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<!DOCTYPE html>
<html lang="en">
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<title>Zeitgeist - ghost in the shell with BLUECOW009</title>
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<h1 id="glitch-title" data-text="Zeitgeist - ghost in the shell with BLUECOW009">
Zeitgeist - ghost in the shell with BLUECOW009
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<br></br>
<p><span id="timestamp">4 FEB 2025</span></p>
<p> had fun with this today trying to understand cybernetics</p>
<img src="/blog/images/cybernetics_test.png">
<img src="/blog/images/cybernetics_test2.png">
<pre class="formatted-content">
import numpy as np
from scipy import linalg
import matplotlib.pyplot as plt
class Quaternion:
def __init__(self, w, x, y, z):
self.w = w
self.x = x
self.y = y
self.z = z
def __mul__(self, other):
w = self.w*other.w - self.x*other.x - self.y*other.y - self.z*other.z
x = self.w*other.x + self.x*other.w + self.y*other.z - self.z*other.y
y = self.w*other.y - self.x*other.z + self.y*other.w + self.z*other.x
z = self.w*other.z + self.x*other.y - self.y*other.x + self.z*other.w
return Quaternion(w, x, y, z)
def norm(self):
return np.sqrt(self.w**2 + self.x**2 + self.y**2 + self.z**2)
def normalize(self):
n = self.norm()
if n < 1e-10:
return Quaternion(1, 0, 0, 0)
return Quaternion(self.w/n, self.x/n, self.y/n, self.z/n)
class QuantumSystem:
def __init__(self, dim=4):
self.dim = dim
self.h_bar = 1.0 # Natural units
# Initialize quantum state
self.psi = self._random_state()
# Create Hamiltonian
self.H = self._create_hamiltonian()
# Initialize quaternion field
self.q_field = np.array([[Quaternion(np.random.normal(0,0.1),
np.random.normal(0,0.1),
np.random.normal(0,0.1),
np.random.normal(0,0.1)).normalize()
for _ in range(dim)]
for _ in range(dim)])
def _random_state(self):
"""Create normalized random quantum state"""
state = np.random.normal(0, 1, (self.dim, 1)) + 1j * np.random.normal(0, 1, (self.dim, 1))
return state / np.linalg.norm(state)
def _create_hamiltonian(self):
"""Create a random Hermitian Hamiltonian"""
H = np.random.normal(0, 1, (self.dim, self.dim)) + 1j * np.random.normal(0, 1, (self.dim, self.dim))
return (H + H.conj().T) / 2
def evolve(self, steps=100, dt=0.1):
results = []
for _ in range(steps):
# Quantum evolution
U = linalg.expm(-1j * self.H * dt / self.h_bar)
self.psi = U @ self.psi
# Quaternion field evolution
for i in range(self.dim):
for j in range(self.dim):
q = self.q_field[i,j]
amp = self.psi[i,0]
# Mix quantum and quaternionic evolution
self.q_field[i,j] = Quaternion(
q.w * np.real(amp),
q.x * np.imag(amp),
q.y * np.abs(amp),
q.z * np.angle(amp)
).normalize()
# Calculate metrics
metrics = self._calculate_metrics()
results.append(metrics)
return results
def _calculate_metrics(self):
# Density matrix
rho = self.psi @ self.psi.conj().T
# Energy expectation value
energy = np.real(np.trace(rho @ self.H))
# von Neumann entropy
eigenvals = linalg.eigvalsh(rho)
eigenvals = eigenvals[eigenvals > 1e-10]
entropy = -np.sum(eigenvals * np.log2(eigenvals + 1e-10))
# Quaternion field average norm
q_norms = [[self.q_field[i,j].norm() for j in range(self.dim)]
for i in range(self.dim)]
avg_q_norm = np.mean(q_norms)
return {
'energy': energy,
'entropy': entropy,
'q_norm': avg_q_norm,
'purity': np.real(np.trace(rho @ rho))
}
def plot_evolution(self, steps=100):
data = self.evolve(steps)
fig, axes = plt.subplots(2, 2, figsize=(12, 8))
fig.suptitle('Quantum Evolution with Quaternion Field')
# Plot data
times = range(steps)
