Mathematics
Functions, calculus, geometry, probability. 245 topics, from beginner to advanced.
Browse all 245 Mathematics topics →Featured
Visual showcases to start withLissajous Figures
Two perpendicular oscillations with related frequencies draw looping figures on oscilloscopes.
Spirograph Curves
A gear rolling inside a ring draws hypotrochoids — the maths behind the classic toy.
Lissajous Knots
Add a phase offset and the figure appears to rotate in three dimensions.
Euler’s Formula
Raising e to an imaginary power produces rotation: e^(iθ) = cos θ + i sin θ.
Roots of Unity
The n solutions of zⁿ = 1 are the corners of a regular n-gon on the unit circle.
Taylor Series
A smooth function is approximated near a point by a polynomial built from its derivatives.
Central Limit Theorem
Averages of many random draws form a bell curve, whatever the original distribution.
Random Walks
Each step left or right at random: typical distance grows like √n.
Estimating π with Monte Carlo
Throw random darts at a square; the fraction inside the quarter circle estimates π/4.
The Galton Board
Balls bouncing left or right through pegs pile into a bell curve.
The Ulam Spiral
Write integers in a square spiral and highlight the primes: diagonal patterns appear.
Harmonograph
Two damped pendulums drawing together make decaying spirograph patterns.
Popular
Classic topics most learners start withLinear Functions
A straight line with constant rate of change. The slope m tilts it and the intercept b shifts it vertically.
Quadratic Functions
Parabolas model projectiles and optimisation. The sign of a decides whether it opens up or down.
The Derivative as a Slope
Zoom in on a curve: the secant line settles into the tangent, whose slope is the derivative.
The Quadratic Formula
The roots of any quadratic come from one formula; the axis of symmetry sits halfway between them.
Pythagorean Theorem
Areas of the squares on the two legs add up to the square on the hypotenuse.
The Golden Ratio
Ratios of consecutive Fibonacci numbers converge to φ ≈ 1.618, the proportion of the golden spiral.
Pascal’s Triangle
Each entry is the sum of the two above. It holds binomial coefficients and many hidden patterns.
Fibonacci Numbers
Each term is the sum of the previous two; squares of those sizes tile into a spiral.
The Normal Distribution
The bell curve is described fully by its mean and standard deviation.
Bayes’ Theorem
Update probabilities with evidence. Even an accurate test gives mostly false alarms for rare conditions.
Compound Interest
Interest earns interest: balance grows exponentially with the compounding rate.
Prime Numbers and the Sieve
The Sieve of Eratosthenes crosses out multiples to leave only primes.
All 245 Mathematics topics
🌱 Beginner
- Linear Functions
- Slope as Rate of Change
- Absolute Value Functions
- Quadratic Functions
- Vertex Form of a Parabola
- Roots and the Discriminant
- Cubic Polynomials
- End Behaviour of Polynomials
- Rational Functions
- Square Root Functions
- Exponential Growth
- Exponential Decay
- Logarithms
- Euler’s Number e
- Inverse Functions
- Piecewise Functions
- Floor and Step Functions
- Even and Odd Functions
- Composition of Functions
- Systems of Linear Equations
- Logistic Growth
- Inverse Proportion
- Direct and Power Variation
- Arithmetic Sequences
- Geometric Sequences
- Triangular Numbers
- Recurrence Relations
- Circles
- Ellipses
- Parabolas as Conics
- Hyperbolas
- Lissajous Figures
- Archimedean Spiral
- Polar Roses
- Cardioid
- Lemniscate of Bernoulli
- Catenary
- Lissajous Knots
- Unit Circle
- The Tangent Function
- Radians
- Amplitude, Period and Phase
- Double-Angle Formulas
- Mean Value Theorem
- Area Under a Parabola
- Area Between Curves
- Determinants and Area
- Slope–Intercept from Two Points
- Point–Slope Form
- Parallel and Perpendicular Lines
- Linear Inequalities
- Completing the Square
- The Quadratic Formula
- Factoring and Zeros
- Polynomial Multiplicity
- Rational Root Intuition
- Horizontal Asymptotes
- Slant Asymptotes
- Cube Root and Odd Roots
- Exponent Rules in Pictures
- Log Rules in Pictures
- Change of Base
- Exponential vs Power Growth
- Compound Interest
- Continuous Compounding
- Doubling Time and the Rule of 72
- Half-Life Decay Model
- Gaussian Bell Curve Shape
- Sigmoid and Logistic Curves
- The Rectified Linear Function
- Sawtooth and Triangle Waves
- Heaviside Step and Delta Idea
- Sign and Absolute Value
- Min and Max Functions
- Ceiling Function and Postage Costs
- Modular Waves
- Inverse Square Functions
