Visual Representations
BitMatrix Spatial Computing: Visual Representations
*Developed by Euan Craig (DigitalEuan.com) New Zealand, 2025*
*With assistance from GPT, Gemini, Grok, and Manus*
This document provides visual representations of key concepts in the BitMatrix Spatial Computing framework. These diagrams help illustrate the multidimensional nature of the architecture and its operational principles.
## 1. Core Architecture Visualizations
### 1.1 3D Bitfield Structure
```
z
|
| y
| /
| /
| /
| /
|/______ x
3D Bitfield (8×8×8)
Layer z=0: Layer z=1: Layer z=2:
+--------+ +--------+ +--------+
|11000011| |10100101| |10011001|
|10000001| |01000010| |01100110|
|10000001| |00100100| |00100100|
|10000001| |00011000| |00000000|
|10000001| |00011000| |00000000|
|10000001| |00100100| |00100100|
|10000001| |01000010| |01100110|
|11000011| |10100101| |10011001|
+--------+ +--------+ +--------+
```
### 1.2 4D Bitfield with Temporal Dimension
```
Time (t)
↑
|
z |
| |
| |
| +----> t
| /|
|/ |
| |
| | y
| | /
| | /
| | /
| | /
| |/______ x
4D Bitfield (8×8×8×4)
t=0: t=1: t=2: t=3:
+---+ +---+ +---+ +---+
| | | | | | | |
| 3D| | 3D| | 3D| | 3D|
|Bit| |Bit| |Bit| |Bit|
|fie| |fie| |fie| |fie|
|ld | |ld | |ld | |ld |
| | | | | | | |
+---+ +---+ +---+ +---+
```
### 1.3 5D KTA Representation
```
Kinetic (k)
↑
|
t |
| |
| |
| +----> k
| /|
|/ |
| |
z |
| |
| | y
| | /
| | /
| | /
| | /
| |/______ x
5D Bitfield (8×8×8×4×4)
k=0: k=1: k=2: k=3:
+---+ +---+ +---+ +---+
| | | | | | | |
| 4D| | 4D| | 4D| | 4D|
|Bit| |Bit| |Bit| |Bit|
|fie| |fie| |fie| |fie|
|ld | |ld | |ld | |ld |
| | | | | | | |
+---+ +---+ +---+ +---+
```
## 2. Spatial Operations Visualizations
### 2.1 Rotation Operation
```
Original Block: After 90° Rotation around Z-axis:
+---+---+---+ +---+---+---+
| 1 | 0 | 0 | | 0 | 0 | 1 |
+---+---+---+ +---+---+---+
| 1 | 1 | 0 | | 1 | 1 | 0 |
+---+---+---+ +---+---+---+
| 1 | 0 | 0 | | 0 | 0 | 1 |
+---+---+---+ +---+---+---+
Rotation Matrix (Z-axis, 90°):
[ cos(90°) -sin(90°) 0 ] [ 0 -1 0 ]
[ sin(90°) cos(90°) 0 ] = [ 1 0 0 ]
[ 0 0 1 ] [ 0 0 1 ]
```
### 2.2 Block Operations
```
Target Bitfield: Block to Insert: Result:
+---+---+---+---+ +---+---+ +---+---+---+---+
| 0 | 0 | 0 | 0 | | 1 | 1 | | 0 | 0 | 0 | 0 |
+---+---+---+---+ +---+---+ +---+---+---+---+
| 0 | 0 | 0 | 0 | | 1 | 1 | | 0 | 1 | 1 | 0 |
+---+---+---+---+ +---+---+---+---+
| 0 | 0 | 0 | 0 | | 0 | 1 | 1 | 0 |
+---+---+---+---+ +---+---+---+---+
| 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 |
+---+---+---+---+ +---+---+---+---+
Insert Position: (1,1)
```
### 2.3 Perspective and Mirroring
```
Original: Perspective Projection: Mirroring (X-axis):
z y z
| | |
| y | | y
| / | | /
| / | | /
| / | | /
| / | | /
|/______ x |_______ x |/______ x
+---+---+---+ +---+---+---+ +---+---+---+
| 1 | 0 | 1 | | 1 | 0 | 1 | | 1 | 0 | 1 |
+---+---+---+ +---+---+---+ +---+---+---+
| 0 | 1 | 0 | | 0 | 1 | 0 | | 0 | 1 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
| 1 | 0 | 1 | | 1 | 0 | 1 | | 1 | 0 | 1 |
+---+---+---+ +---+---+---+ +---+---+---+
Mirror Matrix (X-axis):
[ -1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]
```
## 3. Temporal Operations Visualizations
### 3.1 Wave Propagation
```
Initial State (t=0): Wave at t=1: Wave at t=2:
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 | | 0 | 0 | 0 | 1 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 0 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | | 1 | 0 | 1 | 0 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 | | 0 | 0 | 0 | 0 |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+
