The Lore of Mathematics.

"Read Euler, read Euler. He is the master of us all."
- Pierre-Simon Laplace

A Riemann sum approximates the area under a curve. The definite integral is the exact value these approximations approach as the rectangles become infinitely thin; it is defined as the limit of Riemann sums.

Maxwell–Boltzmann distribution. The air around you is full of tiny molecules, and at room temperature most of them move faster than the speed of sound. They do not all move at the same speed. A few are slow, most cluster around a typical speed, and a rare few are extremely fast. In 1860, James Clerk Maxwell worked out how these speeds are distributed. Ludwig Boltzmann later extended the idea. Here 1,200 beads play the molecules. The faster a bead flies out, the farther it lands, so each slot collects beads of similar speed. Together, the piles trace the Maxwell–Boltzmann curve.

A Coffee Mug Is a Doughnut! In topology, shapes may be stretched and bent, but never cut or glued. Under these rules, a mug's body can be absorbed into its handle, leaving a torus. Both have exactly one hole, so to a topologist they are the same object.

Divide 1 by 9801… and the decimal starts counting: 000, 001, 002… all the way to 99. Except one number never shows up - the number is 98.

“Profound study of nature is the most fertile source of mathematical discoveries.” - Joseph Fourier, The Analytical Theory of Heat (1822).

The golden spiral, unrolled.

Divide 1 by 998001… and the decimal starts counting: 000, 001, 002… all the way to 999. Except one number never shows up - the number is 998.

"Probably 80% of the time you're completely stuck." - Ashoke Sen The other 20% reshaped physics. Prof. Sen has been awarded the 2026 Oskar Klein Medal, the first Indian physicist to receive it, for his work on S-duality, black-hole entropy and string field theory.

Proving a² + b² = c² without a single word.

Solving a Megaminx with graph theory.

One of the most beautiful mathematical expressions for a reason:

2 is the only even prime number.

“Those laws are within the grasp of the human mind; God wanted us to recognize them by creating us after his own image so that we could share in his own thoughts.” - Johannes Kepler

Every conic is a slice of the same cone.

A simple proof of the Pythagorean theorem.

The extraordinary legacy of the Bernoulli family.

A coffee cup and a donut are the same object, topologically. Stretch and bend, but never tear or glue. Each has exactly one hole, so one deforms continuously into the other. They are homeomorphic.

Almost any shape can roll smoothly; you just need the right road.

“Philosophy is written in this grand book, the universe, which stands continually open before our eyes… It is written in the language of mathematics.” - Galileo Galilei

On this day in 1759, Jacob II Bernoulli was born in Basel into the most remarkable family in the history of mathematics: the Bernoulli family.

The Fibonacci Spiral Clock.

“The infinite! No other question has ever moved so profoundly the spirit of man.” - David Hilbert

The circle lore.

Three golden rectangles, set mutually perpendicular, quietly define a regular icosahedron. Their twelve corners are exactly its twelve vertices.

Rotation matrices at work.

