Fibonacci Mystery - Numberphile

Numberphile
18 Sept 201309:48
EducationalLearning
32 Likes 10 Comments

TLDRIn this Numberphile video, the host explores the musical application of the Fibonacci sequence, focusing on the Pisano period—a cyclic pattern of remainders when Fibonacci numbers are divided by a certain integer. Composer Alan Stewart's creative process of translating these remainders into musical notes is discussed. The video delves into the mathematical properties of the Pisano period, including the unique pattern that emerges every 16 numbers when dividing by 7, and touches on the historical discovery by mathematician Lagrange in 1774. The host also corrects a mistake made during the live demonstration, emphasizing the importance of accuracy in mathematical exploration.

Takeaways
  • 🎵 Alan Stewart, the composer for Numberphile, created a musical piece inspired by the Fibonacci sequence by using the remainders of Fibonacci numbers divided by 7.
  • 🔢 The Fibonacci sequence starts with 1, 1, and each subsequent number is the sum of the previous two numbers, e.g., 1+1=2, 1+2=3, 2+3=5, and so on.
  • 🎶 Alan transformed the remainders of the Fibonacci numbers when divided by 7 into musical notes, creating a repeating pattern in the music.
  • 🔄 The pattern of remainders repeats every 16 numbers, which is known as the Pisano period, named after Leonardo Pisano, also known as Fibonacci.
  • 📉 Pisano periods are cyclic patterns that occur when Fibonacci numbers are divided by a certain integer, and the length of the period varies with the divisor.
  • 🌀 The video demonstrates that Pisano periods can be observed with different divisors, such as 5, where the period is 20, and the pattern includes zeros indicating exact divisibility.
  • 👀 The video points out that zeros in the Pisano period correspond to Fibonacci numbers that are exactly divisible by the divisor.
  • 🤔 A mathematical result highlighted is that an nth Fibonacci number divides another Fibonacci number Fm if and only if n divides m.
  • 📊 The Pisano period was first discovered by mathematician Joseph-Louis Lagrange in 1774 while examining patterns in the last digits of Fibonacci numbers when divided by 10.
  • 🔢 When dividing by powers of 10, the Pisano period lengthens, such as 60 for the last digit, 300 for the last two digits, and 1,500 for the last three digits.
  • 🔄 The script explains that the remainders in the Pisano period can be calculated by adding the previous two remainders, similar to the Fibonacci sequence itself.
  • 🔄 The sequence of remainders will always return to the start after a 0 is followed by a 1, which is a key trigger point in the Pisano period.
Q & A

  • What is the Fibonacci sequence and how does it start?

    -The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 1 and 1. It continues with 2 (1+1), 3 (1+2), 5 (2+3), and so on.

  • What was Alan Stewart's approach to incorporating the Fibonacci sequence into music?

    -Alan Stewart divided the Fibonacci numbers by 7 and used the remainders to create musical notes. He noticed a repeating pattern in the remainders which he later identified as a Pisano period.

  • What is a Pisano period?

    -A Pisano period is the sequence of remainders obtained when Fibonacci numbers are divided by a certain integer. It is named after Leonardo Pisano, also known as Fibonacci.

  • Why did Alan choose the number 7 for his musical composition?

    -Alan chose the number 7 because it helped him create a pattern in the remainders that could be translated into musical notes. However, any number would produce a Pisano period, just of a different length.

  • What is the significance of the number 0 in the Pisano period when dividing by 5?

    -The number 0 in the Pisano period indicates that the Fibonacci number is exactly divisible by the divisor in this case, 5. It is interesting because it only occurs when the divisor is a factor of the Fibonacci number.

  • What is the length of the Pisano period when dividing by 5?

    -The Pisano period for the divisor 5 has a length of 20. This means the pattern of remainders repeats every 20 Fibonacci numbers.

  • What mathematical result was mentioned regarding the divisibility of Fibonacci numbers?

    -The result states that an nth Fibonacci number divides another Fibonacci number Fm if and only if n divides m, where n and m are the indices of the Fibonacci numbers.

  • What did Lagrange discover in relation to the Pisano period?

    -Lagrange discovered the Pisano period concept in 1774 when he observed patterns in the remainders of Fibonacci numbers when divided by 10, which had a period of 60.

  • How can one efficiently calculate the Pisano period without writing out the entire Fibonacci sequence?

    -One can calculate the Pisano period by simply adding the previous two remainders to get the next one, similar to the Fibonacci sequence itself, without having to compute the entire sequence.

  • What is the significance of the sequence 0, 1 in the Pisano period?

    -The sequence 0, 1 acts as a reset point in the Pisano period, causing the pattern to restart, similar to how the Fibonacci sequence begins after reaching this pair of numbers.

  • Is there a general formula to determine the length of the Pisano period for any divisor?

    -No, there is no known general formula to determine the length of the Pisano period for any given divisor. Each divisor will produce a unique period length.

Outlines
00:00
🎵 Musical Fibonacci Sequence Exploration

The speaker begins by addressing a previous Numberphile video featuring composer Alan Stewart, who incorporated the Fibonacci sequence into a musical composition. Stewart divided Fibonacci numbers by 7 and used the remainders to create musical notes, observing a repeating pattern every 16 numbers, known as the Pisano period. This period is named after Leonardo Pisano, also known as Fibonacci. The speaker aims to explain the phenomenon and its mathematical significance, highlighting the cyclical nature of the remainders when Fibonacci numbers are divided by any number, with the length of the cycle varying accordingly.

