Visual Intuition & Engineering Guides
Deep dives into the mathematics, algorithms, and visual teaching methods that make complex technical ideas click.
Why Instant Visual Feedback Matters When Learning Hard Subjects
When curiosity strikes, waiting around for a course to load breaks your focus. Here is how PAITS gets you into your first interactive lesson in seconds while full video chapters and 3D models assemble in the background.
Visualizing the Fourier Transform: From Complex Plane Winding to Spectral Decomposition
An interactive visual walkthrough of Euler's formula, rotating epicycles, and frequency decomposition with step-by-step mathematical proofs.
Implementing an O(1) LRU Cache in Python & JavaScript with Live Sandbox Verification
Step-by-step HashMap + Doubly Linked List pointer manipulation, complexity bounds, and runnable in-browser unit tests.
Transformer & LLM Architecture Reference: Scaled Dot-Product Attention, RoPE & KV-Cache Math
Complete mathematical reference and downloadable vector PDF cheat sheet covering multi-head attention, SwiGLU, and KV-cache memory sizing.
Scaled Dot-Product Attention & QKV Geometry: Why We Divide by √d_k
Interactive visual derivation of Query-Key-Value projections, dot-product variance scaling, and softmax Jacobian saturation.
Token Bucket vs. Sliding Window Rate Limiter: Production Redis Lua & Python Implementation
Compare burst tolerance, memory overhead, and atomic distributed rate limiting with runnable test suites.