Knights — Spiral Leaper-Pattern Explorer

Visualize patterns that emerge from placing chess "leaper" pieces (knights and custom variants) along an Ulam spiral on an infinite board. Inspired by a recent Numberphile video.

The basic rule: walk the spiral outward from the origin. At each tile, place a knight if no previously-placed knight attacks it. Team mode adds N colors with custom leapers — same-color pieces cooperate, different colors block.

Build (native)

cmake -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build -j
./build/knights

First configure fetches Dear ImGui and SDL3 via CMake FetchContent and takes about a minute. Subsequent builds are fast.

Build (web, Emscripten)

source $EMSDK/emsdk_env.sh
emcmake cmake -B build-web -DCMAKE_BUILD_TYPE=Release
cmake --build build-web -j
emrun build-web/index.html

Targets WebGL 2 / OpenGL ES 3.0. The build produces index.html, index.js, and index.wasm ready to drop on a static host.

Publish

cmake --install build-web                  # → build-web/dist/
# or, to a custom location:
cmake --install build-web --prefix ./out

Either form copies just the three files needed to serve the page.

Controls

• Left-click + drag inside the canvas: pan
• Mouse wheel: zoom (toward cursor)
• Touch: one-finger drag to pan, two-finger pinch to zoom
• Sidebar:
  • Reset: clear board, keep colors & leapers
  • Center view: snap pan/zoom back to origin
  • Auto-run + Budget / frame: how many placements to compute per frame
  • Step 1k / 10k / 100k: manual stepping
  • Per-color row: color picker, Edit movement..., remove, count, and a Cooperate with same color toggle (off = basic-mode self-attack; on = team-mode cooperation)
  • Add color: appends a new color (knight by default, cooperative)

Movement editor

A 9x9 grid. The center cell is the piece. Click a cell to toggle whether the piece can leap to that offset. Symmetric (default on) mirrors clicks across the origin, so a knight needs 4 clicks instead of 8.

Self-test

A small headless correctness check is built into the binary:

KNIGHTS_SELFTEST=1 ./build/knights

Runs the basic-mode simulation for 2000 placements and verifies no two placed pieces sit a knight's move apart.

Numberphile videos

The class of patterns this explores was popularized in:

• Red & Black Knights — extraordinary result
• Amazing Chessboard Patterns — extra

Layout

• src/spiral.hpp — Ulam spiral iterator
• src/leaper.{hpp,cpp} — Leaper struct + Knight preset
• src/board.{hpp,cpp} — placed-piece store and lookups
• src/simulation.{hpp,cpp} — turn-based per-color cursor; is_blocked
• src/view.hpp — pan/zoom state + screen↔world transforms
• src/renderer.{hpp,cpp} — instanced-quad OpenGL renderer (shared GLSL for GL 3.3 core and GL ES 3.0)
• src/ui.{hpp,cpp} — ImGui sidebar and movement-editor modal
• src/main.cpp — SDL3/GL/ImGui bootstrap and frame loop