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Fibonacci = Pythagoras: Help save a beautiful discovery from oblivion

Mathologer・2 minutes read

The connection between the Pythagorean theorem and the Fibonacci sequence reveals surprising insights, leading to the discovery of new Pythagorean triples and geometric relationships within triangles. Manipulating the Fibonacci sequence generates a Pythagorean triple tree that showcases the connection between Fibonacci and Pythagorean concepts, uncovering unique mathematical properties and relationships.

Insights

  • The Fibonacci sequence, when analyzed in pairs, reveals a connection to the Pythagorean theorem, specifically the 3, 4, 5 triangle, providing insights into the incircle and excircles of this triangle.
  • Manipulating the Fibonacci sequence leads to the creation of a Pythagorean triple tree, generating all primitive Pythagorean triples once and highlighting a natural link between Fibonacci and Pythagorean concepts, showcasing a growth mechanism visually and geometrically.

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Recent questions

  • What is the connection between the Pythagorean theorem and the Fibonacci sequence?

    The Pythagorean theorem involves integer squares creating right-angled triangles, while the Fibonacci sequence is a series where consecutive terms add up to the next term. By examining pairs of Fibonacci numbers, a link is found to the Pythagorean theorem, specifically the 3, 4, 5 triangle, showcasing unexpected connections.

  • How do new Fibonacci numbers lead to the discovery of additional Pythagorean triples?

    Introducing new Fibonacci numbers like 5 and 8 leads to the discovery of additional Pythagorean triples, such as 5, 12, 13 and 16, 30, 34. Manipulating the Fibonacci sequence in a specific manner generates a Pythagorean triple tree, showcasing a natural connection between Fibonacci and Pythagorean concepts.

  • What unique property does the tree of fractions created from the Fibonacci sequence possess?

    The tree of fractions from the Fibonacci sequence reveals a unique property where every reduced positive fraction less than 1 appears exactly once, hinting at deeper mathematical connections. It contains every rational number from 0 to 1 exactly once, surprising due to the density of rational numbers in that interval.

  • How are the children of the Pythagorean triple tree related to the parent triangles?

    The children of the Pythagorean triple tree are geometrically related to the parent triangles, showcasing specific measurements and connections. Scaling triangles by a factor of four reveals the right-angled children triangles, illustrating a growth mechanism purely in geometry.

  • What real-world applications do Pythagorean triples have?

    Real-world applications of Pythagorean triples include a method for checking right angles in tiling and a mathematical gift puzzle. Climbing up the tree reveals Pythagorean triples with special properties, such as the first two numbers differing by 1, leading to isosceles triangles.

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Summary

00:00

Fibonacci and Pythagorean: Unexpected Mathematical Connections

  • In 2007, a connection was discovered between two mathematical concepts: the Pythagorean theorem and the Fibonacci sequence.
  • The Pythagorean theorem involves integer squares adding up to another integer square, creating right-angled triangles with integer sides like the 3, 4, 5 triangle.
  • The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, etc., is a sequence where consecutive terms add up to the next term.
  • By bending the Fibonacci sequence and examining pairs of numbers, a link is found to the Pythagorean theorem, specifically the 3, 4, 5 triangle.
  • Picking pairs of Fibonacci numbers reveals insights about the incircle and excircles of the 3, 4, 5 triangle, showcasing unexpected connections.
  • Six ways of picking pairs of Fibonacci numbers provide significant information about the 3, 4, 5 triangle, including radius values and the area of the triangle.
  • Introducing new Fibonacci numbers like 5 and 8 leads to the discovery of additional Pythagorean triples, such as 5, 12, 13 and 16, 30, 34.
  • The Fibonacci sequence, when manipulated in a specific manner, generates a Pythagorean triple tree, showcasing a natural connection between Fibonacci and Pythagorean concepts.
  • The Pythagorean triple tree, derived from Fibonacci numbers, produces all primitive Pythagorean triples exactly once, capturing the essence of Pythagorean relationships.
  • The tree of fractions created from the Fibonacci sequence reveals a unique property where every reduced positive fraction less than 1 appears exactly once, hinting at deeper mathematical connections.

12:59

"Exploring Pythagorean Triples and Geometric Connections"

  • The tree contains the reduced fraction 1/2, but not 2/4 or 3/6, only reduced fractions less than one.
  • The tree encompasses every rational number from 0 to 1 exactly once, surprising due to the density of rational numbers in that interval.
  • The Pythagorean triple tree showcases every primitive Pythagorean triple once, visually capturing a simple growth mechanism.
  • The children of the Pythagorean triple tree are geometrically related to the parent, with specific measurements and connections.
  • The Feuerbach circle, discovered in 1822, touches the incircle and all three excircles of a triangle, revealing magical properties.
  • Scaling triangles by a factor of four reveals the right-angled children triangles, showcasing a growth mechanism purely in geometry.
  • Connecting the centers of the incircle and Feuerbach circle of a triangle leads to a right-angled triangle, revealing connections to parent triangles.
  • Real-world applications of Pythagorean triples include a method for checking right angles in tiling and a mathematical gift puzzle.
  • Climbing up the tree reveals Pythagorean triples with special properties, such as the first two numbers differing by 1, leading to isosceles triangles.
  • Isosceles triangles with integer sides do not exist due to the irrationality of the square root of 2, showcasing a mathematical concept.

28:07

"Pythagorean Triples Approximate Root 2"

  • Special triangles approximate square root 2 using Pythagorean triples
  • Example: 169 divided by 119 or 120 approximates root 2
  • Fermat's family of triples found in the middle of the tree
  • Triangles become thinner and thinner as they ascend the tree
  • Pythagoras' family: largest number and next largest differ by 1
  • Babylonians knew about Pythagorean triples before Pythagoras
  • Plato's family: largest number is two larger than one of the smaller numbers
  • Euclid's Elements from 300 BC explains primitive Pythagorean triples
  • Fibonacci sequence produces Pythagorean triples, with every second number being a Fibonacci number
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