Wave Equation: B(x,y,z,t+1) = 2B(x,y,z,t) - B(x,y,z,t-1) + c²∇²B(x,y,z,t)
```
### 3.2 Temporal Patterns
```
Sine Wave Pattern:
Amplitude
1.0 | * * * * *
| / \ / \ / \ / \ / \
0.5 |--/---\---/---\---/---\---/---\---/---\--
| \ / \ / \ / \ / \ /
0.0 | * * * * *
+--------------------------------------- Time
0 1 2 3 4 5 6 7 8
Square Wave Pattern:
Amplitude
1.0 | ******* ******* *******
| | | | | | |
0.5 |----+ +-------+ +-------+ +--
| | | | | | |
0.0 | ******* ******* *******
+--------------------------------------- Time
0 1 2 3 4 5 6 7 8
Sawtooth Wave Pattern:
Amplitude
1.0 | * * *
| /| /| /|
0.5 | / | / | / |
| / | / | / |
0.0 | * | * | * |
+--------------------------------------- Time
0 1 2 3 4 5 6 7 8
```
## 4. Neural Network Integration Visualizations
### 4.1 Neural Layer Representation
```
BitMatrix Neural Network Architecture:
Input Layer Hidden Layer Output Layer
(3D Bitfield) (3D Bitfield)
+---+---+---+ +---+---+---+
| 1 | 0 | 1 | | 0 | 1 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
| 0 | 1 | 0 | → | N | N | N | → | 1 | 0 | 1 |
+---+---+---+ +---+---+---+ +---+---+---+
| 1 | 0 | 1 | | 0 | 1 | 0 |
+---+---+---+ +---+---+---+
Neural Layer Equation: y = σ(Wx + b)
```
### 4.2 Convolutional Operations
```
Input Bitfield: 3D Convolutional Filter: Output Feature Map:
+---+---+---+---+ +---+---+ +---+---+
| 1 | 0 | 1 | 0 | | 1 | 0 | | 2 | 1 |
+---+---+---+---+ +---+---+ +---+---+
| 0 | 1 | 1 | 0 | | 0 | 1 | | 2 | 3 |
+---+---+---+---+ +---+---+
| 0 | 0 | 1 | 1 |
+---+---+---+---+
| 1 | 0 | 0 | 0 |
+---+---+---+---+
Convolution Equation: C(x,y) = ∑∑ B(x+i,y+j) · F(i,j)
```
## 5. Quantum-Inspired Visualizations
### 5.1 Superposition Representation
```
Classical Bit: Quantum-Inspired Bit in Superposition:
State: |0⟩ or |1⟩ State: α|0⟩ + β|1⟩ where |α|² + |β|² = 1
Visualization: Visualization (Bloch Sphere):
|1⟩ |0⟩
^ ^
| |
| |
| θ |
| / |
| / |
| • |
| /| |
| / | |
| / | |
| / | |
| / | |
| / | |
|0⟩ ---------------------> |1⟩
Measurement: Always 0 or 1 Measurement: |0⟩ with probability |α|²
|1⟩ with probability |β|²
```
### 5.2 Quantum Gates
```
X Gate (NOT): H Gate (Hadamard): CNOT Gate:
|0⟩ → |1⟩ |0⟩ → (|0⟩+|1⟩)/√2 |00⟩ → |00⟩
|1⟩ → |0⟩ |1⟩ → (|0⟩-|1⟩)/√2 |01⟩ → |01⟩
|10⟩ → |11⟩
Matrix: Matrix: |11⟩ → |10⟩
[ 0 1 ] [ 1/√2 1/√2 ] [ 1 0 0 0 ]
[ 1 0 ] [ 1/√2 -1/√2 ] [ 0 1 0 0 ]
[ 0 0 0 1 ]
[ 0 0 1 0 ]
```
### 5.3 Quantum Search
```
Grover's Algorithm Visualization:
Initial State: After Oracle: After Diffusion:
(Equal Superposition) (Flip Target State) (Amplify Target)
Amplitude Amplitude Amplitude
0.5 | 0.5 | 0.8 | *
| | | |
| | | |
| | | |
0.0 | * * * * * * 0.0 | * * * * * * 0.0 | * * * * *
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |
-0.5 | -0.5 | * -0.5 |
+------------ +------------ +------------
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
Target state: |3⟩
```
## 6. Data Compression Visualizations
### 6.1 Enhanced Compression Pipeline
```
BitMatrix Enhanced Compression Pipeline:
Original KTA Quantum Pattern RLE
Bitfield → Transform → Fourier → Extraction → Encoding →
Transform
LZW
Compressed ← Dictionary ← Metadata ← Encoding
Data
Compression Ratio Comparison:
Method Compression Ratio
Standard RLE 2:1 - 5:1
Standard LZW 10:1 - 20:1
BitMatrix Enhanced 30:1 - 100:1
```
### 6.2 Run-Length Encoding
```
Original Bitfield: RLE Compressed:
1 1 1 1 0 0 0 0 1 1 (1,4), (0,4), (1,2)
Visualization:
Value
1 **** **
| | ||
0 | |**** ||
| || | ||
+---++---+--++--- Position
0 45 9 11
```
## 7. Error Correction Visualizations
### 7.1 Hamming Code
```
