Build your foundations
Begin with familiar observations, then use the links and guided routes to find supporting ideas. A topic can be introduced at several levels, so its ring is a guide rather than a fixed prerequisite order.
FOUNDATION 02 / MATH → PHYSICS → BIOLOGY
Everything obeys
a very short list of rules.
Use the language of Math to explore matter, energy, motion and chance. Then follow those mechanisms into Biology: molecules move, membranes hold gradients, and cells exchange energy with their surroundings. The same physical ideas also connect to simulations and machine learning. Start with the map.
151ideas mappedfrom everyday to unresolved
12branchesone circle, twelve wedges
6guided routesincluding physics → biology
7open questionswith the level each one asks
A universe of connected ideas.
Every star is one idea; each constellation is a branch. Move outward from everyday observations toward deeper theory and research. The rings suggest a study level, and the lines show useful connections across branches. The outer ring includes both established results and open questions.
WALK A ROUTE
151 of 151 ideas lit
Skip to the selected idea ↓Choose a star to explore its connections. Use arrow keys on the map, follow a route, or browse the idea list below. Zoom in to read more closely and scroll around the map.
THE PHYSICAL UNIVERSE12 branches. From matter to living systems.
Thermodynamics & statistical mechanics / Core university
Entropy
A measure of how many microscopic arrangements match what you can actually see.
Where you meet it · Why heat flows one way and eggs do not unscramble.
Where it touches computing · The identical quantity Shannon arrived at independently for information — cross-entropy loss is this idea.
Read the related Physics chapter ↓Rests on
Leads to
Browse every idea · 151 matching
Classical mechanics19
Waves & oscillations13
Fluids & continuum13
Thermodynamics & statistical mechanics18
Electromagnetism15
Optics & photonics10
Relativity8
Quantum mechanics14
Particle & nuclear8
Condensed matter9
Astrophysics & cosmology10
Methods & computation14
Branches are administrative
The wedges are how departments are organised, not how nature is. The harmonic oscillator appears in mechanics, waves, circuits, quantum and condensed matter, and it is the same object every time.
Continue into living systems
Choose Leads into biology to explore diffusion, cellular energy and electrical signals. Each highlighted idea links to a Biology lesson or model. Use the computing filter to explore another connection.
Physics is a method before it is a body of facts.
What makes it work is not that physicists are clever about the world. It is a short list of moves that keep turning out to be the right ones, applied with unusual stubbornness about measurement.
| Move | The idea | Why it pays |
|---|---|---|
| Conservation | Some quantities never change, whatever else happens. Energy, momentum, charge. | You can often answer a question about the end without following the middle. |
| Symmetry | If a situation looks the same after some change, that change protects a quantity. | Noether's theorem turns an aesthetic observation into a calculation. |
| Least action | Of all the paths a system could take, it takes the one that makes the action stationary. | Mechanics, optics and field theory all restate as the same optimisation. |
| Statistics over detail | When there are 10²³ particles, stop tracking them and describe the distribution. | Temperature, pressure and entropy are all properties of a distribution, not a particle. |
| Scale separation | Most phenomena depend on a narrow band of scales, so you can ignore the rest. | It is why you can build a bridge without quantum mechanics — and why turbulence is hard, because it refuses to separate. |
The same laws, across sixty orders of magnitude.
What changes across scales is not the rules but which of them you can afford to ignore. That is the single most useful idea for knowing which physics a problem needs.
- 10-35 mPlanck length
Where our current theories stop making predictions.
- 10-18 mQuark
The smallest distance probed by collider experiments.
- 10-15 mAtomic nucleus
Where the strong force dominates.
- 10-10 mAtom
Chemistry lives here.
- 10-9 mTransistor gate
Modern chips are a few tens of atoms across.
- 10-6 mBacterium
Where viscosity beats inertia and swimming works differently.
- 100 mA person
The scale our intuition was built for.
- 107 mEarth
Gravity, weather, oceans.
- 1011 mEarth to Sun
Light takes eight minutes.
- 1021 mThe galaxy
Where dark matter shows itself in rotation curves.
- 1026 mObservable universe
Limited by how long light has had to travel.
Scale separation is why you can design a bridge without quantum mechanics and model a galaxy without tracking atoms. It is also precisely what fails in the hard problems: turbulence refuses to separate, which is why it is unsolved, and a phase transition is the moment when every scale starts to matter at once.
Find what cannot change, and the rest follows.
