🌌 Cosmic Formula Playground & Sandbox

An airy, interactive visual science museum, tactile formula playground, and physics sandbox across 25 verified physical laws.

Explore Live Hub · Visual Playground (/playground) · Parameter Sandbox (/sandbox) · Syllabus · Practice

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🎯 Educational Philosophy

Cosmic Formula Playground is designed to help curious learners understand astronomy, astrophysics, orbital mechanics, and plasma physics visually and intuitively before confronting formal mathematical notation.

  ┌─────────┐      ┌───────────┐      ┌────────────┐      ┌───────────┐
  │   SEE   │ ───► │   TOUCH   │ ───► │   CHANGE   │ ───► │   WATCH   │
  └─────────┘      └───────────┘      └────────────┘      └───────────┘
                                                                │
  ┌──────────┐      ┌─────────────┐      ┌────────────────┐     ▼
  │ PRACTICE │ ◄─── │  CALCULATE  │ ◄─── │   UNDERSTAND   │ ◄───┘
  └──────────┘      └─────────────┘      └────────────────┘

Every lesson presents concepts in a structured pedagogical order:

1. Everyday intuition & ELI5 concept
2. Animated real-time vector simulation stage
3. Tactile parameter blocks and live sliders
4. Instant numerical recalculation and unit breakdown
5. Formal mathematical derivation (collapsible)
6. Adaptive practice challenges and timed missions

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⚙️ Interactive Parameter Sandbox (/sandbox)

The Formula Sandbox Workbench allows learners to pick foundational physical laws and tweak variables with live sliders, observing instant dimensional recalculations, real-world astronomical presets, and step-by-step substitution breakdowns.

Live Parameter Workbench	Step-by-Step Algebraic Solver
Real-time variable sliders and preset buttons	Automated dimensional expansion and unit checks

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🚀 The 25 Visual Simulation Labs (/playground)

Stage 1 — Measuring the Universe

Distance ladders, cosmic expansion, and cosmological redshift.

Lab 01: Hubble Expansion Flow	Lab 02: Cosmic Distance Ladder
Formula: $v = H_0 d + v_{\text{pec}}$ · Open Lab 01	Formula: $m - M = 5 \log_{10}(d) - 5$ · Open Lab 02

Lab 11: Cosmological Redshift & Wave Stretch
Formula: $1 + z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}}$ · Open Lab 11

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Stage 2 — Finding Hidden Structure

Density clumps, cosmic voids, velocity divergence, and Bayesian inference.

Lab 03: Density Contrast & Cosmic Voids	Lab 04: Velocity Field Divergence
Formula: $\delta = \frac{\rho - \bar{\rho}}{\bar{\rho}}$ · Open Lab 03	Formula: $\nabla \cdot \vec{v} = -a H f \delta$ · Open Lab 04

Lab 05: Bayesian Belief Scale	Lab 06: Measurement Uncertainty
Formula: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$ · Open Lab 05	Formula: $\sigma_{\text{tot}} = \sqrt{\sigma_1^2 + \sigma_2^2}$ · Open Lab 06

Lab 07: Wiener Signal-to-Noise Filter
Formula: $W(k) = \frac{S(k)}{S(k) + N(k)}$ · Open Lab 07

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Stage 3 — Seeing Through Interstellar Dust

Dust attenuation, optical depth, color excess, and 21-cm neutral hydrogen radio emissions.

Lab 08: Interstellar Dust Attenuation	Lab 09: Optical Depth & Extinction
Formula: $I = I_0 e^{-\tau}$ · Open Lab 08	Formula: $A_\lambda = 1.086 , \tau_\lambda$ · Open Lab 09

Lab 10: Color Excess & Reddening	Lab 12: 21-cm Radio Brightness Temperature
Formula: $E(B-V) = (B-V)_{\text{obs}} - (B-V)_0$ · Open Lab 10	Formula: $T_b = T_s (1 - e^{-\tau_{21}})$ · Open Lab 12

Lab 13: Neutral Hydrogen ($H,\text{I}$) Column Mass
Formula: $M_{HI} = 2.36 \times 10^5 , D^2 \int S_v , dv$ · Open Lab 13

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Stage 4 — Predicting Motion & Autonomous Navigation

Spacecraft state propagation, sensor noise modeling, and Kalman filter gain updates.

