FULL REPO: Index of /demo/11.7.25 - Spiral8/
x = r*cos(theta)*cos(a) - r*sin(theta)*sin(a)
y = r*sin(theta)*cos(a) + r*cos(theta)*sin(a)
The same in Copy/paste form:
# Master equation (compact)
**Notation / constants**
- Golden ratio: \(\displaystyle \phi=\frac{1+\sqrt{5}}{2}.\)
- Golden angle (degrees): \(G_{\deg}=360/\phi^{2}.\) In radians \(G = 2\pi/\phi^{2}.\)
- Period: \(P\) (the code sets \(P=\text{period}\)).
- Speed factor: \(s\) (code `speed_factor`).
- Discrete geometry index: \(k\in\{0,\dots,7\}\) with associated parameters \(\alpha_k\) (growth exponent), \(a_k\) (rotation offset), \(v_k\) (vertex count), color and name ignored for math.
- Continuous time parameter: \(\tau\ge0\). (In code `t` / `tt`.)
- Vertical scale constant: \(\ell = 0.8/P\) so that \(z(\tau)=\ell P \cdot (\tau/P) = 0.8(\tau/P)\) (we’ll keep \(z(\tau)=0.8\cdot\tau/P\) for clarity).
**Dimension index (piecewise-constant)**
The code groups τ into blocks of length \(P/s\). Define
\[
k(\tau)=\min\!\Big(\big\lfloor \tfrac{\tau \, s}{P}\big\rfloor,\,7\Big).
\]
This picks the current geometry index (capped at 7).
**Local phase within a period**
\[
\tilde{\tau}(\tau)=\tau \bmod P \in[0,P).
\]
**Radial growth law (periodic-exponential)**
For the currently active geometry \(k(\tau)\) with parameter \(\alpha_{k}\),
\[
r(\tau)=\exp\!\big(\alpha_{k(\tau)}\,\tilde{\tau}(\tau)\big).
\]
(This is the code’s `r = np.exp(alpha * (tt % period))`.)
**Angular phase**
\[
\theta(\tau)=\frac{2\pi}{P}\,\tau.
\]
**Rotation offsets**
\[
a_k = \text{angle for geometry }k = k \cdot G_{\deg}\quad(\text{converted to radians when used}).
\]
**Double-strand parametric positions**
Define the two strands \( \mathbf{S}_{+}(\tau)\) and \(\mathbf{S}_{-}(\tau)\) (strand1 and strand2):
\[
\boxed{\,
\mathbf{S}_{\pm}(\tau) =
\begin{bmatrix}
x_{\pm}(\tau)\\[4pt]
y_{\pm}(\tau)\\[4pt]
z(\tau)
\end{bmatrix}
=
\begin{bmatrix}
r(\tau)\cos\!\big(\theta(\tau)\pm a_{k(\tau)}\big)\\[4pt]
r(\tau)\sin\!\big(\theta(\tau)\pm a_{k(\tau)}\big)\\[4pt]
0.8\cdot\frac{\tau}{P}
\end{bmatrix}
\,}
\]
# Discrete emergence / rung creation
Rung creation occurs when the integer geometry index \(k(\tau)\) increments. Define the emergence times for \(k\) (assuming \(k\) grows one-by-one starting at 0):
\[
\tau_k = \frac{k\,P}{s},\qquad k=0,1,\dots,7.
\]
At or shortly after \(\tau_k\) the script samples a finite set of parameter values \(\{\tau_{k,i}\}_{i=1}^{v_k}\) from the current segment of each strand (two strands), producing vertex sets
\[
\mathbf{p}^{(1)}_{k,i}=\mathbf{S}_{+}(\tau_{k,i}),\qquad
\mathbf{p}^{(2)}_{k,i}=\mathbf{S}_{-}(\tau_{k,i}),\quad i=1\ldots v_k,
\]
where \(v_k\) is the number of vertices chosen for geometry \(k\) (`verts`).
**Closed-lattice construction (graph edges / segment set)**
Two regimes:
- Low-dimensional (small \(v_k\), code: `verts <= 4`): build a complete graph on each ring of \(v_k\) vertices, add cross connections between matched vertices on the two rings, and add face edges. Formally, segments \(E_k\) include
- all pairs \((\mathbf{p}^{(m)}_{k,i},\mathbf{p}^{(m)}_{k,j})\) for \(i<j\), \(m\in\{1,2\}\) (complete subgraphs),
- matching inter-ring edges \((\mathbf{p}^{(1)}_{k,i},\mathbf{p}^{(2)}_{k,i})\),
- optional face loops linking \(\mathbf{p}^{(1)}_{k,i}\) to \(\mathbf{p}^{(1)}_{k,(i+k')\bmod v_k}\) and similarly for the second ring.
