Overview
Sequences
Chunks
Combinators
Visualization
Examples
Theory
Troubleshooting
🌟 Overview
Welcome to the UNNS (Unbounded Nested Number Sequences) Non-Linear Extensions tool!
Key Concepts:
- UNNS Framework: A mathematical system for studying nested number sequences with unbounded growth patterns
- Non-Linear Extensions: Advanced operations that go beyond traditional linear transformations
- Chunk Operations: Divide sequences into contiguous subsequences for analysis
- Combinators: Mathematical functions that combine chunks in non-linear ways
Quick Start:
- Generate a sequence using the Sequence Generator panel
- Apply chunk operations to divide the sequence
- Select a combinator to transform chunks
- Visualize results in 2D and 3D
- Use animation controls to see dynamic behavior
🧬 Sequence Types
Available Generators:
- Fibonacci: Classic recursive sequence (F(n) = F(n-1) + F(n-2))
- Catalan: Combinatorial sequence counting binary trees
- Polynomial: Customizable polynomial recurrence
- Prime Sieve: Sequence of prime numbers
- Chaotic: Logistic map with chaos parameter
Convergence Analysis:
The tool automatically calculates convergence ratios to identify limiting behavior.
📦 Chunk Operations
Parameters:
- Chunk Size: Number of elements in each chunk (2-8)
- Shift Amount: Overlap between consecutive chunks (0 = no overlap)
Example:
Sequence: [1, 2, 3, 4, 5, 6]
Chunk Size: 3, Shift: 1
Result: [1,2,3], [2,3,4], [3,4,5], [4,5,6]
⚡ Non-Linear Combinators
Types:
- Multiplicative: Element-wise multiplication and sum
- Exponential: Power operations with modular reduction
- Modular: Sum modulo chunk length
- Cross-Product: Cartesian product operations
- Polynomial: Custom polynomial transformation
- Chaos Generator: Logistic map application
📊 Visualization Features
2D Canvas:
- Bar chart representation
- Connecting line plot
- Color gradient mapping
- Real-time animation
3D Spiral:
- Three.js powered visualization
- Spiral arrangement of chunks
- Rotation controls
- Multiple camera views
💡 Examples
Example 1: Fibonacci Chunks
- Select "Fibonacci" sequence
- Set length to 20
- Generate sequence
- Set chunk size to 3
- Apply multiplicative combinator
Example 2: Chaos Analysis
- Select "Chaotic" sequence
- Generate with length 30
- Apply chaos generator combinator
- Observe convergence patterns
📐 Mathematical Theory
UNNS Formal Definition:
A UNNS is a sequence S = {s₀, s₁, s₂, ...} where:
- Each element can grow unboundedly
- Nested structure allows recursive definitions
- Non-linear operations preserve algebraic properties
Convergence Criteria:
For a sequence to converge: lim(n→∞) s_{n+1}/s_n = L
🔧 Troubleshooting
Common Issues:
- 3D not displaying: Check WebGL support in browser
- Animation lag: Reduce sequence length
- Buttons unresponsive: Refresh page and check console
Performance Tips:
- Keep sequences under 50 elements for smooth animation
- Use Chrome/Firefox for best WebGL performance
- Close other tabs to free memory