Welcome to OpenCurve — A bit of maths and physics for everyone. This is a collection of conceptual explainers across mathematics and theoretical physics. We do deep dives on classical mechanics, relativity, quantum physics, calculus, deterministic and stochastic dynamics, featuring complete derivations and visual models.

Gravity assist

PHYSICS

How does a planetary push actually work?

PUBLISHED — December 26, 2021

Last Christmas, my father and one of my brothers wondered how a spacecraft’s planetary slingshot works. It’s a well-known manoeuvre to get a big swing forward by passing by a planet. They realized that if the craft falls toward the planet due to its gravitational pull, it will gain momentum. However, as soon as it has flung around the planet, wouldn’t that same gravitation slow down the craft just as much? Great question, of course, so, let’s dive into this gravity assist short and simple.

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Why is glass transparent?

A clear case of quantum apathy

The Copenhagen interpretation

How Niels Bohr and Werner Heisenberg attempted to resolve the quantum measurement problem through wave function collapse

Spaces and dimensions

What mathematicians and physicists actually mean by dimensions

Quantum mechanics in ten ideas for people on the move

Ten essential concepts about particles, wave functions, and reality in accessible language

$Intricate fractal spiral pattern showing a colorful detail of the Mandelbrot set in vibrant shades of blue, gold, and white against a deep blue gradient background.$

Complex numbers: an introduction

An accessible introduction to the complex plane, imaginary numbers, and their fundamental role in physics

Heisenberg’s uncertainty principle

Why Heisenberg’s uncertainty principle isn’t about clumsy measurements or mysticism, but fundamental wave mechanics and Fourier pairs

Quantum entanglement: the EPR paradox and Bell’s Theorem

How Einstein, Podolsky, and Rosen challenged quantum mechanics, and how John Bell devised a theorem to test local realism

Quantum entanglement: non-locality and the state of a two-particle system

What locality and non-locality mean in physics, and how entangled quantum states challenge our classical intuition

Lab centrifuges and prime numbers

How prime factorisation and roots of unity determine whether any given number of test tubes can be evenly balanced in a laboratory centrifuge

The double-slit experiment

How firing particles through two narrow slits broke classical intuition and revealed the wave-particle nature of reality

The Collatz Conjecture

Why one of the simplest mathematical rules remains one of the hardest to prove

This is not an atom

Why the classic planetary model of the atom is wrong, and what quantum mechanics really tells us

Proof that the square root of 2 is irrational

A step-by-step classic proof by contradiction showing why the square root of 2 cannot be expressed as a ratio of two integers

The meaning of E=mc²

Why mass is not converted into energy, why pure energy does not exist, and what Einstein really meant

Rainbows: Alexander’s band

Why the sky between primary and secondary rainbows is noticeably darker

A radioactive smoking gun

Why radioactivity means nuclear instability, and how inhaling radioactive polonium and lead from tobacco smoke exposes lungs to extreme radiation doses

How do polarised sunglasses work?

How quantum fields, oscillating electron clouds, and vertical polarisation filter out blinding glare from roads, water, and screens

Is microwave oven radiation unhealthy?

Why microwave radiation is non-ionising electromagnetic energy that merely causes friction in water molecules without damaging DNA or vitamins

Why, exactly, do glass and liquids refract light?

How Maxwell’s equations, interface boundary conditions, and microscopic electron responses explain optical refraction

Finding the normal force in planar non-uniform circular motion using polar coordinates

Deriving an analytical expression for the normal force on a sliding mass over a cylinder, eliminating second derivatives via integration

Why do wet clothes dry?

How Ludwig Boltzmann, statistical microstates, and the second law of thermodynamics explain why water inevitably evaporates into the air

Deriving the volume of the inside of a sphere using spherical coordinates

A complete step-by-step derivation of the volume of a 3-ball using spherical coordinates, infinitesimal volume elements, and triple integration

Simple problems on relativistic energy and momentum

Deriving the energy-momentum invariant, analyzing massless particles, and calculating relativistic proton speeds at high kinetic energies

How big is the universe?

Understanding the scale of the observable universe, the mechanics of accelerated cosmic expansion, and why Earth is at the centre of its own visible bubble

Happy birthday mister Einstein, happy Pi Day to you!

Celebrating Pi Day, Albert Einstein’s birthday, and the surprising appearances of pi in general relativity, number theory, and colliding blocks

Just a minute: why do large and heavy ships not sink?

How Archimedes’ principle and hydrostatic pressure explain why massive vessels float, and why gravity is both the cause of sinking and floating

Mirror, mirror, what’s up with the mirror writing?

Why mirrors do not reverse left and right, and how geometric reflection differs fundamentally from rotational transformation

Why your coffee does not have tides

The Moon’s influence on tiny things is tiny

Minus minus and negative times negative

Simple algebraic proofs demonstrating why subtracting a negative yields a positive and why multiplying two negatives results in a positive

Energy is neither fundamental nor conserved

Why energy is a mathematical bookkeeping tool rather than a physical substance, and how spacetime expansion breaks global energy conservation

The riddle of birthdays

Why the birthday paradox means you only need 23 people for an even chance of a shared birthday, and 50 people for a 97% certainty

$Conceptual grid diagram illustrating probability fractions: sixteen circles transitioning from empty outlines to blue and green highlights.$

When and why do you multiply probabilities?

An intuitive explanation of independent probabilities and why combined outcomes correspond to taking a fraction of a fraction

What is a spacetime interval?

How Einstein and Minkowski showed that while space and time individually contract and dilate, the spacetime interval remains strictly invariant

Deriving the Lorentz transformations from a rotation of frames of reference about their origin with real time Wick-rotated to imaginary time

A geometric derivation of the Lorentz transformations using Wick rotation, complex spacetime coordinates, and planar Euclidean rotations

Real eigenvalues and eigenvectors of 3x3 matrices, example 3

A step-by-step undergraduate tutorial on calculating real eigenvalues and corresponding eigenvectors for a 3x3 matrix

Real eigenvalues and eigenvectors of 3x3 matrices, example 1

A step-by-step undergraduate tutorial on computing real eigenvalues and eigenvectors for a triangular-leaning 3x3 matrix

Real eigenvalues and eigenvectors of 3x3 matrices, example 2

A step-by-step undergraduate tutorial on finding real eigenvalues and corresponding eigenvectors for a 3x3 matrix with algebraic multiplicity