A Next.js application for visualizing and interacting with 4D and 5D geometric shapes and their transformations.

• 4D/5D Shape Visualization: Render hypercubes, pentachorons, and other higher-dimensional primitives
• Real-time Transformations: Rotate, scale, and translate shapes in higher dimensions
• 3D Projections: View how higher-dimensional objects project into 3D space
• Interactive Controls: Intuitive UI for manipulating shapes and camera perspectives
• Cross-sections: Explore slices through higher-dimensional objects
• Performance Optimized: Efficient calculations and rendering for complex geometry
• Responsive Design: Works across desktop and mobile devices

• 4D Shapes: Tesseract (4D cube), 16-cell, 600-cell, 120-cell
• 5D Shapes: Penteract (5D cube), 5-simplex, 5-orthoplex
• Custom Primitives: User-defined higher-dimensional objects

• Next.js with TypeScript
• Three.js for 3D rendering
• React Three Fiber for React integration
• Custom mathematical libraries for higher-dimensional calculations
• Tailwind CSS for styling

├── backend/
│   ├── index.ts              # API endpoints
│   └── routes/
│       └── calculations.ts   # Mathematical computation endpoints
├── frontend/
│   ├── components/
│   │   ├── App.tsx          # Main application component
│   │   ├── GeometryViewer.tsx # 3D visualization component
│   │   ├── ControlPanel.tsx  # User interface controls
│   │   ├── ShapeSelector.tsx # Shape selection interface
│   │   └── CrossSection.tsx  # Cross-section visualization
│   ├── index.html           # Main HTML template
│   ├── index.tsx            # Frontend entry point
│   └── style.css            # Global styles
├── shared/
│   ├── math/
│   │   ├── geometry4d.ts    # 4D geometric calculations
│   │   ├── geometry5d.ts    # 5D geometric calculations
│   │   ├── projections.ts   # Projection algorithms
│   │   └── transformations.ts # Transformation matrices
│   ├── types.ts             # Shared TypeScript types
│   └── utils.ts             # Utility functions
└── README.md

1. The application runs as an HTTP val
2. Navigate to the main page to start exploring higher-dimensional geometry
3. Use the control panel to select shapes, adjust transformations, and modify viewing parameters
4. Interact with the 3D viewport to explore projections and cross-sections

This application implements advanced concepts from higher-dimensional geometry:

• Orthogonal projections from nD to 3D space
• Rotation matrices in 4D and 5D spaces
• Cross-sectional analysis of higher-dimensional objects
• Real-time transformation calculations optimized for performance