The universe of equations. In your pocket.

Equations are everywhere

See the ideas shaping your world.

They live in the note a string can sing, the curve of a growing population, the price of time, the path of light, and the motion of worlds above us.

Life: Logistic growth

Growth has a rhythm and a limit.

Populations may surge at first, then settle as food, space, or other resources become scarce.

\frac{\mathrm dN}{\mathrm dt}=rN\left(1-\frac{N}{K}\right)

r is the growth rate. K is the carrying capacity.

A compact idea that helps describe populations, epidemics, ecology, and systems that cannot grow forever.

Sound: Standing waves

A string knows which notes it can sing.

A string fixed at both ends can sustain only the modes whose nodes meet those endpoints. Length and wave speed set its harmonic family.

f_n=\frac{nv}{2L}

n is the harmonic number, v the wave speed, and L the string length.

The structure behind harmonics, musical timbre, acoustic design, and the notes that feel naturally related.

Light: Snell’s law

Light changes course when the world changes.

A ray bends toward the normal as it passes from air into a denser material because its speed changes at the boundary.

n_1\sin\theta_1=n_2\sin\theta_2

n₁ and n₂ are refractive indices. Both angles are measured from the normal.

The principle inside lenses, cameras, telescopes, fibre optics, and the apparent bend of a straw in water.

Markets: Compound interest

Time can become part of value.

When each return can earn a return of its own, small changes in rate or time reshape the outcome.

A=P\left(1+\frac{r}{n}\right)^{nt}

P is the starting principal, r the nominal rate, n the compounding frequency, and t elapsed time.

A lens for saving, borrowing, investing, inflation, and the long reach of repeated change.

Information: Entropy

Uncertainty can be measured.

A rare message carries more surprise than an expected one. Together, those surprises define information.

H(X)=-\sum_{x\in\mathcal X}p(x)\log_2p(x)

p(x) is an outcome’s probability. Base-2 logarithms measure entropy in bits.

The idea behind compression, communication, machine learning, and how much a signal can tell us.

Cosmos: Circular-orbit speed

Falling can become an orbit.

Move sideways quickly enough and gravity keeps pulling you around a world instead of down to its surface.

v_{\mathrm{circ}}=\sqrt{\frac{GM}{r}}

For a circular orbit around a dominant mass M, r is the orbital radius.

A relationship that shapes satellites, planetary motion, spaceflight, and our view of the sky.

The daily ritual

Hold the mystery. Keep the meaning.

A small moment of anticipation becomes an idea you can carry into the rest of your day.

NoticeBegin with a question grounded in the world around you.
RevealMeet the equation, its symbols, and the intuition inside it.
CarryKeep the story, favorite the idea, and return when curiosity calls.

Sound: Standing waves

What decides the family of notes a string can sing?

f_n=\frac{nv}{2L}

Standing-wave frequencies

Length and wave speed determine the harmonic notes a string can sustain.

Hold for a moment. Keyboard and assistive technology users can activate once.

See how Principia explains it

Understanding without the textbook

Not a list of formulas. A way of seeing.

Plain language, human history, beautiful notation, and just enough interaction to make each idea stay with you.

Understand the idea, not just the symbols.

Move from intuition to meaning at your own pace, with prose written for curious people whether or not they are specialists.

Meet the people behind the insight.

Every discovery arrived in a real place, at a real moment, through human imagination and persistence.

Ideas become more memorable when their history has a voice.

Keep what moves you.

Favorite an equation and build a personal constellation of ideas worth returning to.

Share the wonder.

Turn an idea into a beautiful card and send it through the iOS share sheet. You choose when it goes.

Let curiosity compound.

A gentle daily rhythm makes room for depth without turning discovery into homework.

Your curiosity is yours

Designed to feel personal. Built to stay private.

There is no Principia account, advertising, analytics SDK, or tracking. Progress stays on your device and, when iCloud is available, in your private iCloud.

Read the app privacy policy

No profileYour optional display name stays on your device.

Private syncYour progress can follow you through your own iCloud account.

Local remindersDaily notifications are scheduled on your phone, not pushed by us.

Feedback by choiceA message and limited diagnostics leave only after you tap Send.

What’s free, what’s Pro

Start free. Keep discovering with Pro.

Principia is free to download, and a new equation arrives every day. Your first two full reveals include the complete insight, the term-by-term breakdown, and the quiz, all free.

Principia Pro opens the complete story and symbol-by-symbol breakdown behind every new daily equation, plus quick quizzes and your searchable universe of past discoveries, favorites, and collections. Monthly and annual plans renew automatically until canceled, and a one-time lifetime purchase is also available. Eligibility, trial terms, and current prices are always shown before purchase. You can manage or cancel a subscription in your Apple Account settings.

Terms of Use Privacy Policy

Principia for iPhone

One beautiful equation. Each day.

Principia is now on the App Store. One equation each day, revealed with its story, its meaning, and the people behind it. No account. No ads. No noise.

Download on the App Store