One neuron, trained live: every mistake nudges the line towards the point it got wrong. On separable data it always stops; on XOR it never can.

Step, sigmoid, tanh, ReLU and GELU side by side with their slopes — and what a stack of layers can draw with each one, and with none.

Trace one input through a 2-3-2 network, multiply by multiply: weighted sums, activations, then softmax turns two scores into probabilities.

Watch a trained network stretch, shift and squash the plane layer by layer until two tangled moons come apart along a straight line, and see why two units per layer can never cut a ring off its core.

Record L = (wx + b − y)² as a tape of tiny operations, each storing its own local derivatives, replay it newest first, then use w twice and watch its gradient become a sum.

Step through every gradient of a real network: upstream × local on each edge, sums where paths meet, each checked against finite differences. Then train XOR and break it.

SGD, momentum, RMSProp and Adam race down the same narrow ravine from the same start. Watch who zig-zags, who overshoots, who reaches the floor first and who cannot stay there, then turn the ravine 45°.

Follow a gradient back through 12 layers and watch it shrink to 10⁻¹¹ with sigmoid, still vanish with ReLU at the default scale, explode at twice He’s, and keep its size with He initialisation.

Batch norm and layer norm, computed on a batch you can read: subtract the mean, divide by the spread, then let γ and β put back what training wants.

Train an oversized network on a few noisy points and watch it memorise them; then switch on dropout or weight decay and watch the boundary calm down.

Pick a dataset, stack layers, choose activations, an optimizer and regularisation, and train it live — with every unit’s picture of the plane on view.