Lab 14: Spacecraft State Predictor	Lab 15: Kalman Filter Trust Gain
Formula: $\hat{x}{k} = F \hat{x}{k-1} + B u_k$ · Open Lab 14	Formula: $K = \frac{P}{P + R}$ · Open Lab 15

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Stage 5 — Traveling Farther: Orbital Mechanics & Spaceflight

Lab 16: Newton's Universal Gravitation	Lab 17: Keplerian Orbit Period
Formula: $F = G \frac{m_1 m_2}{r^2}$ · Open Lab 16	Formula: $T^2 = \frac{4\pi^2}{GM} a^3$ · Open Lab 17

Lab 18: Gravity Well & Escape Trajectory	Lab 19: Tsiolkovsky Rocket Staging
Formula: $v_e = \sqrt{\frac{2GM}{R}}$ · Open Lab 18	Formula: $\Delta v = I_{\text{sp}} g_0 \ln\left(\frac{m_0}{m_f}\right)$ · Open Lab 19

Lab 20: Solar Sail Photon Radiation Pressure	Lab 21: Interplanetary Light Latency
Formula: $P_{\text{rad}} = \frac{(1 + R) I_{\odot}}{c}$ · Open Lab 20	Formula: $t_{\text{light}} = \frac{d}{c}$ · Open Lab 21

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Stage 6 — Space Plasma & Electromagnetic Environments

Lab 22: Electron Plasma Oscillation Frequency	Lab 23: Debye Shielding Sphere
Formula: $\omega_{pe} = \sqrt{\frac{n_e e^2}{\varepsilon_0 m_e}}$ · Open Lab 22	Formula: $\lambda_D = \sqrt{\frac{\varepsilon_0 k_B T_e}{n_e e^2}}$ · Open Lab 23

Lab 24: Magnetohydrodynamic Alfvén Waves	Lab 25: Thermal vs. Magnetic Plasma Beta
Formula: $v_A = \frac{B}{\sqrt{\mu_0 \rho}}$ · Open Lab 24	Formula: $\beta = \frac{P_{\text{thermal}}}{B^2 / (2\mu_0)}$ · Open Lab 25

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📚 Curriculum, Library & Adaptive Practice

Structured Curriculum Syllabus	Interactive Formula Library
6 stages mapped with prerequisites and milestone tracks	Filterable formula cards with KaTeX math rendering

Adaptive Practice & Timed Missions	3D Shape & Geometry Hero
250+ procedural question variants with multi-step hints	Tactile drag rotation & dynamic isometric geometry

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🎨 UI/UX Architecture & Concept Gallery

High-fidelity interface design specifications and pedagogy mockups crafted for the Cosmic Formula ecosystem:

01. Home Hero & 3D Shapes	02. Syllabus & Learning Roadmap
Tactile 3D shape manipulation & Learno guide	Milestone tracks across 25 space formulas

03. Formula Library Catalog	04. Hubble Flow & Velocity Lab
Categorized searchable catalog across 120+ formulas	Cosmological expansion & peculiar velocity

05. Interstellar Dust Attenuation	06. Cosmological Redshift Waves
Radiative transfer & Beer-Lambert dust extinction	Wavelength wave stretching across cosmic time

07. Rocket Equation & Thrust	08. Solar Sail Radiation Pressure
Tsiolkovsky delta-v, fuel mass ratio & staging	Solar sail lightness number & radiation push

09. Space Plasma Physics & Waves	10. Practice Engine & Challenges
Debye shielding bubbles & Alfvén magnetic waves	Adaptive multiple choice questions & XP rewards

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🛠️ Architecture & Local Development

Technology Stack

• Pure Web Standards: Vanilla ES Modules (app.js, v3-model.mjs, formulas.mjs) — Zero framework bloat.
• Rendering: Semantic SVG vectors, HTML5 Canvas, and MathJax/KaTeX math notation.
• Routing: HTML5 History API with static clean-URL fallback entrypoints.
• Offline / PWA: Service Worker (sw.js) with proactive cache lifecycle management and manifest support.
• Styling: Responsive modern Airy CSS tokens (--indigo, --coral, --emerald, --sky).

Run Locally

# Clone the repository
git clone https://github.com/zrt219/Cosmic-Formula-Playground.git
cd Cosmic-Formula-Playground

# Start a local static HTTP server
npx serve .
# or
python -m http.server 8080

Open http://localhost:8080 in your browser.

Verification & Test Suite

# Run syntax checks, QA invariants, and DOM selector tests
npm run check

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📄 License

MIT License © 2026 Cosmic Formula Playground Contributors.