• 4D Shapes: Tesseract (4D cube), 16-cell (4D cross-polytope)
• 5D Shapes: Penteract (5D cube), 5-simplex (5D tetrahedron), 5-orthoplex
• Real-time Rendering: Smooth 60 FPS visualization with Three.js
• Multiple Projections: Perspective, orthogonal, stereographic, and Schlegel projections

• 4D Rotations: 6 rotation planes (XY, XZ, XW, YZ, YW, ZW)
• 5D Rotations: 10 rotation planes with complex matrix operations
• Scaling: Uniform scaling across all dimensions
• Translation: Movement in higher-dimensional space
• Auto-rotation: Animated continuous rotation with customizable speed

• Responsive Control Panel: Tabbed interface with shape, transform, and render controls
• Real-time Sliders: Immediate visual feedback for all transformations
• Shape Selection: Easy switching between different geometric primitives
• Rendering Options: Toggle vertices, edges, faces, wireframe mode, opacity control
• Color Customization: Custom colors for shapes, edges, and vertices

• 4D Cross-sections: Slice through 4D objects with 3D hyperplanes
• Interactive Positioning: Real-time adjustment of hyperplane position
• Visual Analysis: 2D visualization of cross-section results
• Animation Tools: Automated slicing animation through the object
• Mathematical Accuracy: Precise edge-hyperplane intersection calculations

• Real-time Performance Monitoring: FPS counter and vertex count display
• Efficient Rendering: Optimized Three.js usage with proper disposal
• Responsive Design: Works on desktop, tablet, and mobile devices
• Memory Management: Proper cleanup of geometries and materials
• Adaptive Quality: Performance-based rendering adjustments

• Mathematical Documentation: Comprehensive explanation of concepts
• Interactive Help: Built-in keyboard shortcuts and controls guide
• Shape Information: Detailed properties for each geometric primitive
• Visual Learning: Immediate feedback for mathematical transformations
• Progressive Complexity: Start simple and explore advanced features

• TypeScript: Full type safety for mathematical operations
• Modern React: Hooks-based architecture with efficient state management
• Three.js Integration: Professional 3D rendering with WebGL
• API Architecture: RESTful endpoints for shape data and calculations
• Error Handling: Graceful degradation and user-friendly error messages

• Geometry4D: Complete 4D shape generation and transformation
• Geometry5D: Advanced 5D geometric calculations
• Projections: Multiple projection algorithms for nD to 3D conversion
• Transformations: Matrix operations for rotations, scaling, translation
• Cross-sections: Hyperplane intersection algorithms

• React Components: Modular, reusable UI components
• Three.js Rendering: Hardware-accelerated 3D graphics
• State Management: Efficient React state with minimal re-renders
• Event System: Custom events for component communication
• Responsive Design: Mobile-first CSS with desktop enhancements

• Hono.js Server: Fast, lightweight API server
• Static File Serving: Efficient delivery of assets
• Health Monitoring: Application status and performance endpoints
• Shape Metadata: Detailed information about geometric primitives
• Calculation APIs: Server-side mathematical computation support

• Camera Rotation: Intuitive click-and-drag orbit controls
• Zoom: Smooth mouse wheel zooming
• Responsive Feedback: Immediate visual response to interactions

• Space: Toggle auto-rotation
• Ctrl+R: Reset transformations
• 1-4: Quick shape selection
• V/E/F/W: Toggle rendering modes
• Accessibility: Full keyboard navigation support

• Modern UI: Clean, professional interface with glassmorphism effects
• Color Scheme: Carefully chosen colors for optimal visibility
• Typography: Clear, readable fonts with proper hierarchy
• Animations: Smooth transitions and hover effects
• Dark Theme: Eye-friendly dark interface for extended use

• Target FPS: 60 FPS for smooth interaction
• Vertex Handling: Efficiently renders up to 10,000+ vertices
• Memory Usage: Optimized for long-running sessions
• Browser Compatibility: Works on all modern browsers with WebGL

• Precision: Double-precision floating-point calculations
• Numerical Stability: Robust algorithms that handle edge cases
• Validation: Mathematical results verified against known properties
• Error Handling: Graceful handling of degenerate cases

• Spatial Reasoning: Develop intuition for higher dimensions
• Mathematical Visualization: See abstract concepts concretely
• Interactive Exploration: Learn through hands-on experimentation
• Progressive Understanding: Build from simple to complex concepts

• Students: High school to graduate level mathematics
• Educators: Teaching aids for geometry and topology courses
• Researchers: Visualization tool for mathematical research
• Enthusiasts: Anyone curious about higher-dimensional mathematics

• Real-time Slicing: Interactive hyperplane positioning
• 3D Visualization: See the interior structure of 4D objects
• Animation: Automated slicing sequences
• Mathematical Insight: Understand topology through cross-sections

• Perspective: Distance-based scaling for realistic depth
• Orthogonal: Simple coordinate dropping for clarity
• Stereographic: Avoid singularities with sphere-to-plane mapping
• Adaptive: Automatically choose best projection method

• Regular Polytopes: All major 4D and 5D regular shapes
• Detailed Properties: Vertex, edge, face, and cell counts
• Mathematical Accuracy: Precise geometric construction
• Extensible: Framework for adding custom shapes

• TypeScript: Full type safety and IntelliSense support
• Modular Architecture: Clean separation of concerns
• Documentation: Comprehensive inline and external documentation
• Testing: Mathematical functions validated for accuracy

• Plugin Architecture: Easy to add new shapes and projections
• API Design: RESTful endpoints for external integration
• Configuration: Customizable settings for different use cases
• Open Source: MIT license for educational and research use

1. First-Class 5D Support: One of the few tools supporting 5D visualization
2. Real-time Interaction: Immediate feedback for all transformations
3. Educational Focus: Designed specifically for learning and teaching
4. Mathematical Accuracy: Precise calculations with proper error handling
5. Cross-Platform: Works on any device with a modern browser
6. Performance Optimized: Smooth interaction even with complex shapes
7. Open Source: Free for educational and research use

• 6D and Higher: Extend to even more dimensions
• Custom Shape Builder: User-defined geometric primitives
• VR/AR Support: Immersive higher-dimensional exploration
• Collaboration Features: Share and discuss visualizations
• Advanced Analytics: Detailed geometric property analysis

• Guided Tours: Step-by-step concept exploration
• Interactive Tutorials: Learn-by-doing mathematical lessons
• Assessment Tools: Quiz and problem-solving features
• Curriculum Integration: Alignment with educational standards

This Higher Dimensional Geometry Visualizer represents a significant advancement in mathematical visualization tools, combining cutting-edge web technologies with rigorous mathematical foundations to create an unparalleled educational and research platform.