Math
Riemann surfaces
The sheets of the complex logarithm and the square root, joined along their branch cuts.
Available for Windows
Makarov Math Suite
160 interactive modules bring geometry, topology, fractals, calculus, algebra, probability and algorithms into view. Rotate a surface, change a construction or follow an algorithm one step at a time.
160 modules · Page 2 / 8
Math
4D polytopes
The tesseract, the 24-cell and the 600-cell, rotated in four dimensions and projected into three.
Math
Platonic and Archimedean solids
Five perfect solids, their truncations and their duals: cut the corners and the football appears.
Math
Minimal surfaces
Catenoid, helicoid, Enneper, Scherk, Schwarz P and the gyroid: the shapes a soap film settles into.
Math
Möbius strip
One side, one edge, and an ant that comes back mirrored: it paints the band as it walks the core, and after one lap on the Möbius strip it is painting what looked like the other side.
Math
Hyperbolic geometry
The Poincaré disc and the hyperboloid: tilings that never run out of room, sliding by isometries.
Math
Knots
Torus knots and links: the trefoil, the cinquefoil, Solomon's link, and the crossings a picture cannot get rid of.
Math
Linear transformations
The whole space deformed by a matrix: the unit cube carries the determinant and turns inside out through zero volume, the unit ball becomes the ellipsoid of the singular values, and the eigenvectors are the rods that do not turn.
Math
Eigenvectors
Explore power iteration: repeated multiplication can align vectors with a dominant eigenvector. Compare the measured angular error with the asymptotic spectral ratio, and inspect the Rayleigh quotient for symmetric matrices.
Math
Singular value decomposition
Any matrix as a rotation, a stretch and another rotation: the unit sphere turns, stretches into the ellipsoid of the singular values and turns into place, one factor at a time, and the one tripod whose images stay orthogonal is measured, not drawn.
Math
Optimisation on a landscape
Gradient descent, the heavy ball, the Nelder–Mead simplex, simulated annealing and a genetic population go down the same surface - the bowl shows the condition number, the banana shows the crawl, Himmelblau shows the basins and Rastrigin shows who gets stuck.
Math
Fourier series
Epicycles assembling a drawing, sines assembling a wave, and the overshoot at a jump that never goes away.
Math
Vector fields
Thousands of tracers reveal a three-dimensional field with luminous fading paths. Orbit the cloud, change its field and emitter, then measure divergence, curl, flux and circulation.
Math
Logistic map and bifurcation
The period-doubling cascade as a glowing curtain coloured by the Lyapunov exponent, with a live cobweb orbit at the marker - doublings at 3 and 1 + √6 exactly, chaos at the end.
Math
Monte Carlo
Random darts estimate π, two hundred throwers show the spread predicted by 1/√N and the 95 % band, and one integral is taken three ways - the sampler that leans like the curve wins twentyfold.
Math
Percolation
An inlet, an outlet and one sample of a random lattice: predict at which p this sample first lets a flood through, raise p or open single sites by hand, and watch the reachable region spread - then run many samples at several sizes and see the crossing point spread out around p_c and narrow as the lattice grows.
Math
Graph theory
Shortest paths, spanning trees and flows on one graph, drawn as they are computed: Dijkstra's ripple, Kruskal's islands, Edmonds–Karp's paths and the cut that equals the flow.
Math
Voronoi and Delaunay
Nearest-neighbour cells that reshape as you drag their sites, the Delaunay triangulation dual to them with its empty circles, discs that grow into the cells, and Lloyd's relaxation that turns them into hexagons.
Math
Travelling salesman
A tour that shortens while you watch - greedy, 2-opt uncrossing edge by edge, annealing climbing over the wall - against the exact optimum for thirteen cities and brute force over every tour for eight.
Math
Collatz conjecture
Halve if even, triple and add one if odd: does every positive integer reach one? Explore a number's flight, stopping times up to 4096, and the different behaviour of 3n − 1 and 5n + 1. The general 3n + 1 claim remains a conjecture.
Some modules belong to more than one Suite. Shared experiments have one description here.