Visualisations

Computer vision

• Receptive field of a CNNHow the kernel size and stride at each layer grow the input region a single neuron sees.
• Epipolar geometry and multiple viewsAn orbitable two-camera scene tying together the epipolar plane, the essential and fundamental matrices, triangulation and homography.
• Camera projectionHow the extrinsics [R|t] and the intrinsics K turn a 3D world point into a pixel.
• Correspondences and RANSACDescriptor matching with Lowe's ratio test, then RANSAC fitting a model and rejecting the geometric outliers.
• Optical flow and the aperture problemWhy a single moving edge pins down only one velocity component while a corner recovers both.
• The SigLIP sigmoid lossThe image-text similarity matrix as n² independent sigmoid classifications, pulling the diagonal up and every other pair down, with a bias that absorbs the positive/negative imbalance — against CLIP's per-row softmax.
• HOG as a convolutional networkThe Histogram of Oriented Gradients read as a tiny fixed CNN: oriented edge filters (conv), pooling responses into per-cell orientation histograms (pool), L2-normalising them (layer-norm) and flattening to a descriptor, with a clickable rose-grid and per-cell stage breakdown.

Continuous optimisation

• The trust-region subproblemThe model contours, the trust region and the secular equation that solves the boundary case.
• The KKT conditionsThe four conditions read as a force balance, with controls to make each one hold or fail in turn.
• Critical cone and feasible directionsThe set of linearised feasible directions and the critical cone drawn as a scatter of candidate step directions.
• Barrier method and the central pathThe log-barrier objective and the central path traced from the analytic centre to the optimum as the penalty rises.
• Armijo and Wolfe conditionsThe sufficient-decrease and curvature conditions read off the line-search slice f(x+αs), with the acceptable step lengths shaded where both hold.

Optimisation for data science

• Convexity, strong convexity and L-smoothness boundsThe tangent-line floor, the γ-strong-convexity quadratic floor and the L-smoothness quadratic ceiling that trap a function at any anchor point.
• The proximal operator and Moreau envelopeThe prox step as a sliding proximity bowl meeting the penalty, read off as the soft-threshold/projection/shrinkage map and as the Moreau-smoothed envelope.

Galois theory

• The Galois group acting on the rootsThe Galois group as exactly the permutations of the roots that preserve every algebraic relation between them, shown in the complex plane.
• A gallery of Galois groupsOne small plot per automorphism of a chosen polynomial, drawing how it permutes the roots, so the grid is the whole group. After Chris Grossack's pretty pictures.
• The Galois correspondenceThe order-reversing bijection between subgroups and intermediate fields, drawn as twin Hasse diagrams with the normal subgroups picked out.
• Extending the identityBuilding the automorphisms one generator at a time, so the branching degree at each tower step multiplies out to the order of the Galois group.
• Cyclic Galois groupsPower-map Galois groups: the cyclotomic action on the roots of unity and the Frobenius cycle in a finite field.