Hamming(7,4) Code:
Data bits: d₁, d₂, d₃, d₄
Parity bits: p₁, p₂, p₃
Codeword layout:
p₁ p₂ d₁ p₃ d₂ d₃ d₄
Parity equations:
p₁ = d₁ ⊕ d₂ ⊕ d₄
p₂ = d₁ ⊕ d₃ ⊕ d₄
p₃ = d₂ ⊕ d₃ ⊕ d₄
Error detection:
Error position = (p₁ error)×1 + (p₂ error)×2 + (p₃ error)×4
```
### 7.2 Error Correction Process
```
Original Data: Encoded Data: Transmitted Data:
1 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1
p₁p₂d₁p₃d₂d₃d₄ p₁p₂d₁p₃d₂d₃d₄
^ Error
Error Detection: Corrected Data: Decoded Data:
p₁ check: 1⊕1⊕1=1 ✓ 1 1 1 0 0 1 1 1 0 1 1
p₂ check: 1⊕0⊕1=0 ✗ p₁p₂d₁p₃d₂d₃d₄
p₃ check: 0⊕0⊕1=1 ✓
Error position = 0×1 + 1×2 + 0×4 = 2
Bit at position 2 is flipped
```
## 8. 5D KTA Visualizations
### 8.1 Dimensional Transformation
```
Original 3D Space: After Dimension Shift (X→Z):
z z
| |
| y | y
| / | /
| / | /
| / | /
| / | /
|/______ x |/______ x
Coordinates (2,3,1) Coordinates (1,3,2)
Dimension Shift Operator: D_{x→z}[B](x,y,z) = B(z,y,x)
```
### 8.2 Kinetic Energy Distribution
```
Kinetic Energy Distribution in 3D Space:
z=0: z=1: z=2:
y y y
^ ^ ^
| | |
3 | 0.1 0.2 0.3 3 | 0.2 0.3 0.4 3 | 0.3 0.4 0.5
| | |
2 | 0.2 0.3 0.4 2 | 0.3 0.4 0.5 2 | 0.4 0.5 0.6
| | |
1 | 0.3 0.4 0.5 1 | 0.4 0.5 0.6 1 | 0.5 0.6 0.7
| | |
0 +--+---+---+--- 0 +--+---+---+--- 0 +--+---+---+---
0 1 2 3 0 1 2 3 0 1 2 3
x x x
Kinetic Energy Function: K(x,y,z) = (x+y+z)/10
```
### 8.3 Kinetic Wave Propagation
```
Kinetic Wave Equation:
∂²B/∂t² = c² · ∂²B/∂k² + ∇²B
Visualization of Wave Propagation through Kinetic Dimension:
k=0, t=0: k=1, t=0: k=2, t=0:
+---+---+---+ +---+---+---+ +---+---+---+
| 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
| 0 | 1 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
| 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
k=0, t=1: k=1, t=1: k=2, t=1:
+---+---+---+ +---+---+---+ +---+---+---+
| 0 | 1 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
| 1 | 0 | 1 | | 0 | 1 | 0 | | 0 | 0 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
| 0 | 1 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |
+---+---+---+ +---+---+---+ +---+---+---+
```
## 9. Application Visualizations
### 9.1 Enhanced Data Compression
```
Original Image: After BitMatrix Compression:
+---+---+---+---+ High-Magnitude Components:
| ■ | ■ | □ | □ | (0,0): 0.8∠0°
+---+---+---+---+ (1,1): 0.5∠90°
| ■ | ■ | □ | □ | (2,0): 0.3∠180°
+---+---+---+---+
| □ | □ | ■ | ■ | Compressed Size: 24 bytes
+---+---+---+---+ Original Size: 64 bytes
| □ | □ | ■ | ■ | Compression Ratio: 2.67:1
+---+---+---+---+
```
### 9.2 Neural Pattern Recognition
```
Input Pattern: Neural Network: Recognition Result:
+---+---+---+---+ Pattern: "Letter A"
| □ | ■ | ■ | □ | Input Hidden Out Confidence: 92%
+---+---+---+---+ Layer Layer Layer
| ■ | □ | □ | ■ | ○ ○ ○ Features Detected:
+---+---+---+---+ ○ ○ ○ - Diagonal lines
| ■ | ■ | ■ | ■ | ○ ○ ○ - Horizontal middle
+---+---+---+---+ ○ ○ ○ - Vertical sides
| ■ | □ | □ | ■ |
+---+---+---+---+
```
### 9.3 Quantum-Inspired Computing
```
Problem: Find pattern "101" in bitfield
Bitfield: Quantum Search: Result:
1 0 1 0 1 1 0 1 0 1 Initial state: Pattern found at:
|ψ⟩ = (1/√10)∑|i⟩ - Position 0
- Position 4
Quantum Circuit: Oracle: - Position 7
Marks positions 0,4,7
|0⟩ --H--
| Diffusion:
|0⟩ --H-- Oracle -- Diffusion
| Amplifies marked states
|0⟩ --H--
```
These visual representations help illustrate the key concepts of the BitMatrix Spatial Computing framework, making it easier to understand the multidimensional nature of the architecture and its operational principles.