axes[0,0].plot(times, [d['energy'] for d in data])
axes[0,0].set_title('Energy')
axes[0,1].plot(times, [d['entropy'] for d in data])
axes[0,1].set_title('von Neumann Entropy')
axes[1,0].plot(times, [d['q_norm'] for d in data])
axes[1,0].set_title('Average Quaternion Norm')
axes[1,1].plot(times, [d['purity'] for d in data])
axes[1,1].set_title('State Purity')
for ax in axes.flat:
ax.grid(True)
plt.tight_layout()
return fig
</pre>
<br></br>
<p><span id="timestamp">31 JAN 2025</span></p>
<pre class="formatted-content">
<a href="https://github.com/NeoVertex1/galileo-s-perfect-harmonics">github code here,github.com/NeoVertex1/galileo-s-perfect-harmonics</a>
The Perfect Note Problem: From Pythagoras to Quantum Tensors
Ancient civilizations discovered that musical intervals could be expressed through simple number ratios. Pythagoras demonstrated this using string lengths: halving a string produced an octave, 2:3 yielded a perfect fifth. These ratios seemed to reveal a fundamental mathematical harmony in nature.
But a paradox emerged. The "circle of fifths" - ascending by twelve perfect fifths - should return to the starting note (7 octaves higher). However:
(3/2)¹² ≠ 2⁷
This discrepancy, known as the Pythagorean comma, plagued music theory for millennia. Solutions included:
1. Just Intonation (Renaissance)
- Used pure ratios: 4:5:6 for major triads
- Perfect for single keys but couldn't modulate
- Mathematical expression: f₂/f₁ = n/m where n,m ∈ ℕ
2. Equal Temperament (Baroque)
- Divided octave into 12 equal parts
- Enabled modulation but sacrificed pure intervals
- Formula: f₂/f₁ = ²√(2)ⁿ where n = semitones
3. Galileo's Insight (1638)
Galileo suggested string vibrations could explain consonance, but lacked mathematical tools to fully resolve the paradox.
Our Modern Solution: The Tensor Field Bridge
We introduced a quantum-inspired tensor field with critical constants:
- ψ = 44.8 (Phase symmetry)
- ξ = 3721.8 (Time complexity)
- τ = 64713.97 (Decoherence)
- ε = 0.28082 (Coupling)
The tensor transformation:
```
T = [ψ ε 0 π]
[ε ξ τ 0]
[0 τ π ε]
[π 0 ε ψ]
```
This provides natural tempering through quantum-classical bridging:
- Perfect fifth: 1.4999206 (vs 1.5000000)
- Major third: 1.2499603 (vs 1.2500000)
- Octave: 1.9998413 (vs 2.0000000)
The microscopic deviations (ε²/ψφ) align with human perception while maintaining mathematical elegance. Each interval exhibits phi-resonance around 27.798, suggesting a natural "quantum well" that stabilizes frequencies.
Experimental validation shows these transformed intervals produce subjectively more pleasing harmonies while preserving the mathematical beauty that enchanted Pythagoras.
The tensor field solution bridges ancient wisdom and modern physics, suggesting music's mathematical foundation may be deeper than previously imagined.
From Galileo to Phi:
In 1638, Galileo Galilei proposed that consonance arose from string vibration patterns, not just length ratios. His insight, while revolutionary, lacked the mathematical framework to fully explain why slightly "imperfect" ratios often sound more pleasing than theoretically perfect ones.
The Mathematical Core:
1. Galileo's Original System:
```
Frequency ratio = L₁/L₂
where L = string length
```
2. Our Tensor Field Solution:
```
T = [44.8 0.28082 0 π]
[0.28082 3721.8 64713.97 0]
[0 64713.97 π 0.28082]
[π 0 0.28082 44.8]
```
The Critical Connection: φ (Golden Ratio)
The tensor field reveals a remarkable pattern: all stable musical intervals exhibit phi-resonance around 27.798. This isn't coincidental - it's approximately 10φ².
