AlgeBench · Blog Open the app →

Interactive Mathematics

Grounded, AI-driven mathematical proof system

Every step machine-verified, then animated — built into AlgeBench.

Most math on the web is a static wall of symbols. The seven figures below are different: each is a live, interactive derivation — step through it, hover any term to see what it is, and watch the expression morph line by line. Every step also carries a confidence badge showing how it was checked. Play with them first — the how and why is just underneath.

1 · Where the quadratic formula comes from

Everyone memorizes \( x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), but few see why it is true. It is just completing the square on \( ax^2 + bx + c = 0 \), carried out symbol by symbol. Watch the discriminant \( b^2 - 4ac \) assemble itself under the root, and the \( \pm \) split into the two roots.

Figure 1 — deriving the quadratic formula by completing the square. Interactive.
The point of these proofs isn't the animation — it's that every step is graded, and you can read the grades at a glance. The pill in the card's top-right corner is the overall badge — a single verdict for the whole proof. Click it and it stays pinned open, and every step lights up with its own per-step badge across the step navigator, each number underlined in its tier color. Hover any badge to read the verdict in words. The colors and icons run from strongest to weakest: ★ Grounded (symbolically proven from the line above) · ✓ Verified (strong CAS evidence) · ✦ Domain (vouched for by a domain expert) · ◇ Plausible (valid, but the CAS couldn't decide) · ○ Unchecked (nothing to convert) · ✗ Refuted (disproven — caught and never shipped). In this proof almost every step glows ★ gold: completing the square is pure algebra the CAS proves line by line. The full grading pipeline is spelled out in How it works, below.

2 · How a spinning station fakes gravity

Spin a space station and the floor pushes its occupants outward — artificial gravity, no mass required. The felt strength, \( g_a = \omega^2 R \), drops straight out of circular motion: a rim point at radius \( R \) moves at \( v = \omega R \), and its centripetal acceleration \( v^2/R \) becomes the weight you feel. Watch the rim speed substitute itself in, line by line.

Figure 2 — deriving artificial gravity \( g_a = \omega^2 R \) for a rotating habitat. Interactive.
Every symbol is live. Hover any term — say \( \omega \) or \( R \) — and every copy of it lights up across the whole derivation. A small ✦ AI button fades in beside it: click it to jump into the full app and chat with an AI tutor that already has the proof and the exact step you were on loaded as context. Here it opens the live 3D rotating habitat — spin it up and watch the gravity arrow grow, just as you derived it.

3 · A spacecraft hitting the atmosphere

Now something you would normally only meet in an aerospace course: the Allen–Eggers model for a body entering an atmosphere. Starting from the equation of motion through an exponential atmosphere, the derivation separates variables, integrates, and solves for the velocity as a function of altitude. Notice the mixed badges — some steps the CAS verifies outright, the genuinely uncheckable integration moves are vouched for as domain knowledge.

Figure 3 — Allen–Eggers entry velocity. Open ⓘ Explore for the follow-ups.

4 · How much “push” a rocket really has

The most important equation in rocketry is the Tsiolkovsky rocket equation: the total velocity change \( \Delta v \) a stage can deliver depends only on its exhaust speed \( v_e \) and how much of its mass is propellant. It falls straight out of conservation of momentum — separate the variables, integrate, and the mass ratio becomes a logarithm. This is the “\( \Delta v \) you have” — watch for the teal Domain badges on the honest integration steps.

Figure 4 — the Tsiolkovsky rocket equation, \( \Delta v = v_e \log(m_0/m_f) \) (natural log, as the CAS writes it). Interactive.

5 · The burn that sends you to the Moon

So that’s the \( \Delta v \) you have; here’s the \( \Delta v \) you need. Artemis II's trans-lunar injection (TLI) is a single push, and its size is just a difference of two orbital speeds: the perigee speed on the transfer ellipse (from the vis-viva equation) minus the circular parking-orbit speed. Watch \( \Delta v = v_p - v_c \) take on the two velocity formulas and collapse to one clean expression.

Figure 5 — the trans-lunar injection Δv, from \( v_p - v_c \) to the factored form. Interactive.

6 · Why a qubit is a point on a sphere

One from quantum computing to finish. A qubit's state is two complex amplitudes — four real numbers — yet every textbook draws it as a single point on the Bloch sphere. This derivation shows why: normalization removes one degree of freedom, the unobservable global phase removes another, and what survives is exactly two angles, \( \lvert\psi\rangle = \cos(\theta/2)\lvert 0\rangle + e^{i\phi}\sin(\theta/2)\lvert 1\rangle \). The badge mix tells the honest story again: the phase-factoring algebra is Grounded, while genuinely physical moves — writing amplitudes in polar form, discarding the global phase — carry teal Domain badges.

Figure 6 — parameterizing a qubit onto the Bloch sphere, \( (\theta, \phi) \). Interactive.
Hover \( \theta \) or \( \phi \) to see what each angle means, then click the ✦ AI button to land in the quantum-states lesson — the proof and the step you were on already loaded next to the live Bloch sphere.

7 · Why trapped energy comes in \( n^2 \) steps

Confine a particle — an electron in an atom, light in a cavity — and its energy can no longer take just any value: it climbs in discrete levels. This derivation shows where those levels come from, and why they grow as \( n^2 \) rather than \( n \). Only whole half-wavelengths fit between the walls (\( \lambda_n = 2L/n \)); de Broglie turns that into a momentum ladder \( p_n = nh/2L \) that rises linearly in \( n \); then kinetic energy \( E = p^2/2m \) squares it into \( E_n = n^2 h^2 / 8mL^2 \). The badge mix tells the honest story once more: the squaring-and-simplifying algebra is Grounded, while the two physical inputs — the de Broglie relation and the kinetic-energy formula — carry teal Domain badges.