- Hyperbola as Inverse Variation
- Reciprocal Trig Overview
- Phase Shift of a Sine Wave
- Frequency and Pitch
- Half-Angle Identity
- Tangent Addition and Slopes
- Sinusoidal Models of Tides and Seasons
- Beats from Close Frequencies
- Chirp: Sweeping Frequency
- Sinc Function and Diffraction
- Derivative of a Parabola
- Tangent to a Cubic
- Area Under a Line
- Area Under a Cosine Hump
- Surface Area of Revolution
- Average Value of a Function
- Angle Between Vectors
🔧 Intermediate
- Function Transformations
- Geometric Series
- Cycloid
- Spirograph Curves
- Pythagorean Identity
- Inverse Trigonometric Functions
- Secant, Cosecant, Cotangent
- Hyperbolic Functions
- Damped Oscillations
- Sum of Sines
- Law of Sines and Cosines
- Euler’s Formula
- Complex Number Multiplication
- Roots of Unity
- The Derivative as a Slope
- Tangent of a Sine Wave
- Power Rule
- Product Rule
- Chain Rule
- Derivative of eˣ
- Continuity and Discontinuity
- L’Hôpital’s Rule
- Optimisation
- Critical Points and Concavity
- Newton’s Method
- Riemann Sums
- Fundamental Theorem of Calculus
- Exponential Growth ODE
- Volumes of Revolution
- Related Rates
- Rotation Matrices
- Scaling Matrices
- Vector Addition
- Scalar Multiplication
- Dot Product
- Vector Projection
- Cross Product and Area
- Reflection Across the x-axis
- Pythagorean Theorem
- Angle Sum of a Triangle
- Area of a Circle
- Pascal’s Triangle
- Linear Regression
- Graph Theory Basics
- Sine of a Sum
- Product-to-Sum Identity
- Integration by Parts Picture
- Antiderivatives and Constants
- Newton’s Cooling as an ODE
- Slope Fields Intuition
- Euler’s Method Error
- Radius of Convergence
- Arc Length of a Curve
- Centre of Mass of a Rod
- Gamma Function Preview
- Epicycloid
- Hypocycloid and Astroid
- Rose with Many Petals
- Limaçon
- Logarithmic Spiral
- Fermat’s Spiral
- Trefoil Projection
- Butterfly Curve
- Heart Curve
- Superellipse
- Rotating Ellipse
- Projectile Path as Parametric Curve
- Lissajous 3:2 Figure
- Harmonograph
- Reflection Matrices
- Projection Matrices
- Orthogonal Vectors
- Vector Components
- Parallelogram Area and Determinant
- Triangle Centre and Medians
- Area and Perimeter Trade-offs
- Volume of a Cylinder and Cone
- Sphere Volume and Surface Area
- Square–Cube Law
- Law of Cosines Generalises Pythagoras
- Inscribed Angle Idea
- Golden Rectangle Ratio
- Binomial Coefficient Row Shape
- Prime Gaps Grow on Average
- Chi-Square Shape
- Regression to the Mean
- Linear vs Exponential Trend Fits
🚀 Advanced
- Limits
- The Definite Integral
- Taylor Series
- Maclaurin Series for eˣ
- Improper Integrals
- Matrices as Transformations
- Shear Transformations
- Eigenvectors and Eigenvalues
- Regular Polygons
- Geometric Transformations
- The Golden Ratio
- Prime Numbers and the Sieve
- The Ulam Spiral
- The Collatz Conjecture
- Fibonacci Numbers
- Euclid’s Algorithm for GCD
- Modular Arithmetic
- Prime Number Theorem
- Law of Large Numbers
- Sums of Dice
- Central Limit Theorem
- Random Walks
- Estimating π with Monte Carlo
- The Galton Board
- The Birthday Paradox
- The Normal Distribution
- Binomial Distribution
- Bayes’ Theorem
- Exponential Distribution
- Standard Error
- Combinations
- Permutations
- Venn Diagrams and Inclusion–Exclusion
- Big-O Growth Rates
- Differential Equation Solutions
- Logistic Differential Equation
- Harmonic Series Diverges
- p-Series Convergence
- Alternating Series
- Power Series of 1/(1−x)
- Taylor Series for Cosine
- Taylor Series for ln(1+x)
- Work as an Integral
- Sum of the First n Integers
- Sum of Squares
- Number of Subsets
- Factorial Growth
- Stirling’s Approximation
- Catalan Numbers Growth
- Zeta Function and the Basel Problem
- Euler’s Totient Intuition
- Big Numbers: Powers of Ten
- Poisson Distribution
- Uniform Distribution
- Exponential Survival Function
- Cumulative Distribution of the Normal
- Student t vs Normal
- Confidence Interval Width
- Margin of Error for Polls
- Expected Value of a Game
- Variance of a Sum
- Entropy of a Coin
- Gambler’s Ruin
- Monty Hall Switching
Frequently asked questions
What will I learn in Mathematics?
Functions, calculus, geometry, probability. Each topic has a live animation, a lesson and a quick quiz.
Where should I start?
Try “Linear Functions” from the Popular list, or any beginner topic in Browse.
Are the topics in a set order?
Browse groups topics by level, with beginner topics first. Each topic’s FAQ names the topic to read before it.
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