1³ + 2³ + … + n³ = (1 + 2 + … + n)²

The Schrödinger Equation: The Heart of Quantum Mechanics By the early 1900s, physics had a problem. Classical physics, including Newton’s laws, worked well for planets, cars and falling objects, but it could not explain how atoms behave. Experiments showed that light can act both as a wave and as a particle, and that each type of atom gives off light only at certain fixed colours. Classical physics could not explain these facts. In 1924, the French physicist Louis de Broglie proposed a bold idea: if light can behave like particles, then particles such as electrons might also behave like waves. The Austrian physicist Erwin Schrödinger, then a professor at the University of Zurich, took this idea further. Around Christmas 1925, during a stay in the Swiss mountain town of Arosa, he worked out an equation describing how these “matter waves” behave. He published his theory in 1926 in a series of four papers. What does the equation do? It describes a quantity called the wavefunction, written as Ψ (psi). The wavefunction contains all the information that can be known about a quantum system, such as an electron in an atom. Schrödinger’s equation tells us how that wavefunction changes over time, in the same way Newton’s laws tell us how an object moves. What does the wavefunction mean? Schrödinger first thought it described the electron’s charge spread out in space, but that idea did not hold up. In 1926, the German physicist Max Born gave the explanation still used today: the wavefunction gives probabilities. It does not say exactly where an electron is, only how likely it is to be found in each place. At the smallest scales, nature is described in terms of chances, not certainties. Its first great success Schrödinger applied his equation to the hydrogen atom, the simplest atom. The equation produced the atom’s allowed energy levels, which match the colours of light hydrogen is observed to emit. Earlier, Niels Bohr’s model had obtained the same levels only by adding special rules. In Schrödinger’s equation, they followed naturally from the mathematics. Two theories, one physics A year earlier, in 1925, Werner Heisenberg, working with Max Born and Pascual Jordan, had developed a very different-looking version of quantum mechanics based on tables of numbers called matrices. In 1926, Schrödinger showed that the two approaches give the same results. They were two ways of describing the same physics. Why it matters Quantum mechanics explains why atoms are stable, how chemical bonds form, and why some materials conduct electricity while others do not. Schrödinger’s equation is its central working tool. The understanding behind transistors, computer chips, lasers and LEDs depends on quantum mechanics, and so does the operation of quantum computers. The equation does have limits. It applies to particles moving much slower than light. In 1928, Paul Dirac found an equation for electrons moving close to the speed of light. Recognition In 1933, Schrödinger and Dirac shared the Nobel Prize in Physics. Two years later, Schrödinger described his famous thought experiment of a cat that, according to quantum theory, would have to be considered both alive and dead until observed. He meant it to show how strange quantum theory becomes when applied to everyday objects. A hundred years later, his equation remains one of the foundations of modern physics.

e^π > π^e.

This one definition underlies the whole of calculus: velocity, growth, tangent lines, curvature, optimisation, approximation, series expansions, vector fields, and the differential equations that describe nearly every law of nature.

Pythagorean theorem, written on the unit circle: 1+ cot²θ = cosec²θ

Pythagorean theorem, written on the unit circle: 1+ tan²θ = sec²θ

Pythagorean theorem, written on the unit circle: sin²θ + cos²θ = 1.

The Aizawa attractor: It's a distinctive three-dimensional chaotic shape traced out by a system of three nonlinear differential equations with six parameters.

"What we observe is not nature itself, but nature exposed to our method of questioning." - Werner Heisenberg

“The cosmos is within us. We are made of star-stuff.” - Carl Sagan

From folded paper to space instruments: origami lets large structures launch small and open in orbit.

"The eternal silence of these infinite spaces frightens me." - Blaise Pascal

Among the infinite number of 3D shapes you can imagine or draw, there are exactly five Platonic Solids in the universe: the Tetrahedron, Hexahedron, Octahedron, Dodecahedron, and Icosahedron.

In 1666, as the bubonic plague shut down Cambridge, a young Isaac Newton returned to his family home in Woolsthorpe. There, using a simple glass prism, he showed that white light is not pure but a blend of colours. He passed sunlight through one prism to split it into a spectrum, then through a second prism to recombine it into white light. This proved that the colours came from the light itself, not from the glass.

"God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe." - Paul Dirac

"The universe is written in the language of mathematics." – Galileo Galilei.

The Rubik's Cube is one of the most iconic math toys ever, invented by Ernő Rubik in 1974. It has 43,252,003,274,489,856,000 possible configurations, and you can solve every single one in 20 moves or fewer.

Fibonacci: From curves to cubes and back to curves.

The Archimedean spiral is one of the oldest curves in mathematics, with turns that stay evenly spaced. Here, a straight line slowly turns into this spiral, and four hundred connected points follow its shape.

A conic section is the curve formed when a plane intersects a double cone. The angle of the cutting plane alone determines which curve appears. Circle: The plane is perpendicular to the axis of the cone. Eccentricity e = 0. Ellipse: The plane is inclined to the axis and cuts completely through one nappe. 0 < e < 1. Parabola: The plane is parallel to the side of the cone. e = 1. Hyperbola: The plane cuts through both nappes of the cone. e > 1.

