The Most Beautiful Equation in Mathematics, Ranked

Voting rules: Choose the equation you think is the most beautiful!

Updated on May 19, 2024 06:18

Mathematics holds an array of equations, each serving as a fundamental building block in understanding our universe. Yet, amidst the formulas and complexities, some equations strike a remarkable balance of elegance, simplicity, and profound insight. These standout equations often spark debates among enthusiasts and scholars alike regarding their aesthetic and functional virtues. Here, your opinions help shape the collective perception of mathematical beauty. By casting your vote on what you consider the most beautiful equation, you contribute to a dynamic, community-driven ranking that reflects a diverse array of perspectives. This engaging process not only highlights popular favorites but also introduces lesser-known equations that deserve recognition.

What Is the Most Beautiful Equation in Mathematics?

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AstroOgier · CC BY-SA 4.0

Euler's identity

Leonhard Euler

Euler's identity is e^(iπ) + 1 = 0. This equation is often called the most beautiful equation in mathematics because it elegantly combines five of the most important mathematical constants, e, i, π, 1, and 0, into one equation, and it connects them in a way that seems almost magical.
Euler's identity is a mathematical equation that connects five of the most fundamental mathematical constants: the number 0, the number 1, the irrational number e (the base of the natural logarithm), pi (the ratio of a circle's circumference to its diameter), and the imaginary unit i (which is defined as the square root of -1). It is considered one of the most beautiful equations in mathematics because it combines these five constants using basic arithmetic operations and exponentiation.

Equation: e^(iπ) + 1 = 0

Symbols involved: e, π, i, 1, 0

Euler's number (e): approximately 2.71828

Pi (π): approximately 3.14159

Imaginary unit (i): defined as the square root of -1
Euler's identity in other rankings
- Position 1 of 10 in Most beautiful equation in Physics
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Wapcaplet · CC BY-SA 3.0

Pythagorean theorem

Pythagoras of Samos

Pythagorean theorem is a^2 + b^2 = c^2. This equation relates the sides of a right triangle, and it has been known for thousands of years. It is simple yet profound, and it has many applications in geometry, physics, and engineering.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Triangle type: Right-angled triangle

Formula: c^2 = a^2 + b^2

Variables: c: Hypotenuse, a, b: Other sides

Geometric interpretation: Relates the areas of squares built on each side of the right triangle

Applicability: Used in various fields like geometry, trigonometry, physics, engineering
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This W3C-unspecified vector image was created with Inkscape . · CC BY-SA 3.0

Maxwell's equations

James Clerk Maxwell

These four equations describe the behavior of electromagnetic fields and have revolutionized our understanding of light, electricity, and magnetism. They are complex and elegant, and they have led to many technological advances, from radio and TV to cell phones and computers.
Maxwell's equations are a set of fundamental equations in electromagnetism that describe the behavior of electric and magnetic fields. They were formulated by the Scottish physicist James Clerk Maxwell in the 19th century.

Gauss's law for electric fields: Relates electric fields to electric charges and their distributions.

Gauss's law for magnetic fields: Relates magnetic fields to magnetic charges and their distributions.

Faraday's law of electromagnetic induction: Describes how changing magnetic fields induce electric fields.

Ampère's law with Maxwell's addition: Relates magnetic fields to electric currents and changing electric fields.

Differential form: Maxwell's equations are often expressed in their differential form, which represents how the fields change continuously in space and time.
Maxwell's equations in other rankings
- Position 2 of 10 in Most beautiful equation in Physics
- Position 6 of 10 in Most popular equation
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Navier-Stokes equation

This equation describes the motion of fluids, and it is one of the most important equations in physics and engineering. It is notoriously difficult to solve, but it has many applications, from designing airplanes and rockets to predicting weather patterns and ocean currents.
The Navier-Stokes equation is a set of partial differential equations that describe the motion of viscous fluid substances. It is widely used in fluid dynamics and is considered one of the most important and beautiful equations in mathematics.

