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Drag two conics. Sometimes you see four crossings. Sometimes two. Sometimes none. Bezout says the count is still four. Where did the missing intersections go — and what does a graphics engine do about it?

Lossy compression isn't bound by entropy — it picks what to discard. JPEG changes basis (DCT) so the picture becomes sparse, throws away the coordinates that don't matter (quantization), and Huffman-packs the rest. Three steps, one savings: fewer coefficients, smaller values, longer zero runs.

Two same-size images: one shrinks, one doesn't. The histogram tells half the story; the spatial structure tells the rest. The entropy module names the floor — this page shows what that floor isn't bounding.

A designer drags four handles. A movie character gets a smooth cheek, a car gets a perfect hood, a letter gets its curve. How does a computer turn a few points into a smooth path?

A pendulum left alone slows and stops; the same pendulum pushed on the beat swings higher and higher. One equation — ẍ + 2γ·ẋ + ω₀²·x = F(t) — runs both. Damping decides how it stops; resonance decides when an outside push wins.

Double the swing. Why does the clock barely change? A real pendulum's period depends on amplitude — but for small swings it almost doesn't. Linearize sin θ ≈ θ, and the period becomes constant. The clock stands on that approximation.

Gravity pulls forever, but a falling raindrop doesn't speed up forever. The reason is one first-order equation — a derivative balanced against a force that grows with speed. Terminal velocity is not a maximum; it's an equilibrium.

Throw a ball. Ignore air. Its horizontal motion keeps time; its vertical motion loses to gravity. Why does that make a parabola?

A model is wrong. You can measure how wrong. But which knob should move, and by how much? The derivative gives the direction; the learning rate gives the distance. Repeat. That five-line loop trains every modern ML model.

A model does not know it is right. It has scores. Softmax turns those scores into numbers that look like probabilities. Cross-entropy punishes the probability the model gave to the correct answer. The trap: a bad score can still become a very confident probability.

A model says '70% confident.' Does that mean, across many such predictions, seven in ten are right? The number on the screen and the long-run frequency are two different quantities. The reliability diagram makes the gap visible; one scalar — temperature — rotates the curve back to the diagonal.

Why does Google show those results in that order? For thirty years the decision rule barely changed. TF-IDF ranks documents by a score that looks like a sum of probabilities — and the log inside it is the same log as in entropy. Rare words carry many bits; common words carry near zero. Stopwords aren't a list, they're the zero set of a function.

A future dollar is not a present dollar. Discounting and integration price every cash-flow stream — from rent to bonds to perpetuities — through one identity: future cash times the discount factor, summed across time.

On May 22, 2010, a Florida programmer paid 10,000 BTC for two Papa John's pizzas — about $41. Sixteen years later, those coins are worth $1 billion. The most expensive meal in history.

Two risky assets, mixed, can be less risky than either alone. The mathematics is one cross-term in a quadratic — and it's the same identity that makes a vector sum's length more (or less) than the sum of its parts. Variance, covariance, correlation; the dot product of returns.

Anyone can see your Bitcoin address. No one should see your private key. So how can the network verify that you authorized a payment without learning the secret that authorizes all payments?