"The sole aim of science is the honour of the human spirit." - Carl Jacobi

"We are made of star-stuff." - Carl Sagan

"The profound study of nature is the most fertile source of mathematical discoveries." - Joseph Fourier

"If you want to find the secrets of the universe, think in terms of energy, frequency and vibration." - Nikola Tesla

Nature's Only Perfect Shapes: Five Platonic Solids.

People enjoy discussing mathematics.

The Hidden Beauty of Fractals in Motion.

A glimpse into the extraordinary life of Leonhard Euler.

Solving a Rubik's Cube with graph theory.

Beauty is inseparable from mathematics...ᝰ.

17 equations that changed the world.

Viviani’s theorem demonstrates spatial conservation, showing that the three perpendicular lines behave like a zero-sum game: moving closer to one side requires corresponding movement away from the other two. Local changes maintain a global constant.

The golden spiral, set in motion.

Among two-digit numbers, excluding palindromes, only two pairs have the property where the number and its reverse both square to numbers that are also reverses of each other.

Leonhard Euler (1707–1783) is among the most prolific mathematicians in history. He wrote more than 850 papers and books across analysis, number theory, mechanics and optics. Nearly blind in later life, he kept working, and his work remains central to mathematics today.

Beauty in mathematics often reveals unexpected simplicity where none seemed to exist. The Gaussian integral stands as one of the cleanest examples.

Matrix multiplication seems complex until you observe it step by step. Watch how each of the nine entries is formed: one row from matrix A, one column from matrix B, three products, and one sum.

a³−b³ = (a−b)(a²+ab+b²)

He will remain the last human to solve a Millennium Prize problem without AI.

Proof of the limit of sinx/x as x approaches zero is 1 using the Squeeze Theorem.

(𝑎 + 𝑏)³ = 𝑎³ + 3𝑎²𝑏 + 3𝑎𝑏² + 𝑏³

Size is not a property of the object. It is a ratio of the object to whatever you set beside it.

A square wave in time is a set of amplitudes in frequency, and the Fourier series is the bridge between the two.

(a + b + c)² = a² + b² + c² + 2(ab + ac + bc)

The Law of Cosines.

The Law of Cosines.

A plane sliding through a double cone traces every conic section.

"There are things which seem incredible to most men who have not studied mathematics." - Archimedes

Augustin-Louis Cauchy, shortly before his death, 1857.

Every real number ≥1 has its own infinite nested radical.

Twelve nines, then it gives up! e^(π√163) misses being a whole number by less than a trillionth. It is not an integer. It is not even rational. It is transcendental. It just happens to stand this close to 262537412640768744 and then walk away.

The same curve has a finite signed area but an infinite total area.

The beauty of mathematics is undeniable.

Leonhard Euler produced roughly half his output after going completely blind in 1766. He dictated to assistants and could hold enormous calculations in his head. His collected works run to more than 80 volumes and still aren't fully published.

"The mind is not a vessel to be filled, but a fire to be kindled." - Plutarch

If the length of an arc of a circle equals its radius, the angle subtended is defined as one radian.

The distance a circle travels in a single revolution without slipping equals π times its diameter, regardless of the wheel's size.

You Can Watch Euler's Formula, Unrolled.

Johann Bernoulli posed the problem to the mathematicians of Europe in 1696; it remains vital today in quantum mechanics and many other fields. If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp's curve, such that the descent time is a minimum? Intuition says that it should be a straight line. That would minimise the distance, but the minimum time happens when the ramp curve is a cycloid.

Unrolling Euler's Formula.

No error, this image is the answer.

Niels Henrik Abel was one of the greatest young mathematicians of the 19th century, but he spent much of his life struggling financially. He made major discoveries in algebra, including proving that a general fifth-degree equation cannot be solved by radicals. In 1829, only days after Abel died at the age of 26, a letter arrived offering him an academic position in Berlin. One of mathematics’ most tragic “too late” stories.