The most powerful technique in the subject is not solving the equations. It is noticing a quantity that the equations cannot alter, and using it to skip the middle entirely.
Energy
Conserved because the laws do not change from one moment to the next. A ball at the bottom of a ramp has a speed you can state without following the ramp.
Momentum
Conserved because the laws do not change from one place to another. Collisions become arithmetic.
Angular momentum
Conserved because the laws do not care which way you face. Skaters, gyroscopes and galaxies all follow.
Noether's theorem makes the pattern exact: every continuous symmetry of the action produces a conserved quantity. This is the deepest idea in classical physics, and it is the same argument that justifies equivariant neural networks — bake in the symmetry and you no longer have to learn the invariance from data.
Nothing acts at a distance.
Newton described gravity precisely and was openly uncomfortable that it appeared to reach across empty space instantly. The resolution took two centuries: there is no reaching. There is a field, defined everywhere, and things respond to it locally.
Maxwell's four equations were the moment this became unavoidable. They unified electricity and magnetism, then predicted a self-sustaining wave whose speed fell out of two constants measured in laboratory experiments that had nothing to do with light. The number matched the measured speed of light, and the conclusion was inescapable.
Stop tracking the particles.
A litre of air holds around 10²⁵ molecules. No computer will ever track them, and it would not help: the useful quantities — temperature, pressure, entropy — are properties of the distribution, not of any molecule.
That move, from following individuals to describing a distribution, is the whole of statistical mechanics. It also produces the most consequential shared idea between physics and machine learning. Boltzmann's result that the probability of a state falls exponentially with its energy over temperature is the softmax function, with logits in place of negative energies and sampling temperature in place of temperature.
The rules change, and the predictions get better.
By 1900 classical physics was describing heat radiation with an answer that went to infinity. The fix — energy in discrete packets — was introduced reluctantly and turned out to be how the world is.
Discreteness
Energy, angular momentum and charge come in fixed amounts. Atoms are stable and have spectra because of it.
Amplitudes, not probabilities
Quantum states add as complex amplitudes, which can cancel. Interference of possibilities has no classical counterpart.
Limits on joint knowledge
Position and momentum cannot both be sharp. This is a property of the objects, not of the apparatus.
It is worth being precise about where the difficulty actually lies. The mathematics is learnable and the predictions are the most accurately confirmed in science. What remains genuinely unsettled is what a measurement is and why one outcome occurs. Anyone who tells you that part is resolved is describing an interpretation they prefer.
A practical note: the wavefunction of n particles lives in a space whose dimension grows exponentially with n. That single fact is why quantum chemistry is hard, why quantum computers are interesting, and why tensor networks and neural wavefunctions exist.
From physical mechanisms to living systems.
Biology adds organization, heredity and evolution to the physical picture. These three connections give you a starting point for the next foundation.
Small steps, changing concentrations
Random molecular motion can produce net diffusion. At cellular distances it is an important transport mechanism; over larger distances, bulk flow can move materials faster.
Try the diffusion model →Gradients can power a cell
Electron transport can build a proton gradient across a membrane. Coupling the return flow to ATP synthase connects an electrochemical potential difference to ATP production.
Explore molecules and energy →Membranes have electrical properties
Charge separation and selective ion channels shape membrane voltage. Changes in conductance help produce the electrical signals of nerve and muscle cells.
Explore bodies, brains and feedback →Further reading: molecular motors, flow and viscosity, and the experimental study of Brownian motion.
Physics and machine learning share mathematical tools.
It is easy to overclaim here, and popular accounts usually do. Below are the places where the connection is an actual shared formalism, with the one well-known analogy that is not established marked as such.
| In physics | In machine learning | What is shared |
|---|---|---|
| Boltzmann distribution | Softmax and temperature sampling | The probability of a state falling exponentially with energy over temperature is exactly the softmax over negative logits. Sampling temperature is not a metaphor — it is the same parameter in the same position. |
| Non-equilibrium thermodynamics | Diffusion models | Adding noise until structure is gone and learning to run it backwards was proposed directly from this literature, and is now written as a stochastic differential equation. |
| Spin glasses | Loss landscapes | Disordered systems with conflicting interactions and vast numbers of near-degenerate minima are the closest physical analogue to a deep network's landscape, and one of the few sources of real theory about it. |
| The Ising model | Hopfield networks and energy-based models | Spins preferring to agree with neighbours, with the couplings learned rather than given. The 2024 Nobel Prize in Physics recognised this lineage. |
| Noether's theorem | Equivariant architectures | Build a symmetry into the model and the corresponding invariance comes free, rather than being learned from data that may not contain enough of it. |
| Hamiltonian dynamics | Hamiltonian Monte Carlo | Sampling a distribution by treating its negative log-density as a potential energy and rolling a particle across it, using a symplectic integrator to keep the energy honest. |
| Variational principles | Variational inference | Both replace an intractable problem with the search for the best member of a tractable family. The shared word is not a coincidence. |
| Renormalisation group | Depth and coarse-graining | An appealing account of what successive layers do to detail. Suggestive, repeatedly proposed, and not established — worth knowing as an open analogy rather than a result. |
The parts nobody has finished.