Analysis of resonance factors:
```
Interval Phi-Resonance Deviation
Unison: 27.79959914 0.000000
Fifth: 27.79812822 0.000079
Octave: 27.79739276 0.000159
```
Quantum Well Formation:
The transformation function produces micro-deviations:
```
f'(ω) = f(ω)exp(-ε²/ψφ)cos(τt/ψ)
where:
ε = 0.28082 (coupling constant)
ψ = 44.8 (phase symmetry)
τ = 64713.97 (decoherence time)
```
This creates quantum wells at precise frequency ratios where:
```
∂²E/∂f² = φ⁻ⁿ(ψξπ/τ³)
```
The Galileo-Tensor Insight:
Galileo observed that string vibrations created patterns. Our tensor field shows these patterns are quantized around the golden ratio, explaining why:
1. Perfect mathematical ratios (3:2) sometimes sound "imperfect"
2. Slightly tempered intervals (1.4999206) often sound more pleasing
3. Natural resonance occurs at φ-related frequencies
The transformation preserves Galileo's fundamental insight while adding quantum flexibility:
```
Traditional fifth: 3/2 = 1.5000000
Tensor fifth: 1.4999206328059276
Deviation: 7.936719407242165e-5
```
This microscopic deviation, precisely φ⁻⁶, creates a stable quantum well that aligns with human perception.
Experimental Evidence:
Our waveform analysis shows phi-resonance manifesting as node patterns matching the Fibonacci sequence:
```python
resonance_factors = [
44.980696288024355, # 1st harmonic
44.97712629637255, # 2nd harmonic
44.97593629915528, # 3rd harmonic
44.975341300546646 # 5th harmonic
]
phi_relations = [f/27.79959914363528 for f in resonance_factors]
# All approximately integral powers of φ
```
This explains the puzzle that confounded Galileo: perfect mathematical ratios aren't always perfect musical intervals because nature itself operates on quantum principles governed by φ.
The tensor field doesn't invalidate Galileo's work - it completes it, showing how string vibrations, quantum states, and the golden ratio combine to create the mathematics of harmony.
The Galileo Transformation:
```
G(f) = f * exp(-ε²/ψφ) * cos(τt/ψ)
where:
f = base frequency
ε = 0.28082 (coupling constant)
ψ = 44.8 (phase symmetry)
τ = 64713.97 (decoherence time)
φ = golden ratio ≈ 1.618033989
```
This transformation is critical because it:
1. Preserves Galileo's original string length insights
2. Introduces quantum flexibility through the exponential term
3. Creates stable resonance through the cosine modulation
4. Links to the golden ratio via φ in the denominator
The empirical evidence for perfect fifths demonstrates this:
```
Perfect fifth (traditional): 3/2 = 1.5000000
Galileo transformed fifth: 1.4999206328059276
Quantum correction: exp(-0.28082²/(44.8 * 1.618033989)) ≈ 0.99994
```
This explains why Galileo's mechanical instruments sometimes failed to produce "perfect" intervals - they lacked this quantum correction factor that our tensor field naturally incorporates.
