A network of neurons. Each one takes a weighted sum of its inputs, squashes it through a nonlinearity, and passes it forward. Two inputs — the coordinates of a point. One output — which class it belongs to. Between them, a hidden layer that can represent whatever geometry the data requires.
At the start, the weights are random. The network's answer is nonsense. But after every guess, the error is computed — how far the output was from the truth — and propagated backward through the network, layer by layer, in the exact direction each weight should move to reduce the error. Gradient descent: the same logic that water finds the bottom of a valley. No one designs the decision boundary. The network finds it by correcting its own mistakes, one step at a time, until the errors stop.
Watch the boundary morph from a flat plane to two interlocking spirals. Increase the hidden layer and watch it fit tighter. Crank the learning rate and watch it overshoot. Reset and watch a different network find the same answer from a different start.
Frank Rosenblatt built the first neural network in 1958 — the Perceptron, a single layer. It couldn't solve XOR. The insight that hidden layers unlock nonlinearity took thirty more years. The order here — a boundary that separates two classes — is not designed. It is found, by a system that knows nothing except its own error. Nobody's in charge. The gradient is.