: image/document to LMS/Drive as a pdf; file-naming convention provided.

Historical Context and References

• Leonardo of Pisa (Fibonacci) biography:
  • Lived circa 1170–1250; regarded as a premier European medieval mathematician.
  • Traveled to Africa and parts of Asia and interacted with mathematicians there.
• Liber Abaci (Book of Calculation, 1202):
  • Introduced Hindu-Arabic numerals to Europe; promoted the decimal system and arithmetic using base-10 numerals.
• Fibonacci sequence origin:
  • Earlier descriptions appear in Indian mathematics; the sequence appears in his rabbit problem.
• References cited in the module (selected):
  • Baltazar, Ragasa, Evangelista. Mathematics in the Modern World. C & E Publishing, 2024.
  • Sobecki et al. Math in Our World, 4th Edition. McGraw-Hill Education, 2019.
  • Lecture notes in Fibonacci Sequence.
  • J. Chasnov. Fibonacci Numbers and The Golden Ratio. Creative Commons Attribution 3.0 Hong Kong License, 2016.

Summary of Key Formulas and Concepts (for quick reference)

• Fibonacci sequence: $F<em>n = F</em>{n-1} + F<em>{n-2},\, F</em>1 = 1,\ F_2 = 1.$
• Initial terms: $F<em>1 = 1, F</em>2 = 1, F<em>3 = 2, F</em>4 = 3, F<em>5 = 5, F</em>6 = 8, F<em>7 = 13, F</em>8 = 21, F<em>{9} = 34, F</em>{10} = 55, \dots$
• Golden ratio: $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618…$
• Convergence: $\lim<em>{n\to\infty} \frac{F</em>{n+1}}{F_n} = \phi.$
• Fibonacci/rabbits connection example: $\frac{F<em>6}{F</em>5} = \frac{8}{5} = 1.6.$
• Golden rectangle property: length/width = $\phi \approx 1.618.$
• Golden spiral property: radius multiplies by the golden ratio after every $\frac{\pi}{2}$ radians turn.

Note on symbols

• Golden ratio symbol is typically $\phi$ (not to be confused with other symbols in texts).

End of Notes