Type: Partial differential equations

Domain: Fluid dynamics

Variables: Velocity, pressure, density, viscosity

Equation form: Conservation of momentum and mass

Nonlinearity: Nonlinear equations
Navier-Stokes equation in other rankings
- Position 1 of 10 in Most difficult equation to solve
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The golden ratio

The golden ratio is φ = (1 + √5)/2. This number is often considered the most beautiful ratio in mathematics because it appears in many natural and artistic phenomena, from the spirals of seashells and galaxies to the proportions of the Parthenon and the Mona Lisa. It has many interesting properties, and it has inspired many artists, architects, and mathematicians.
The golden ratio is a mathematical concept that represents a special number approximately equal to 1.6180339887. It is often denoted by the Greek letter phi (φ). The ratio follows a unique pattern where a number divided by its larger part is equal to the ratio itself. This creates a captivating aesthetic appeal and has been revered by artists, architects, and mathematicians throughout history.

Ratio representation: Approximately 1.6180339887

Mathematical representation: φ = (1 + sqrt(5)) / 2

Geometric representation: A rectangle with sides in the golden ratio is said to be visually pleasing.

Natural occurrence: The golden ratio appears in various natural phenomena, such as the spiral patterns of shells, the branching of trees, and the proportions of human faces and bodies.

Fibonacci sequence: The golden ratio is closely linked to the Fibonacci sequence, where each number is the sum of the two preceding numbers.
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The Black-Scholes equation

This equation is used to price options and other derivatives, and it has had a profound impact on finance and economics. It was developed by Fischer Black and Myron Scholes in the 1970s, and it won them the Nobel Prize in Economics in 1997. It is a beautiful and elegant equation that has revolutionized the way we think about risk and reward.
The Black-Scholes equation, also known as the Black-Scholes-Merton equation, is a partial differential equation used to model the pricing of financial derivatives, particularly European-style options. It provides a mathematical framework for determining the fair price of options based on various inputs such as the current price of the underlying asset, the strike price, the time to maturity, the risk-free interest rate, and the volatility of the underlying asset's returns.

Type: Partial differential equation

Application: Pricing of European-style options

Inputs: Underlying asset price, strike price, time to maturity, risk-free interest rate, volatility

Assumptions: Efficient markets, constant volatility, no dividends, risk-free rate constant over time

Formula: ∂V/∂t + (1/2)σ²S²∂²V/∂S² + rS∂V/∂S - rV = 0
The Black-Scholes equation in other rankings
- Position 7 of 10 in Most popular equation
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The Fourier transform

Jean-Baptiste Joseph Fourier

This equation is used to decompose a signal into its frequency components, and it has many applications in signal processing, image analysis, and data compression. It was developed by Joseph Fourier in the 19th century, and it is a beautiful and powerful tool that has transformed many fields of science and engineering.
The Fourier transform is a mathematical technique that decomposes a function into its constituent frequencies. It transforms a time-domain signal into its frequency-domain representation.

Domain: Transforms time-domain signals into frequency-domain signals

Linearity: It follows the principles of linearity

Orthogonality: The basis functions used in Fourier transform are orthogonal to each other

Inverse transform: It has an inverse transform that reconstructs the original signal

Convolution: It simplifies convolution operations in the frequency domain
The Fourier transform in other rankings
- Position 9 of 10 in Most beautiful equation in Physics
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The Schrödinger equation

Erwin Schrödinger

This equation describes the behavior of quantum systems, and it has revolutionized our understanding of atoms, molecules, and other small particles. It was developed by Erwin Schrödinger in the 1920s, and it is a beautiful and elegant equation that has led to many important discoveries in physics and chemistry.
The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function of a physical system evolves over time.

Type: Partial differential equation

Application: Quantum mechanics

Time evolution: Deterministic

Wave function: Describes physical state

Complex-valued: Yes
The Schrödinger equation in other rankings
- Position 3 of 10 in Most beautiful equation in Physics
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The Riemann hypothesis

Bernhard Riemann

This equation is one of the most famous unsolved problems in mathematics, and it has intrigued mathematicians for over a century. It relates to the distribution of prime numbers, and it has many important implications for number theory and cryptography. It was proposed by Bernhard Riemann in 1859, and it remains one of the most beautiful and mysterious equations in mathematics.
The Riemann hypothesis is a conjecture in mathematics formulated by the German mathematician Bernhard Riemann. It relates to the distribution of prime numbers and provides insights into the behavior of the Riemann zeta function.

