The Pendulum 2.0

Balance a real inverted pendulum. 7 game modes, AI assistance, 100+ scientific metrics, Apple Health integration, and adaptive difficulty powered by your own biometrics.

The Pendulum is a rigorous inverted pendulum simulation built on real physics — not an approximation. Balance an unstable system using touch controls while the app tracks over 100 scientific metrics across your gameplay, health, and learning progress.

REAL PHYSICS, REAL CHALLENGE The simulation solves nonlinear differential equations in real time using a 4th-order Runge-Kutta integrator. Adjust mass, length, gravity, damping, spring constant, and moment of inertia to explore how parameter changes affect system dynamics. This is the same inverted pendulum problem studied in university control theory courses — now in your hands.

7 GAME MODES

• Free Play: No levels, no timer. Balance at your own pace.
• Progressive: Increasing balance duration with each level.
• Spatial: The balance zone narrows as you advance.
• Jiggle: Random perturbations shake the pendulum with increasing intensity.
• Timed: Race against a countdown while maintaining balance.
• Random: Physics parameters change unpredictably every level.
• Golden Mode: AI-powered adaptive difficulty shaped by your Apple Health

AI ASSISTANCE Five AI modes let you learn, compete, or observe:

• Off: Pure player control.
• Helper: Gentle nudges when you're struggling.
• Competition: The AI actively tries to destabilize you.
• Tutorial: Step-by-step guided lessons with directional hints.
• Demo: Watch the AI balance the pendulum to study optimal control.

100+ SCIENTIFIC METRICS Track your performance across six categories:

• Basic: Stability, efficiency, session time, push count.
• Advanced: Overcorrection rate, reaction time, directional bias.
• Scientific: Phase space coverage, energy management, Lyapunov exponent.
• Topological: Winding number, basin stability, periodic orbits, Betti
  numbers.
• Educational: Learning curve slope, skill retention between sessions.
• Performance: Frame rate, computation efficiency.

An interactive dashboard displays charts, time-filtered metrics, and health-performance correlations — all updating in real time.

APPLE HEALTH INTEGRATION Connect Apple Health to discover how your resting heart rate, HRV, sleep,
steps, and active calories correlate with gameplay performance. Sessions are
logged as mindfulness minutes. Golden Mode uses this data to adapt difficulty to your current physical and cognitive state.

CROSS-APP SYNERGY The Pendulum shares data with The Maze through a secure App Group connection. Balance control patterns combine with maze-solving cognitive metrics for a
unified picture of motor and mental performance.

BEAUTIFUL & CUSTOMIZABLE Choose from 9 nature photography themes — Acadia, Guatemala, Joshua Tree, Sachuest, New England, and more — or play on the default golden parchment. Toggle sound effects, haptic feedback, and control styles. Export all session data as CSV for your own analysis.

BUILT FOR EDUCATION & RESEARCH Suitable for physics coursework, nonlinear dynamics research, control theory demonstrations, cognitive science studies, and human-machine interaction experiments. Every metric is computed from first principles, not estimated.

10+ hand-crafted levels with infinite procedural generation beyond.