Physics is often taught as a completed edifice. It is not, and the gaps are large enough to be worth naming — roughly 95% of the universe's energy content is currently labelled rather than explained.
What is dark matter?
Roughly five times more mass than everything visible, detected only through gravity. Decades of direct searches have returned nothing.
Level: Entry: orbital mechanics. Frontier: particle astrophysics, detector design.
Why is the expansion accelerating?
Called dark energy, which names the observation rather than explaining it. It is about 70% of the universe's energy budget.
Level: Entry: Hubble's law. Frontier: general relativity, observational cosmology.
How do gravity and quantum mechanics fit together?
Our two best theories use incompatible mathematics and disagree wherever both matter — inside black holes, and at the first instant.
Level: Entry: special relativity and quantum states. Frontier: quantum field theory in curved spacetime.
What actually happens in a measurement?
The formalism predicts outcomes with extraordinary precision and does not say what selects one. Interpretations differ; experiments so far do not.
Level: Entry: superposition. Frontier: decoherence, foundations.
Can turbulence be solved?
The equations have been known since 1845. Their behaviour across scales still resists both analysis and complete simulation.
Level: Entry: Reynolds number. Frontier: PDE analysis, spectral methods, learned closures.
What makes high-temperature superconductors work?
The low-temperature theory is settled and does not explain the materials discovered in 1986. Nearly forty years on, there is no consensus.
Level: Entry: band structure. Frontier: strongly correlated electron systems.
Why is there more matter than antimatter?
The Standard Model allows an asymmetry far too small to explain why anything exists at all.
Level: Entry: particle conservation laws. Frontier: CP violation, neutrino physics.
A workable order, and how to know you are ready.
The common mistake is starting with whatever sounds most exciting — usually quantum or relativity — and bouncing off, because both assume mechanics you have not built yet.
- STEP 01Get the units habit first
Before any topic, make dimensional analysis automatic. If both sides of an equation do not describe the same kind of thing, nothing after that matters.
Ready when: You catch your own errors by inspection rather than by checking the answer.
- STEP 02Mechanics, slowly and with your hands
Position, velocity, acceleration, force, momentum, energy. Do the problems. This is where physical intuition is actually built and there is no shortcut past it.
Ready when: You can predict roughly what will happen before you calculate.
- STEP 03Then waves and oscillations
The harmonic oscillator is the most reused model in the subject and it appears in every branch after this one. Time spent here pays back everywhere.
Ready when: You recognise a restoring force when you see one.
- STEP 04Electromagnetism, for the field idea
The content matters, but the bigger prize is learning to think in fields — a quantity defined everywhere, acting locally.
Ready when: “Action at a distance” starts to sound like the strange idea it is.
- STEP 05Thermodynamics before quantum
Entropy, temperature and the statistical view of many particles. It is also the branch that connects most directly to machine learning.
Ready when: You can explain why entropy in physics and entropy in information theory are the same quantity.
- STEP 06Now quantum, and expect discomfort
The mathematics is learnable and the interpretation is genuinely unsettled. Learn to calculate first and argue about meaning afterwards.
Ready when: You are comfortable saying “the formalism predicts this and nobody agrees what it means”.
Where to learn it
The Feynman Lectures on Physics
The whole thing, free from Caltech. Still the best explanation of why anybody would care.
MIT 8.01 Classical Mechanics
A complete first mechanics course with problem sets and exams.
MIT 8.02 Electricity and Magnetism
The follow-on, and where the field idea is properly built.
MIT 8.04 Quantum Physics I
A serious undergraduate introduction rather than a popularisation.
3Blue1Brown
Not physics-specific, but the differential-equations and Fourier series are the right intuition for most of this atlas.
arXiv physics
Where the research actually appears, months before journals.