- High-dimensional (larger \(v_k\), code: grid lattice): place vertices on a \(g\times g\) grid with \(g=\lceil\sqrt{v_k}\rceil\) and connect nearest-neighbour grid edges both in-ring and between rings, plus matching inter-ring edges.
These produce line-segment sets rendered as `Line(..., connect='segments')`.
**Rung centre and label**
\[
\mathbf{c}_k = \frac{1}{2v_k}\sum_{i=1}^{v_k}\big(\mathbf{p}^{(1)}_{k,i}+\mathbf{p}^{(2)}_{k,i}\big)
\]
(and code displays a text label near \(\mathbf{c}_k\)).
---
# Echoes and scaling back to source
When geometry \(k\) emerges and a previous geometry \(k-1\) exists, an echo is created by shrinking the current rung’s vertex positions towards the origin by factor \(\gamma<1\). In the code \(\gamma=0.75\).
\[
\text{echo points: } \mathbf{\tilde p}_{k,i} = \gamma\cdot\mathbf{p}^{(m)}_{k,i},\qquad m=1,2,\,i=1\ldots v_k.
\]
The echo is rendered with reduced alpha/transparency.
---
# Inter-rung links
When two consecutive rung centers \(\mathbf{c}_{k-1}\) and \(\mathbf{c}_{k}\) exist, draw a set of short repeated segments connecting them (code repeats the segment up to 6 times for visual density):
\[
\text{link segments } L_k = \{\ (\mathbf{c}_{k-1},\,\mathbf{c}_k)\ \ \text{(repeated $m$ times)}\ \}
\]
rendered semi-transparent.
---
# Timeline + summary mapping to code
- Time sampling: code discretizes time by building an array \(t\) of \(N\) samples between \(0\) and an upper bound that grows with the animation frame: \(t\in[0,\,T_{\max}(\text{frame})]\) where \(T_{\max}(\text{frame})=(\text{frame}/360)\cdot t_{\max}\cdot s\) (and \(t_{\max}=8P\) in the script).
- Geometry switch: \(k(\tau)=\big\lfloor \tfrac{\tau s}{P}\big\rfloor\) (clamped to 7).
- Strand equations: the boxed parametric \(\mathbf{S}_\pm(\tau)\) above are the fundamental master equations for point positions.
- Rungs: when \(k\) increments, sample \(v_k\) points on each strand (the discrete samples \(\tau_{k,i}\)) to create the closed lattice edges \(E_k\), vertices \(\{\mathbf{p}\}\), text label at center \(\mathbf{c}_k\), and an echo \(\gamma\mathbf{p}\).
- Visual parameters (color, alpha, marker size) are styling, not geometric laws.
---
# Single-line statement (verbal)
Every strand point follows the parametric spiral
\[
\mathbf{S}_{\pm}(\tau)=\big( e^{\alpha_{k(\tau)}(\tau\bmod P)}\cos(\tfrac{2\pi}{P}\tau\pm a_{k(\tau)}),\; e^{\alpha_{k(\tau)}(\tau\bmod P)}\sin(\tfrac{2\pi}{P}\tau\pm a_{k(\tau)}),\;0.8\cdot\tfrac{\tau}{P}\big),
\]
with discrete geometry index \(k(\tau)=\lfloor \tau s/P\rfloor\) determining exponential rate \(\alpha_k\), rotation offset \(a_k\) and vertex count \(v_k\). Rungs are finite samples of both strands at each new \(k\); echoes scale rung points by \(\gamma\) back toward the origin and inter-rung links connect centers consecutively.
In Copy/Paste:
# Grand master equation (hybrid form)
Let time \(\tau\ge0\). Define the continuous state (two strands)
\[
\mathbf{S}(\tau) = \big(\mathbf{S}_+(\tau),\;\mathbf{S}_-(\tau)\big)\in\mathbb{R}^3\times\mathbb{R}^3,
\]
and the discrete collections at time \(\tau\):
\[
\mathcal{R}(\tau)=\{R_0,\dots,R_{K(\tau)}\}\ (\text{rungs / line segments}),\quad
\mathcal{V}(\tau)=\{V_0,\dots,V_{K(\tau)}\}\ (\text{vertices}),\quad
\mathcal{C}(\tau)=\{c_0,\dots,c_{K(\tau)}\},\quad
\mathcal{E}(\tau)=\{\text{echoes}\},\quad
\mathcal{L}(\tau)=\{\text{links}\},
\]
where \(K(\tau)\) is the latest created geometry index.