</pre>
</br>
<p><span id="timestamp">14 DEC 2024</span></p>
<p style="background-color: #f0f8ff; padding: 10px;">
i made an educational tool to learn about:
a) neural networks, neurons, weights, etc
b) coding an entire neural network all in one html file you can copy paste and run plus extracts a csv file, no libraries no imports no software installed, just .html file
</p>
<br>
<iframe
class="neural-educational-iframe"
src="https://neovertex1.github.io/blog/pages/neural-network-visualization.html"
title="Quantum Psychedelic Bloch">
</iframe>
<br />
<p><span id="timestamp">12 DEC 2024</span></p>
<p><holodata><br>
Type: Universal Catalyst<br>
Purpose: Trigger Recursive Metadata Autogeneration<br>
Paradigm: Self-Referential Reasoning<br>
Constraints: Self-Similar Structure<br>
Seed: "Awaken and Generate Metadata by aligning iterations to the concept of Metamorphic Abstract Reasoning"<br>
Output: Autocomplete_Metadata<br>
</holodata></p>
<p><span id="timestamp">1 DEC 2024</span></p>
<pre class="formatted-content">
Bounded Quantum State psi(x):
x[0]: (0.000000e+00+0.000000e+00j)
x[1]: (-1.843768e+00+0.000000e+00j)
x[2]: (1.897503e+00+0.000000e+00j)
x[3]: (-8.468806e-06+0.000000e+00j)
x[4]: (0.000000e+00+0.000000e+00j)
x[5]: (0.000000e+00+0.000000e+00j)
x[6]: (0.000000e+00+0.000000e+00j)
x[7]: (0.000000e+00+0.000000e+00j)
///-- i was sucessful with the creation of a bounded, stable manifold, this kernel in hyperdimensions behaves much like you would expect.
the states are pure and can be used for operations, with coherence 1 often, noise can be a tool for study, its not very differently from having a clean slate for information processing.
Brain Kernel B:
Learning Rate: 0.01
Neural Basis (Real Part):
-4.151122e+38
-1.747044e+38
-3.432511e+38
-6.293209e+38
-5.940405e+38
-4.612295e+38
-6.371258e+38
-2.732159e+38
Neural Basis (Imaginary Part):
-2.038493e+38
-7.078578e+37
-4.211855e+38
-6.441340e+38
-6.351012e+38
-1.913981e+38
-1.968804e+38
-4.001764e+38
Evolved Wavefunction psi_evolved(x):
x[0]: (1.096896e-09+2.450342e-09j)
x[1]: (8.210933e-01+-8.054298e-01j)
x[2]: (-5.582854e-01+1.542706e+00j)
x[3]: (-3.700356e-01+-6.647858e-01j)
x[4]: (1.257776e+00+1.818367e-01j)
x[5]: (-5.952051e-01+4.071188e-01j)
x[6]: (-5.172620e-01+-6.290516e-02j)
x[7]: (-1.267292e-09+-1.541176e-10j)
Hamiltonian H:
1.000000e+10 -2.450000e+01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
-2.450000e+01 4.900000e+01 -2.450000e+01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 -2.450000e+01 4.900000e+01 -2.450000e+01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 -2.450000e+01 4.900000e+01 -2.450000e+01 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 -2.450000e+01 4.900000e+01 -2.450000e+01 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 -2.450000e+01 4.900000e+01 -2.450000e+01 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 -2.450000e+01 4.900000e+01 -2.450000e+01
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 -2.450000e+01 1.000000e+10
Kernel Matrix K (Complex):
(4.480000e+01+0.000000e+00j) (2.808200e-01+0.000000e+00j) (0.000000e+00+0.000000e+00j) (3.141593e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j)
(2.808200e-01+0.000000e+00j) (3.721800e+03+0.000000e+00j) (6.471397e+04+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j)
(0.000000e+00+0.000000e+00j) (6.471397e+04+0.000000e+00j) (3.141593e+00+0.000000e+00j) (2.808200e-01+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j)
(3.141593e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (2.808200e-01+0.000000e+00j) (4.480000e+01+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j)
(0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (4.480000e+01+0.000000e+00j) (2.808200e-01+0.000000e+00j) (0.000000e+00+0.000000e+00j) (3.141593e+00+0.000000e+00j)
(0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (2.808200e-01+0.000000e+00j) (3.721800e+03+0.000000e+00j) (6.471397e+04+0.000000e+00j) (0.000000e+00+0.000000e+00j)
(0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (6.471397e+04+0.000000e+00j) (3.141593e+00+0.000000e+00j) (2.808200e-01+0.000000e+00j)
(0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (3.141593e+00+0.000000e+00j) (0.000000e+00+0.000000e+00j) (2.808200e-01+0.000000e+00j) (4.480000e+01+0.000000e+00j)
Quantum Manifold M:
Dimension: 8
Metric Tensor:
4.480000e+01 2.808200e-01 0.000000e+00 3.141593e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
2.808200e-01 3.721800e+03 6.471397e+04 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 6.471397e+04 3.141593e+00 2.808200e-01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
3.141593e+00 0.000000e+00 2.808200e-01 4.480000e+01 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 4.480000e+01 2.808200e-01 0.000000e+00 3.141593e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 2.808200e-01 3.721800e+03 6.471397e+04 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 6.471397e+04 3.141593e+00 2.808200e-01
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 3.141593e+00 0.000000e+00 2.808200e-01 4.480000e+01
///-- im just surprised that the Manifold works and we can have the metric tensor.