05:04
🔢 Fibonacci Divisibility and Pisano Periods

In the second paragraph, the speaker delves deeper into the mathematics behind the Pisano period, discussing Fibonacci numbers' divisibility properties. They explain that if the nth Fibonacci number divides another Fibonacci number Fm, it is only when n divides m. The speaker provides examples to illustrate this rule, such as every fifth Fibonacci number being divisible by 5. They also mention that the Pisano period was first discovered by Lagrange in 1774 while examining patterns in division by 10, which has a period of 60. The speaker further explains that the period's length depends on the divisor and that there is no general formula for its duration. They demonstrate how the remainders can be calculated using the Fibonacci sequence's additive property, pointing out that a sequence of 0 and 1 acts as a reset to the beginning of the pattern.

Mindmap
Keywords
💡Fibonacci sequence
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. In the context of the video, the sequence is integral to the theme as it is used to generate a pattern of remainders when divided by a number, which is then translated into musical notes. The script describes the sequence's progression starting with 1, 1, 2, 3, 5, and so on.
💡Pisano period
The Pisano period is the sequence of remainders obtained when Fibonacci numbers are divided by a certain integer. Named after Leonardo Pisano, who is also known as Fibonacci, this concept is central to the video's exploration of patterns in mathematics and music. The script explains that Alan noticed a repeating pattern every 16 numbers when dividing Fibonacci numbers by 7, which is an example of a Pisano period.
💡Musical notes
In the video, the concept of musical notes is used to represent the remainders obtained from dividing Fibonacci numbers by 7. This creative application of mathematics to music is a key element of the video's narrative, showing how mathematical patterns can be translated into artistic expression. Alan's process of turning remainders into musical notes is a direct example from the script.
💡Remainder
A remainder is what is left over after division of one number by another. In the script, remainders are used to explore the Pisano period by examining the leftover values when Fibonacci numbers are divided by 7. The video uses remainders to demonstrate the cyclical nature of the sequence and how it can be applied to create music.
💡Divisibility
Divisibility in the video refers to the property of a number being evenly divisible by another without a remainder. The script discusses the divisibility of Fibonacci numbers by specific integers, such as 5 or 8, and how this relates to the Pisano period and the occurrence of zeros in the sequence.
💡Lagrange
In the video, Lagrange is mentioned as the mathematician who first discovered the Pisano period in 1774. His work focused on patterns observed when dividing Fibonacci numbers by 10, which led to the understanding of the 60-cycle pattern in the last digits of the sequence. The script uses Lagrange's discovery to illustrate the historical context and significance of the Pisano period.
💡Pattern
A pattern in the video refers to the repeating sequence of numbers or events. The script describes how the remainders of Fibonacci numbers when divided by different integers form recognizable patterns, such as the 16-number cycle when divided by 7 or the 20-number cycle when divided by 5. These patterns are crucial to understanding the Pisano period.
💡Zero
In the context of the video, a zero indicates a point in the sequence where a Fibonacci number is exactly divisible by the divisor. The script notes the significance of zeros in the Pisano period, as they mark points where the sequence resets, and they can only occur in specific quantities within a period.
💡Last digits
The term 'last digits' in the script refers to the units place of a number when considering its divisibility by a certain integer. The video discusses how the last digits of Fibonacci numbers form a pattern when divided by powers of 10, such as 10, 100, and 1000, leading to periods of 60, 300, and 1500, respectively.
💡Wrapping back
Wrapping back in the video is a concept used to describe how the sequence of remainders resets after reaching the divisor's value. When the sum of two remainders exceeds the divisor, it 'wraps back' to 0, continuing the cycle. The script uses this concept to explain the cyclical nature of the Pisano period and its relation to the Fibonacci sequence.
Highlights

Introduction of a video response to a previous Numberphile video featuring Alan Stewart.

Alan Stewart's challenge involving the Fibonacci sequence in music composition.

Explanation of the Fibonacci sequence and its mathematical significance.

Alan's method of using Fibonacci numbers modulo 7 to create musical notes.

Discovery of a repeating pattern in remainders when dividing Fibonacci numbers by 7, known as the Pisano period.

The Pisano period is named after Leonardo Pisano, also known as Fibonacci.

Different Pisano periods can be observed with different divisors.

Example calculation of Pisano periods with division by 5, revealing a 20-cycle pattern.

Mistake made during the live demonstration and its correction regarding the Pisano period for division by 5.

Explanation of the mathematical result that Pisano periods can only have one, two, or four zeros.

Connection between the Pisano period and the divisibility of Fibonacci numbers by their index.

Lagrange's discovery of Pisano periods when dividing Fibonacci numbers by 10, with a 60-cycle pattern.

Generalization of Pisano periods for different divisors and their respective cycle lengths.

Introduction of a property to calculate Pisano periods without writing out the entire Fibonacci sequence.

Demonstration of how the remainders in Pisano periods follow the Fibonacci sequence pattern.

The significance of the 0, 1 sequence in triggering a restart in Pisano period patterns.

Acknowledgment that there is no general formula for the length of Pisano periods.

Conclusion of the video with a summary of the findings on Pisano periods and Fibonacci numbers.

Transcripts

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Fibonacci SequencePisano PeriodMathematicsMusic CompositionNumberphileAlan StewartBrady HaranSequence PatternsDivide RemainderLagrangeEducational