The Golden Ratio (why it is so irrational) - Numberphile
Numberphile・18 minutes read
The text explores the connection between the golden ratio and flower seed placement, showcasing how different fractions of turns result in unique patterns of spokes or spirals. It also delves into the concept of irrational numbers, showcasing how the golden ratio, Phi, is the most irrational number due to its continued fraction construction, leading to Fibonacci numbers and sunflower seed arrangements.
Insights
- Different fractions of turns create varying patterns of seed placement in flowers, with rational numbers producing spokes and irrational numbers leading to spiral arrangements.
- The Golden Ratio, Phi, represented by (1 + √5) / 2 or (1 - √5) / 2, results in a unique pattern of spokes crossing in both directions, making it the most efficient fraction of a turn for flower seed placement and generating Fibonacci numbers akin to sunflower seed arrangements.
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Recent questions
What is the golden ratio?
The golden ratio is a famous number often associated with mystique.
How does the golden ratio relate to flowers?
The connection between the golden ratio and flowers is explored through a mathematical model.
How are seeds in a flower represented in a mathematical model?
Seeds in a flower are represented by blobs, with the placement determined by turning a fraction of a turn.
What is the significance of the square root of 2 in flower seed placement?
The square root of 2 demonstrates irrational behavior in seed placement.
How does the golden ratio influence Fibonacci numbers?
The golden ratio is considered the most efficient fraction of a turn for flower seed placement, leading to Fibonacci numbers and resembling sunflower seed arrangements.
Related videos
Summary
00:00
"Golden Ratio and Flower Seed Patterns"
- The golden ratio is a famous number often associated with mystique.
- The connection between the golden ratio and flowers is explored through a mathematical model.
- Seeds in a flower are represented by blobs, with the placement determined by turning a fraction of a turn.
- Different fractions of turns result in varying patterns of seed placement, creating spokes.
- The denominator of the fraction of a turn controls the number of spokes in the flower.
- Rational numbers produce spokes, while irrational numbers lead to spiral patterns in the flower model.
- The square root of 2 and 1/Pi demonstrate irrational behavior in seed placement.
- The golden ratio, represented by the square root of 5 minus 1 over 2, results in a unique pattern of spokes crossing in both directions.
- The golden ratio is considered the most efficient fraction of a turn for flower seed placement, leading to Fibonacci numbers and resembling sunflower seed arrangements.
- A continued fraction representation of Pi showcases how rational approximations can be derived from irrational numbers.
08:53
"Golden Ratio: Irrational Number with Reciprocal Property"
- Pi can be well approximated by a rational number early on due to its continued fraction representation.
- The most irrational number would have a continued fraction without large numbers, making it badly approximated when truncated.
- The Golden Ratio, Phi, is the most irrational number due to its continued fraction construction.
- Phi can be calculated as (1 + √5) / 2 or (1 - √5) / 2, resulting in 1.6180339 or -0.6180339, respectively.
- The Golden Ratio's property allows for reciprocals of itself, leading to different but related numbers.