Conjecture type: Mathematical conjecture

Field: Number theory

Importance: Among the most important unsolved problems in mathematics

Connection to prime numbers: Provides information about the distribution of prime numbers

Key concept: Riemann zeta function
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The wave equation

Jean le Rond d'Alembert

This equation describes the behavior of waves, from sound waves to light waves to ocean waves. It was developed by Jean le Rond d'Alembert and Leonard Euler in the 18th century, and it is a simple and elegant equation that has many applications in physics, engineering, and mathematics.
The wave equation describes how waves propagate through a medium. It is a second-order partial differential equation that relates the second derivative of a quantity (such as displacement or voltage) with respect to both time and space. The equation represents a fundamental concept in physics and has applications in various fields including acoustics, optics, and electromagnetism.

Mathematical form: ∂²u/∂t² = c²∇²u

Wave speed: c

Space dimensions: 1, 2, or 3

Time dimension: 1

Solution types: Plane waves, standing waves, spherical waves, etc.

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Discussion

Ranking factors for beautiful equation

Simplicity

A beautiful equation should ideally be concise, with few terms and symbols, yet elegantly convey a profound mathematical truth. The simpler an equation, the easier it is to understand and appreciate its beauty.
Symmetry

Symmetry in an equation often reflects a balance or harmony among its components, which can add to its aesthetic appeal. Symmetric equations can also reveal underlying connections and relationships in mathematics.
Depth

An equation is more likely to be considered beautiful if it has broad applications and deep implications across mathematics and other fields, such as physics, engineering, and computer science. Equations that unlock powerful results or have strong foundational significance are often celebrated for their beauty.
Generality

Equations that can be applied across a wide range of situations and encompass many specific cases are often considered more beautiful than those applicable in limited circumstances. Such equations may show a level of unification in mathematics and inspire further research.
Surprise

A beautiful equation may have an unexpected or counterintuitive result that challenges conventional wisdom or offers new insights. Such equations can invoke a sense of wonder or excitement for mathematicians and enthusiasts alike, adding to their appeal.
Historical context

The history and development of an equation can play a role in its perceived beauty. Equations that marked major breakthroughs in mathematical understanding or that have been pondered upon and contributed to by many great minds over time can carry a certain mystique that adds to their attractiveness.
Personal preferences

Ultimately, the beauty of an equation can be subjective and may vary from person to person based on individual interests, experiences, and perceptions. What one mathematician finds beautiful may not resonate as strongly with someone else.

About this ranking

This is a community-based ranking of the most beautiful equation in mathematics. We do our best to provide fair voting, but it is not intended to be exhaustive. So if you notice something or equation is missing, feel free to help improve the ranking!

Statistics

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234 votes
10 ranked items

Voting Rules

A participant may cast an up or down vote for each equation once every 24 hours. The rank of each equation is then calculated from the weighted sum of all up and down votes.

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More information on most beautiful equation in mathematics

Mathematics is a fascinating subject, and one of the things that makes it so interesting is the beauty of its equations. From simple formulas to complex equations, there is no denying the elegance and symmetry that can be found in mathematics. But what exactly makes an equation beautiful? Is it the simplicity of its form, the complexity of its solution, or something else entirely? Many mathematicians and scientists have pondered this question, and there is no definitive answer. However, there are some equations that are widely considered to be the most beautiful in mathematics. These equations are not only aesthetically pleasing, but they also have significant practical applications in fields such as physics, engineering, and economics. Some of the most beautiful equations in mathematics include Euler's identity, which relates the mathematical constants e, i, and pi, and the Pythagorean theorem, which describes the relationship between the sides of a right triangle. Other notable equations include Maxwell's equations, which describe the behavior of electric and magnetic fields, and the Schrödinger equation, which is used to describe the behavior of quantum particles. Despite the different opinions on what makes an equation beautiful, there is no denying the beauty and power of mathematics. Whether you are a mathematician, a scientist, or simply someone who appreciates the elegance of numbers, there is something undeniably captivating about the most beautiful equations in mathematics.