Map of Mathematics
The abstract rules beneath all science — from infinite fractal geometry, the golden ratio, prime numbers to probability & infinity.
abstract · number · shape · infinityFractal Geometry Hình học Fractal
The Mandelbrot & Julia sets: zoom in forever and they repeat themselves (self-similarity). Born from one complex formula z² + c.
Golden Ratio Tỉ lệ vàng & Fibonacci
The sequence 1,1,2,3,5,8… → the ratio φ ≈ 1.618 appears in seashells, sunflowers, spiral galaxies.
Prime Numbers Số nguyên tố
The building blocks of arithmetic — the Sieve of Eratosthenes, the Ulam spiral, a thinning distribution & cryptographic uses.
Probability Xác suất & Ngẫu nhiên
A Galton board: a thousand beads falling at random → self-arrange into a bell curve (the normal distribution).
Geometry Hình học & Định lý
Pythagoras by area, the triangle angle sum, π from polygons — proven with animation.
Calculus Hàm số & Giải tích
The derivative = instantaneous slope, the integral = area under the curve — visualizing the limit.
Trigonometry Lượng giác
The rotating unit circle generates the sin & cos waves; tan, radians and the identity sin²+cos²=1.
Graph Theory Lý thuyết đồ thị
The 7 bridges of Königsberg & Euler paths, map coloring, BFS/DFS traversal and the minimum spanning tree.
Number Bases Hệ cơ số
The same number in base 2/8/10/16: place value, binary, hex by nibble and two’s complement for negatives.
Topology Tô-pô
The geometry of stretching: a mug = a donut, the one-sided Möbius strip, the Klein bottle — invariants by number of holes & Euler’s χ.
Fourier / Epicycles Chuỗi Fourier & Bánh răng quay
Many nested rotating circles, the pen tip drawing any shape — ‘analysis’ = summing many sine waves. Add harmonics → the shape sharpens.
Linear Transforms Ma trận & Biến đổi tuyến tính
The coordinate grid scaling/rotating/shearing by a 2×2 matrix; eigenvectors stay on their axis, the determinant = the area ratio.
Complex Plane Số phức & Mặt phẳng phức
Multiplying complex numbers = rotate + scale; Euler’s formula e^iθ circling the unit circle, generating sin & cos.
Voronoi / Delaunay Sơ đồ Voronoi & Delaunay
Scatter seeds → each cell is the ‘nearest’ region; move a seed and the cells shift instantly. Cell towers, cells, crystals all follow this rule.
Tessellation / Penrose Lát mặt phẳng & Đối xứng
Tiling with no gaps: only 3 regular shapes tile; Penrose tiles non-periodically with 5-fold symmetry by the golden ratio.
Bézier Curves Đường cong Bézier & Nội suy
Drag the control points → the de Casteljau algorithm builds the curve. How fonts & design software draw every curve.
Pascal / Combinatorics Tam giác Pascal & Tổ hợp
Each cell = the sum of the two above: binomial coefficients, counting paths. Color odd/even → the Sierpinski triangle appears.
Gradient Descent Tối ưu & Xuống dốc gradient
A ball rolling down the ‘loss valley’ along the steepest slope — how neural networks learn. Too big a step overshoots, too small crawls.
Infinity / Cantor Vô hạn & Đếm được
Hilbert’s hotel of infinite rooms, Cantor’s diagonal: ℕ & ℚ are ‘countable’ but ℝ is not — there are many sizes of infinity.
Conic Sections Đường conic & Quỹ đạo
Slice a cone at different angles → ellipse, parabola, hyperbola. Eccentricity decides a planet’s orbit shape (Kepler’s laws).