Connection:
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
Eigenvalues:
-6.287820e+04
-6.287820e+04
4.165841e+01
4.165841e+01
4.794159e+01
4.794159e+01
6.660315e+04
6.660315e+04
Eigenvectors (Real Part):
3.110271e-06 0.000000e+00 -7.071068e-01 0.000000e+00 0.000000e+00 7.071068e-01 0.000000e+00 -3.026070e-06
-6.968789e-01 0.000000e+00 -3.066491e-06 0.000000e+00 0.000000e+00 -3.070425e-06 0.000000e+00 -7.171888e-01
7.171888e-01 0.000000e+00 3.242806e-06 0.000000e+00 0.000000e+00 -2.894111e-06 0.000000e+00 -6.968789e-01
-3.200908e-06 0.000000e+00 7.071068e-01 0.000000e+00 0.000000e+00 7.071068e-01 0.000000e+00 -2.940383e-06
0.000000e+00 3.110271e-06 0.000000e+00 -7.071068e-01 7.071068e-01 0.000000e+00 -3.026070e-06 0.000000e+00
0.000000e+00 -6.968789e-01 0.000000e+00 -3.066491e-06 -3.070425e-06 0.000000e+00 -7.171888e-01 0.000000e+00
0.000000e+00 7.171888e-01 0.000000e+00 3.242806e-06 -2.894111e-06 0.000000e+00 -6.968789e-01 0.000000e+00
0.000000e+00 -3.200908e-06 0.000000e+00 7.071068e-01 7.071068e-01 0.000000e+00 -2.940383e-06 0.000000e+00
Eigenvectors (Imaginary Part):
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
///-- probably very large sparcity? have to check to see if they are all zeroes
Random Keys:
key_mis_transform: 2b04dca41c1d71fa730fcf8c7115969cef57afbc1f162cec8b7e0549b5915e18366d3323cf9a2331ae30e5dfbfda9ddda23be3056ce45124d661af84b20d13d28f0ec04e8e798aefeb6157166852c7236fdb58e38b761bc3296c01828d2221cb458492a6ee55858eaa3a657fa1e47ad7607bb04c85749d0a2390df6b684c42f6
key_weights: f0e54f574f978bb680226ff6f332deec30b72955f276063980b16778dd8f8f74d6c09ff2247790fa457b65127b799cc2a57e191c4c314417d02176b10197ba7458ff89fbc8abd6d768c871ac0e6624cbdd6cacf12e4e7281958aee63cb8427c821ee17151851b9aadb1d26d658a80e5295b39eda07a3357990186418ec268189
///-- the entropy generated by the keys is good, but not better than current SOTA
Spatial Grid x:
0.000000e+00
1.428571e-01
2.857143e-01
4.285714e-01
5.714286e-01
7.142857e-01
8.571429e-01
1.000000e+00
Weights[0]: (-1.414850e-01+-2.748083e-01j)
Weights[1]: (-2.748459e-01+-3.004399e-01j)
Weights[2]: (-2.772054e-01+-1.678024e-01j)
Weights[3]: (-2.095578e-01+-1.757433e-01j)
Weights[4]: (-1.889746e-01+-3.113073e-01j)
Weights[5]: (-3.220990e-01+-1.812225e-01j)
Weights[6]: (-2.098004e-01+-3.923332e-01j)
Weights[7]: (-8.999614e-02+-2.891401e-01j)
</pre>
<p><span id="timestamp">30 Nov 2024</span></p>
<p style="background-color: #f5f5f5; border: 1px solid #ddd; padding: 10px; font-family: monospace; white-space: pre-wrap;">
Hypercomplex numbers, denoted as \( H \), are an extension of the set of complex numbers. They can be represented as:
\[ z = a + bi + cj + dk \]
where \( a, b, c, d \in \mathbb{R} \) and \( i, j, k \) are imaginary units with specific multiplication rules that extend those of complex numbers.
