L2: school
When two things land on the same value
MATHThe birthday problem looks like a party trick and is actually the standard tool for reasoning about collisions. The second step is where computer scientists meet it again, under a different name.
Computing
A computer does exactly what you said, fast enough to make the difference between a good algorithm and a bad one impossible to ignore. These tools run both and let you watch the gap open.
Newest in this category
- Floating point Type a decimal and watch a computer convert it: sign, binary, normalise, bias, round. The value it ends up holding, the error, and the distance to the next number it can hold at all.
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Recursion Tree Explorer
Count the calls, distinct subproblems and stack depth of a naive recursive call tree for Fibonacci, Tower of Hanoi, Pascal's rule, merge sort and stair climbing, and see exactly where a memo table pays.
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Screen Resolution and Viewing Distance
How much of a screen's resolution actually reaches your eye, from its diagonal, its pixel count and where you sit. At 20/20 vision the answer is the field of view measured in arcminutes, and on a living-room television it usually lands below 1080p.
Chapters
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
L2: school
How a video gets encoded
COMPA video encoder is not one algorithm but eight stages in a fixed order, and the order is not arbitrary: each stage exists because the one before it made its job possible. This tool models one of them. The chain below links to the other seven.
L3: undergraduate
How an LLM picks the next word
COMPTwo tools that open up a large language model at the moment it chooses a word. The second literally runs on the first: the candidates it ranks are entries from the vocabulary the first one builds.
L3: undergraduate
Counting work, not seconds
COMPA stopwatch measures your laptop. Counting operations measures the algorithm. Two tools built on that one habit.
L3: undergraduate
Compression by hand
COMPFour encoders you could run on paper, ordered by what each one can see that the last one could not. The fourth compresses nothing whatsoever, and it is the reason the first one works on text.
Problems solved in full
- The Sierpiński triangle at depth 5 with 243 pieces
- Five live cells on a 30 × 30 grid
- The real chromatic number of the Petersen graph
- Two iterations to find a marked item among 8
- A 75% full table averaging 3.5 probes per lookup
- The saving from encoding "hello world" with Huffman coding
- The cheapest edit sequence to turn kitten into sitting
- Sum of minterms for the expression !(A & B)
- Six tokens of "abracadabra" with a search window of 16 characters
- The search window derived from 225 positions per block
- Where the four entropy figures on the panel come from
- The boundary for a 10-cycle pattern sampled 22 times
- Run-length encoding AAAAAABBBCCDDDDD into four pairs
- An RSA key pair for p = 61 and q = 53
- Twelve candidate continuations that cover 13.6% of what actually follows
- Compression when sixty-eight characters become thirteen tokens
- Configurations the palindrome program passes through on 1 0 1 0 1
- Compression of two minutes of 1080p30 at 6 Mbit/s
- Spreads for 2 walkers over 10 steps
- Gradient magnitude at pixel (31, 3) of the checkerboard test image
- Why no finite machine can match up 0s and 1s, however many states you give it
- A half adder built from NAND gates alone, and why five is the floor
- The target mismatch gap of 0.453 over 180 words
- Predicting bubble sort's swap count exactly on 64 reversed elements
What you find out here
- BFS finds the shortest path only when every edge costs the same BFS vs DFS Visualizer
- One adder does subtraction too Binary Arithmetic & Two's Complement Lab
- At δ = π the two halves cancel exactly Bloch Sphere Explorer
- Without padding, a deep network shrinks the image away Convolution and Feature Maps Playground
- Pick the variable that explains the most Boolean Expression Solver
- At 20/20, the columns you can use are the field of view in arcminutes Screen Resolution and Viewing Distance
Articles written from these tools
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Half of written Japanese is 158 characters. The next 45% costs a thousand more.
Word frequencies follow the same lopsided law in every language anyone has checked. It makes the first stage of learning feel miraculous and the last stage feel impossible, and both feelings are the same curve.
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Why sorting cannot get faster
Sorting a hundred items takes at least 525 comparisons and no future algorithm will improve on it. The proof does not examine any algorithm at all: it counts how many arrangements exist and how much a yes-or-no question can possibly tell you.
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Why a language model struggles to count the letters in a word
It never sees the letters. Text is split into tokens before the model reads anything, and a great many puzzling failures follow directly from where the splits land.
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No compressor can shrink every file, and the proof takes one paragraph
Lossless compression works by exploiting predictability, so a file with none of it cannot be made smaller at all. The proof is counting: nine things will not fit into eight boxes.
Every tool on this shelf
Algorithms & Data Structures
BFS vs DFS Visualizer
Compare breadth-first and depth-first graph traversal side by side on various graph topologies.
Big-O Complexity Explorer
Visualize how O(1), O(log N), O(N), O(N²), O(2ᴺ) and O(N!) growth rates compare at different scales.
BST / AVL Tree Visualizer
Build and visualize Binary Search Trees and AVL trees with auto-balancing rotations.
Dijkstra's Shortest Path
Find shortest paths in weighted graphs with step-by-step Dijkstra visualization.
Graph Coloring Explorer
Color graph vertices, test chromatic constraints, and explore clique lower bounds.
Hash Table Visualizer
See how hash tables store data with chaining, linear probing, and quadratic probing collision resolution.
Levenshtein Distance Visualizer
Compute the edit distance (insertions, deletions, substitutions) between two strings and visualize the comparison.