Figure 7 — deriving the quantized energy levels \( E_n = n^2 h^2 / 8mL^2 \). Interactive.
Step through it and watch the linear momentum ladder \( p_n \propto n \) get squared into the \( n^2 \) energy spectrum, then click the ✦ AI button to jump into the photons lesson — the proof docks beside the live standing-wave scene where these exact energy levels are drawn, with the AI ready to explain why they climb as \( n^2 \).

8 · Why moving clocks run slow

The most famous result in special relativity falls out of one gadget and one refusal. The gadget: a light clock — a photon bouncing between two mirrors a height \( h \) apart, one tick per strike. The refusal: light will not inherit your speed. Watch the clock slide sideways and the photon is forced onto a diagonal — a right triangle with the mirror gap \( h = c\,T_0 \) as one leg and the slide \( v\,\Delta t \) as the other. Since the diagonal is covered at the same \( c \), Pythagoras is all it takes: \( (c\,\Delta t)^2 = (c\,T_0)^2 + (v\,\Delta t)^2 \), and solving for \( \Delta t \) delivers time dilation, \( \Delta t = \gamma\,T_0 \). No postulate about time anywhere — just a triangle a runaway mirror drew, and \( \gamma \) as the price of the chase.

Figure 8 — Lorentz time dilation from the light-clock triangle. Interactive.
Click the ✦ AI button to jump into the full Special Relativity lesson — the proof docks beside the live light-clock race, where two clocks tick side by side, flashes bloom as planted wavefronts, and the tutor is ready for the question everyone asks: why does this slow all clocks, not just this one? From there the lesson runs all the way to the twin paradox and the Rindler horizon of a 1g starship.

How it works: written by an AI, checked by a computer

Here is the unusual part. Each derivation above is written by a language model — and then checked, step by step, by a computer-algebra system (CAS). The AI proposes the next line; the CAS tries to prove that line genuinely follows from the one before it. The outcome of that check is stamped onto every step as a colored confidence badge, so you never have to take the AI's word for anything. Hover the pill in a card's top-right corner (or the colored dots under the steps) to read each verdict:

Want the full picture — the AI derivation, the CAS grading pipeline, and how the confidence tiers are decided? Learn how it works →

When the CAS can't decide: domain rescue

Real derivations contain honest moves that no CAS can verify as a pure symbolic identity — separating the variables of a differential equation, an integration step, a physical modeling assumption. Left alone, the CAS stamps these Unchecked, which is unfair: they are correct, just not mechanically provable as an identity. So a second pass asks an AI acting as a domain expert — “is this step valid, in context?” — and only when it vouches with high confidence is the step promoted to the teal Domain tier. It's a deliberate middle ground: an expert accepts this; the CAS could not prove it — strictly more honest than hiding the gap. It's exactly why the Allen–Eggers proof above reads as a mix: green grounded algebra, with teal Domain badges on the separation-and-integrate moves.

How these figures are built — and how to embed them

Each proof is embedded as a plain <iframe> — the same snippet you could drop into your own blog. After it loads it makes no further calls home, so it is safe to publish anywhere. Three things to try in any figure:

Step or play — use the numbered buttons or ▶ Play to morph between lines · Hover a term — point at any symbol (even one buried inside a √ or fraction) to light it up and read what it means · ⓘ Explore — open prerequisites and follow-up questions.

The terms are live — and every one is a door to an AI tutor

Every symbol in these proofs is interactive: hover it to light up each place it appears and read a one-line description. But the real exit is the AI button. Three things open a conversation, not just another formula — the AI ask button on a term or step, a Prerequisite chip, or a follow-up question in the ⓘ Explore panel. Click any of them and AlgeBench opens the full app, landing you in a chat with an AI tutor that already knows the context: which proof you were reading, which step, which symbol — so you pick up the thread instead of re-explaining it.

You can also select more than one term before pressing the AI button — ⌘-click (Ctrl on Windows) to add each one. The tutor then explains your last-selected term in relation to the others you picked, so you can probe how symbols connect — say, how the discriminant \( b^2 - 4ac \) ties back to \( a \), \( b \) and \( c \) — not just what each one means on its own.

For Allen–Eggers that doorway leads somewhere special: the interactive 3D atmospheric-entry scene loads with the tutor listening. Rotate the trajectory, drag the entry angle, and simply ask — “why does deceleration peak so high up?” — and talk it through.

Once you're in the app, open the SCENES tab for the topic's 3D interactive visualization — rotate it, drag the sliders, and watch the geometry behind the algebra move. (The quadratic, for one, opens onto its parabola and the saddle surface beneath it.)

Open the 3D scene + AI tutor →

So each figure is a doorway, not a dead end: a figure on a blog → an interactive, verified proof → a 3D scene where you can question the model directly. No account, no install — just follow the link.

Curious how the whole system works under the hood? Learn how it works →

Every figure on this page is a live AlgeBench embed (an <iframe> plus a tiny auto-resize script). The math renders through KaTeX with a strict trust filter, and the page makes no backend calls after the proof loads. Want one of these in your own post? Open any proof in AlgeBench and hit the </> button to copy the embed snippet.