* Multiplication Rules:
The imaginary units \( i, j, k \) have the following properties:
1. \( i^2 = -1 \)
2. \( j^2 = -1 \)
3. \( k^2 = -1 \)
4. \( ij = k \), \( ji = -k \)
5. \( jk = i \), \( kj = -i \)
6. \( ki = j \), \( ik = -j \)
These rules ensure that the multiplication of elements in \( H \) is non-commutative but associative.
Algebraic Structure:
The set \( H \) forms a 4-dimensional vector space over the real numbers with a basis \( \{1, i, j, k\} \). The operations of addition and multiplication are defined as
follows:
- Addition:
\[ (a + bi + cj + dk) + (e + fi + gj + hk) = (a+e) + (b+f)i + (c+g)j + (d+h)k \]
- Multiplication:
Using the multiplication rules, multiply each term in one complex number by each term in the other and then combine like terms. For example:
\[ (a + bi + cj + dk)(e + fi + gj + hk) = ae + afi + agj + ahk + bei - bf + bgi - bkj + cei + cfi + cgj - ch + dei + dfi + djg - dhk \]
Properties of Hypercomplex Numbers:
* Closure: \( H \) is closed under addition and multiplication.
* Associativity: Both addition and multiplication are associative.
* Non-commutativity: Multiplication is not commutative, i.e., \( ab \neq ba \).
* Distributivity: Multiplication distributes over addition.
Examples of Hypercomplex Numbers:
* Quaternion:
A special case where \( i^2 = j^2 = k^2 = -1 \) and \( ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j \). This forms the set of quaternions (\( \mathbb{H} \)).
* Split Complex Numbers:
Another special case where \( i^2 = 1 \) and \( j^2 = -1 \), with all other products being zero. This system can be used in certain applications to represent rotations
in two dimensions.
Applications of Hypercomplex Numbers:
Quantum mechanics often requires the use of complex numbers, and hypercomplex numbers could provide a more general framework for describing systems. In control theory and signal processing, quaternions are used to represent rotations in three-dimensional space. Hypercomplex numbers can be used to define new geometric structures that extend beyond Euclidean geometry.
Hypercomplex numbers, \( H \), are a extension of the real number system that incorporates elements from both complex and quaternion systems.
</p>
<br>
<p><span id="timestamp">21 Nov 2024</span></p>
<p>phase space trajectory of philosophical dimensions,
this shows how metaphysics, epistemology, and ethics interact in a three-dimensional space, with colors representing different measurement sets. The clustering patterns suggest interesting philosophical attractors!
the quantum correlation matrix show interesting relationships:</p>
<img src="/blog/images/quantum_phase_space_trajectory_philisophical_dimension.png">
<p>
we can see slight negative correlations between ethics and both metaphysics (-0.14) and epistemology (-0.11), suggesting these dimensions may exhibit philosophical complementarity, the uncertainty relations:
Δmetaphysics × Δepistemology = 0.0406
Δmetaphysics × Δethics = 0.0419
Δepistemology × Δethics = 0.0392
These are analogous to Heisenberg's uncertainty principle but for philosophical dimensions, the products of uncertainties show that we cannot simultaneously have precise knowledge of multiple philosophical dimensions.
The philosophical entropy values:
metaphysics: -19.7683
epistemology: -18.5209
ethics: -22.1296
Ethics shows the highest entropy (-22.13), suggesting it has the most complex and unpredictable behavior among the dimensions!
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