Pathfinding Algorithm Visualizer
Watch A*, Dijkstra, Greedy Best-First, and BFS search a grid, step by step.
Recursion Tree Explorer
Count the calls, distinct subproblems and stack depth of a naive recursive call tree for Fibonacci, Tower of Hanoi, Pascal's rule, merge sort and stair climbing, and see exactly where a memo table pays.
Sorting Algorithm Race
Compare sorting algorithms on the same input.
Information Theory & Encoding
Binary Arithmetic & Two's Complement Lab
Explore bit weights, signed representations (Two's Complement, One's Complement, Sign & Magnitude), and step-by-step binary addition, subtraction, and multiplication.
Burrows-Wheeler Transform (BWT)
See how BWT reorders text into runs that compress better, and reverse it losslessly.
Encoding/Base Detector & Converter
Detect likely encoding or structured format, then decode and explain what it probably is.
Entropy Coding Playground
Relate symbol distributions to Shannon entropy, expected compression gain, and next-token prediction in AI/LLMs.
Floating point
Type a decimal and watch a computer convert it: sign, binary, normalise, bias, round. The value it ends up holding, the error, and the distance to the next number it can hold at all.
Huffman Encoding Visualizer
Build a Huffman tree from text input and see the optimal prefix-free binary codes and compression ratio.
LZ77 Compressor Demo
Step through sliding-window back-references used in DEFLATE, ZIP, and gzip.
NLP Entropy Lab
Token entropy, cross-entropy, perplexity and KL divergence — the four numbers a language model is trained and judged on.
Run-Length Encoding (RLE)
Compress repeated symbols into count-value runs and inspect best/worst cases.
Signal & Media Processing
Chroma Subsampling Lab
Compare 4:4:4, 4:2:2, and 4:2:0 to see color-detail loss in image and video compression.
Color Contrast Checker (WCAG Luminance)
Check text/background color contrast against WCAG 2 accessibility thresholds — and see how the luminance formula behind it descends from 1953 broadcast television's need to squeeze color into an existing black-and-white signal.
Convolution and Feature Maps Playground
Explore convolution kernels, stride, and padding with live input/output feature maps.
GOP Structure Visualizer
Understand I/P/B frame patterns, seek latency, and compression tradeoffs in video streams.
Motion Estimation Simulator
Inspect block matching, motion vectors, and residual energy in inter-frame coding.
Pixel Sampling Lab
Understand sampling, Nyquist, and aliasing with signal and pixel-grid visualizations.
Quantization Lab
Explore bit depth, quantization error, and banding with live visual examples.
Rate Control Simulator
Compare CRF and CBR behavior under changing scene complexity and buffer limits.
Sobel vs Canny Edge Detector
Apply Sobel gradients and Canny hysteresis to compare edge-detection pipelines.
Video Encoding Lab
Learn bitrate, codec, CRF/CBR, and quality tradeoffs with live visual estimates.
Automata & Cryptography
Boolean Expression Solver
Type Boolean logic expressions and generate a full truth table with intermediate sub-expression columns.
Finite State Machine Animator
Step through DFA transitions with a live state diagram — watch which state is active after each symbol and whether the input is accepted.
Logic Gates Explorer
Interactive logic gate and small-circuit visualizer with signal pulses, timing strips, and compact truth tables.
RSA Encryption Toy
Explore RSA public-key cryptography with small primes - see key generation, encryption, and decryption step by step.
Turing Machine Simulator
Simulate a Turing machine with customizable tape and transition rules - the foundation of computation.
Demos & Utilities
Bloch Sphere Explorer
Explore qubit states on the Bloch sphere with theta/phi controls, basis-aware probabilities, and H, X, Y, Z gate transformations.
Conway's Game of Life
Simulate Conway's Game of Life with classic patterns - gliders, guns, and oscillators.
Fractal Explorer
Explore Sierpinski, Koch, fractal trees, and dragon curves at adjustable recursion depth.
Grover's Search Algorithm
Step through Grover's quantum search algorithm iteration by iteration. Watch the target amplitude grow while others shrink - O(√N) vs classical O(N).
HTML Asset, Image & Markdown Extractor
Paste HTML and extract linked files by category, with image-first gallery output and a Markdown text export.
Lorem Ipsum Generator Lab
Generate seeded placeholder text with entropy controls, Markov mode, and Zipf frequency diagnostics.
Quantum Random Walk
Compare classical and quantum random walks side by side. See how quantum interference creates a double-peaked distribution that spreads quadratically faster.
Other tools
Bitrate vs Size Calculator
Convert video/audio bitrate and duration into realistic file size budgets.
BPE Tokenizer Explorer
See how byte-pair encoding splits text into the tokens a language model actually reads. Fixed vocabulary, live inference — watch English cost 13 tokens where the same German sentence costs 36.
Screen Resolution and Viewing Distance
How much of a screen's resolution actually reaches your eye, from its diagonal, its pixel count and where you sit. At 20/20 vision the answer is the field of view measured in arcminutes, and on a living-room television it usually lands below 1080p.
Temperature and Sampling
Watch a real next-token distribution get reshaped by temperature and truncated by top-k and top-p. Temperature never changes which token leads — only by how much, from 278x down to 1.76x.
Token Vectors and Similarity
Vectors built by counting which tokens appear near which, over this site's own prose. The pair that scores highest is ␣the and ␣is at 0.505 — higher than ␣Earth and ␣Sun at 0.299, and it means nothing.