New: Quantum Computing, Cryptography & Nuclear Physics — Three Categories Live

⚛️ Quantum Computing — 6 Simulations

Quantum computing is the hardest category to make visually intuitive. The mathematics lives in complex Hilbert spaces; the "particles" are probability amplitudes; the outcomes are stochastic even with perfect operations. The six simulations in this category each tackle a different layer of the quantum stack.

Qubit & Bloch Sphere

The entry point to the category. A Three.js sphere shows the qubit state vector [α, β] in full 3D — you can watch it move as you click the X, Y, Z, Hadamard, S, and T gates. The gate buttons apply 2×2 unitary matrices, and the state smoothly lerps to the new position on the sphere so you can see what each gate actually does geometrically rather than just seeing numbers change.

Quantum Entanglement

Canvas 2D simulation of all four Bell states (Φ⁺, Φ⁻, Ψ⁺, Ψ⁻). Clicking "Measure" collapses the state and shows correlated outcomes for both particles, even though they're drawn on opposite sides of the screen. The CHSH inequality test runs automatically and shows S ≈ 2.828 — a value classically impossible, demonstrating that quantum correlations violate Bell's inequality.

// CHSH S value — quantum mechanics gives 2√2 > 2
const S = Math.abs(E(0, π/8) - E(0, 3*π/8))
        + Math.abs(E(π/4, π/8) + E(π/4, 3*π/8));
// S ≈ 2.828 for maximally entangled Bell states

Grover's Algorithm

The most pedagogically satisfying of the three. Grover's algorithm searches an unsorted N-item database in O(√N) steps, quadratically faster than the classical O(N). The simulation shows N=16 items with an animated amplitude histogram — each Grover iteration you see the target amplitude grow while others shrink. After 3 optimal iterations, the target item has near-unit probability.

🔐 Cryptography — 4 Simulations

The cryptography category covers the mathematical foundations of modern secure communication — from classical substitution ciphers to the asymmetric algorithms that protect every HTTPS connection.

Caesar & Vigenère Ciphers

The historical entry point. Two side-by-side panels let you type plaintext and see the ciphertext update in real time, with a live 26-column frequency histogram comparing your ciphertext distribution against the standard English letter distribution. The Index of Coincidence metric shows exactly why Vigenère defeated simple frequency analysis for two centuries — and why Kasiski's test eventually broke it.

RSA Key Exchange

Step-by-step RSA: choose two primes p and q, compute n = pq and φ(n) = (p-1)(q-1), select public exponent e, calculate private key d via the extended Euclidean algorithm — all shown as animated steps with the complete arithmetic. Encryption and decryption use BigInt modular exponentiation so the maths is exact even for realistic key sizes.

Diffie-Hellman Key Exchange

Two-column layout showing Alice and Bob's parallel computations, alongside a colour-mixing analogy (pioneered by Whitfield Diffie himself as a teaching tool). The "Eve" panel on the right shows why recovering the shared secret requires solving the discrete logarithm problem — computationally infeasible for large primes.

Elliptic Curve Cryptography

A full-screen interactive plot of y² = x³ + ax + b. You can drag two points P and Q and watch the addition P + Q computed visually (reflect the third intersection point through the x-axis). Switch to "Scalar Multiplication" mode and watch k·P built up iteratively — the final point could be a public key.

☢️ Nuclear Physics — 4 Simulations

Nuclear physics sits at the intersection of the very small (nuclear structure) and the very large (energy output, chain reactions). Three core phenomena — fission chain reactions, radioactive decay, and the binding energy curve — plus a fourth covering fusion conditions.

Nuclear Fission Chain Reaction

Canvas 2D simulation of a U-235 fission chain reaction. Neutrons (blue = thermal, white = fast) travel through a field of nuclei. The enrichment slider, moderator slider and neutron multiplicity ν̄ all affect the live k-factor. Below k=1 the reaction dies out; above k=1 it grows exponentially. The sim transitions cleanly between subcritical, critical and supercritical regimes.

Radioactive Decay

Stochastic decay of N₀ starting nuclei against the theoretical exponential N₀·e^(−λt). The split-screen layout shows a grid of nuclei on the left (each one decaying with probability λ·dt per timestep) and a live graph on the right showing the simulated count vs the exact curve. With enough nuclei the stochastic result hugs the theory; reduce N₀ to a handful and the randomness becomes stark.

Binding Energy Curve

The Bethe-Weizsäcker semi-empirical mass formula plotted for A = 2 to 250. Sixteen key nuclides are labelled (H, He, C, Fe, U and others). Adjustable coefficients let you explore what changes when the volume, surface, Coulomb, asymmetry and pairing terms are modified. A vertical marker at Fe-56 (the binding energy peak) delineates the fusion/fission divide.

All Simulations Available in Ukrainian

Every simulation in all three new categories is available in both English and Ukrainian — including the UI labels, the educational descriptions, the error messages, and all structured metadata. Ukrainian learners can explore quantum gates, RSA key generation, and nuclear chain reactions entirely in their native language.

Each Ukrainian page is at /uk/sim-name/ and linked from its English counterpart via hreflang="uk". The JSON-LD inLanguage field, og:locale and page lang attribute are all uk / uk_UA throughout.