Fibonacci Sequence

The mathematical pattern that appears throughout nature, art, science, and financial markets

Introduction

Mathematical Theory

Golden Ratio

In Nature

Trading & Finance

Applications

FAQ

What is the Fibonacci Sequence?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The sequence begins:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377...

This sequence is named after Leonardo Fibonacci, an Italian mathematician from the Middle Ages who introduced it to Western European mathematics in his book Liber Abaci (1202).

Did you know? November 23rd is celebrated as Fibonacci Day because when written as 11/23, it represents the first four numbers in the sequence!

Historical Background

Fibonacci introduced the sequence to model rabbit population growth. He considered the growth of an idealized rabbit population:

Start with one pair of newborn rabbits
After the first month, they mature
In the second month, they produce a new pair
Every subsequent month, they produce another pair
New pairs mature and reproduce after one month

The resulting population growth follows the Fibonacci sequence.

The Fibonacci sequence actually appears in Indian mathematics much earlier, in connection with Sanskrit prosody. Knowledge of the sequence was expressed as early as Pingala (c. 450 BC–200 BC) in the context of enumerating patterns of Sanskrit poetry.

Fibonacci Calculator

Fibonacci(10) = 55

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First 15 Fibonacci Numbers

Mathematical Theory

Recursive Definition

The Fibonacci sequence is formally defined by the recurrence relation:

F₀ = 0, F₁ = 1,
Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 2

Binet's Formula

Although defined recursively, Fibonacci numbers have a closed-form solution known as Binet's formula:

Fₙ = (φⁿ - (-φ)⁻ⁿ) / √5
where φ = (1 + √5)/2 ≈ 1.61803

This formula demonstrates the connection between Fibonacci numbers and the golden ratio.

Mathematical Properties

Cassini's Identity

Fₙ₊₁·Fₙ₋₁ - Fₙ² = (-1)ⁿ

Sum of Squares

F₁² + F₂² + ⋯ + Fₙ² = Fₙ·Fₙ₊₁

GCD Property

gcd(Fₘ, Fₙ) = F_gcd(m,n)

Every 3rd Number

Every third Fibonacci number is even, and every fourth number is a multiple of 3.

The Golden Ratio

The golden ratio (φ ≈ 1.6180339887) is an irrational number with fascinating mathematical properties. It has been studied since antiquity for its aesthetic properties and appears frequently in art, architecture, and nature.

Connection to Fibonacci

The ratio of consecutive Fibonacci numbers approaches the golden ratio:

lim (n→∞) Fₙ/Fₙ₋₁ = φ = (1 + √5)/2

When we take any two successive Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1.618034... In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation.

Convergence to Golden Ratio

The Golden Spiral

A Fibonacci spiral approximates the golden spiral, a logarithmic spiral whose growth factor is φ. This spiral appears frequently in nature.

Fibonacci in Nature

Often called 'nature's secret code,' the Fibonacci sequence is found all over nature, including in the spirals of storm systems like tornadoes and hurricanes. The golden ratio that's inherent in the Fibonacci sequence is also seen in plants like the spiral aloe, the spiral shape of a nautilus shell, facial proportions, and the shapes of eggs and galaxies.

Phyllotaxis

The arrangement of leaves on a stem often follows Fibonacci numbers to maximize sun exposure.

Sunflower Seeds

The seeds in a sunflower head form spirals with Fibonacci numbers of spirals in each direction.

Storm Systems

Hurricanes and tornadoes often display Fibonacci spirals in their structure.

Why Fibonacci in Nature?

These patterns emerge because they represent the most efficient way to pack structures (like seeds) while maximizing exposure to light and nutrients. The golden angle (≈137.5°), derived from the golden ratio, minimizes overlap and allows for optimal growth.

Fibonacci in Trading & Finance

Trading Disclaimer: Trading involves significant risk. The information provided here is for educational purposes only and should not be considered financial advice.

Fibonacci Retracement in Trading

Fibonacci retracements and extensions are used by traders to identify possible support and resistance levels in situations when such levels are difficult to identify. Traders use them to determine critical points where an asset's price momentum is likely to reverse.

Fibonacci Trading Calculator

High Price:

Low Price:

Key Fibonacci Retracement Levels

Key Fibonacci retracement levels, derived from mathematical relationships within the sequence, include 23.6%, 38.2%, 50%, 61.8%, and 78.6%.

23.6%

Shallow pullback

38.2%

Common retracement

50%

Psychological level

61.8%

Golden ratio - Key level

78.6%

Deep retracement

Applications in Different Markets

You can apply Fibonacci retracements to forex, stocks, commodities and even cryptocurrencies. Since they rely on mathematical ratios rather than specific asset characteristics, they're versatile across different markets.

Stock Market: Used to identify potential reversal points in trending stocks
Cryptocurrency: Traders use Fibonacci retracement and extension levels to forecast price movements and determine optimal entry and exit points in the highly volatile cryptocurrency market.
Forex Trading: Applied to currency pairs to spot support and resistance levels
Commodities: Used in gold, oil, and other commodity markets

How Traders Use Fibonacci

Identify a major high and low in the market. Draw the Fibonacci retracement tool from low to high for an uptrend or high to low for a downtrend. The Fibonacci levels will highlight potential areas where the price may stop or bounce.
Wait for price to retrace to a key Fibonacci level
Look for confirmation signals (candlestick patterns, volume, other indicators)
Enter trade with stop loss below/above the Fibonacci level
Set profit targets using Fibonacci extensions

Applications

Computer Science & Programming

Fibonacci Heaps: Data structure for priority queues with efficient operations
Dynamic Programming: Generating a Fibonacci sequence is an extremely popular programming question
Algorithm Analysis: Used to analyze the time complexity of algorithms
LeetCode Problems: Common in coding interviews and competitive programming

Art and Architecture

The golden ratio has been used in design for centuries. Examples include:

The Parthenon in Athens
Leonardo da Vinci's Vitruvian Man
Salvador Dalí's The Sacrament of the Last Supper
Modern logo designs (Twitter, Apple, Pepsi)

Modern Technology

UI/UX Design - Golden ratio for layouts
Data compression algorithms
Computer graphics and animation
Blockchain and cryptocurrency algorithms

Frequently Asked Questions

What are the first 20 Fibonacci numbers?

The first 20 Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181

How is Fibonacci used in crypto trading?

The Fibonacci Retracement is a very popular tool used by many technical traders. It identifies strategic points for placing transactions, target prices, or stop-losses. After significant price fluctuations, the new support and resistance levels are often at or near these lines.

Why is 61.8% so important in Fibonacci trading?

The 61.8% level is known as the "Golden Ratio" and is considered the most important Fibonacci retracement level. It often acts as a strong support or resistance level where price reversals are more likely to occur.

Can beginners use Fibonacci in trading?

Yes, but it's important to practice with demo accounts first. Fibonacci tools are available on most trading platforms and are relatively easy to apply, but understanding market context and combining with other indicators is crucial for success.

Where does the Fibonacci sequence appear in everyday life?

Beyond nature, you can find Fibonacci patterns in music composition, photography (rule of thirds), architecture, human body proportions, and even in social media algorithms and data structures in computer science.

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