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Math
Fibonacci Sequence and the Golden Spiral
Numbers, Squares, and a Famous Curve
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The Fibonacci sequence is a simple number pattern that appears in mathematics, art, architecture, and many natural growth patterns. It begins with 1 and 1, and each new term is found by adding the two terms before it. When squares with Fibonacci side lengths are arranged together, they form a rectangle that can be used to draw a smooth spiral. This visual link helps students see how a number sequence can create a geometric shape.
Key Facts
- Fibonacci rule: F_n = F_(n - 1) + F_(n - 2)
- Common starting terms: 1, 1, 2, 3, 5, 8, 13, 21
- Golden ratio: phi = (1 + sqrt(5)) / 2 ≈ 1.618
- Ratios of consecutive Fibonacci numbers approach phi: F_(n + 1) / F_n → phi
- Area of a Fibonacci square with side length s is A = s^2
- A golden spiral grows by a factor of about phi every quarter turn
Vocabulary
- Fibonacci sequence
- A sequence of numbers in which each term is the sum of the two previous terms.
- Golden ratio
- An irrational number approximately equal to 1.618 that appears in many geometric proportions.
- Golden spiral
- A spiral that grows outward by a constant factor related to the golden ratio.
- Fibonacci square
- A square whose side length is a Fibonacci number, often used to build a spiral diagram.
- Consecutive ratio
- The ratio found by dividing one term in a sequence by the term immediately before it.
Common Mistakes to Avoid
- Adding a constant difference between terms is wrong because the Fibonacci sequence is not arithmetic. Each new term comes from adding the two previous terms, not from adding the same number each time.
- Assuming every spiral in nature is exactly a golden spiral is wrong because many natural spirals are only approximate. Biological growth is affected by genetics, environment, and physical constraints.
- Using phi = 1.6 as an exact value is wrong because the golden ratio is irrational. The value 1.6 is only a rough estimate and can cause noticeable error in calculations.
- Drawing equal-sized squares for the spiral is wrong because the square sizes must follow Fibonacci side lengths. The changing square sizes are what make the spiral grow.
Practice Questions
- 1 Starting with 1, 1, write the next six terms of the Fibonacci sequence.
- 2 Compute the ratios 13/8, 21/13, and 34/21. Which ratio is closest to phi ≈ 1.618?
- 3 Explain why arranging Fibonacci squares can create a spiral that looks similar to growth patterns in shells, plants, or design layouts.