The Pendulum 2.0 vs Bus Loop

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.

Relax, match, and let the loop flow. Bus Loop is a calming yet engaging puzzle game where colorful vehicles smoothly carry bubbles away. Simply choose the right vehicle, match the colors, and watch the satisfying movement unfold. There’s no rush—just thoughtful decisions and smooth motion.  Different vehicle sizes create gentle challenges, encouraging you to slow down, observe, and enjoy the process of solving each puzzle. With clean visuals, intuitive controls, and endlessly satisfying loops, Bus Loop is perfect for unwinding while keeping your mind lightly engaged. Game Features ・Simple color-matching puzzle mechanics ・Smooth vehicle movement and loop-based gameplay ・Satisfying bubble transport animations ・Gradually evolving puzzles with gentle difficulty ・Relaxing visuals and stress-free experience ・Easy to play, enjoyable for all ages Whether you want to relax for a few minutes or get lost in the flow, Bus Loop offers a peaceful puzzle journey you’ll want to return to again and again.