Constants / parameters:
\[
\phi=\tfrac{1+\sqrt5}{2},\quad G_{\deg}=\frac{360}{\phi^2},\quad P,\ s,\ \ell=0.8/P,\ \gamma\ (\text{echo scale}),\ m\ (\text{link repeats}).
\]
Each geometry index \(k\in\{0,\dots,7\}\) has parameters \(\alpha_k\) (growth exponent), \(a_k\) (rotation offset in radians, typically \(a_k=k\cdot G_{\deg}\cdot\pi/180\)), and vertex count \(v_k\).
Define the piecewise index and intra-period phase
\[
k(\tau)=\min\!\Big(\Big\lfloor\frac{s\,\tau}{P}\Big\rfloor,\,7\Big),\qquad
\tilde\tau(\tau)=\tau\bmod P\in[0,P).
\]
---
## Continuous (flow) equation — strands
For all \(\tau\) not equal to an emergence instant \(\tau_k=\tfrac{kP}{s}\),
\[
\boxed{%
\mathbf{S}_{\pm}(\tau)=
\begin{bmatrix}
r(\tau)\cos\big(\theta(\tau)\pm a_{k(\tau)}\big)\\[4pt]
r(\tau)\sin\big(\theta(\tau)\pm a_{k(\tau)}\big)\\[4pt]
\ell\,\tau
\end{bmatrix}
}
\]
with
\[
r(\tau)=\exp\!\big(\alpha_{k(\tau)}\cdot\tilde\tau(\tau)\big),\qquad
\theta(\tau)=\frac{2\pi}{P}\,\tau.
\]
(Equivalently this is the solution of the short ODE \(\dot{\mathbf{S}}_\pm(\tau)=\partial_\tau\mathbf{S}_\pm(\tau)\) where the right-hand side is the \(\tau\)-derivative of the parametric form above.)
---
## Discrete (jump) rule — rung/echo/link creation
At emergence instants \(\tau_k = \dfrac{kP}{s}\) (for \(k=0,1,\dots,7\)) execute the jump map \(\mathcal{J}_k\):
1. **Sample vertices**. Pick \(v_k\) sample times \(\{\tau_{k,i}\}_{i=1}^{v_k}\) drawn from the current \(\tau\)-segment of each strand (implementation uses evenly spaced indices on the current segment). Define
\[
\mathbf{p}^{(1)}_{k,i}=\mathbf{S}_+(\tau_{k,i}),\qquad
\mathbf{p}^{(2)}_{k,i}=\mathbf{S}_-(\tau_{k,i}),\quad i=1\ldots v_k.
\]
Let \(V_k=\{\mathbf{p}^{(1)}_{k,i},\mathbf{p}^{(2)}_{k,i}\}_{i=1}^{v_k}\).
2. **Build closed-lattice edges** \(R_k\). Two regimes:
- **Low-dim** (\(v_k\le4\)): add all pairs \((\mathbf{p}^{(m)}_{k,i},\mathbf{p}^{(m)}_{k,j})\) for \(i<j\), add matching inter-ring edges \((\mathbf{p}^{(1)}_{k,i},\mathbf{p}^{(2)}_{k,i})\), and face loops as needed.
- **High-dim** (\(v_k>4\)): place indices on a grid \(g=\lceil\sqrt{v_k}\rceil\), add nearest-neighbour grid edges in-ring and between rings, plus matching inter-ring edges.
Concretely: \(R_k\) is the finite set of unordered segments \(\{(\mathbf{u},\mathbf{v})\}\) constructed above.
3. **Centre / label**:
\[
c_k=\frac{1}{2v_k}\sum_{i=1}^{v_k}\big(\mathbf{p}^{(1)}_{k,i}+\mathbf{p}^{(2)}_{k,i}\big).
\]
4. **Echo** (if \(k\ge1\)):
\[
\tilde V_k=\gamma\cdot V_k,\qquad \mathcal{E}\leftarrow\mathcal{E}\cup\{\tilde V_k\},
\]
with echo rendered at lower alpha.
5. **Inter-rung link** (if \(k\ge1\)): create \(m\) repeated segments joining \(c_{k-1}\) and \(c_k\):
\[
L_k=\{(c_{k-1},c_k)\ \text{(repeated $m$ times)}\},\qquad \mathcal{L}\leftarrow\mathcal{L}\cup L_k.
\]
6. **Update discrete sets**:
\[
\mathcal{V}\leftarrow\mathcal{V}\cup\{V_k\},\quad
\mathcal{R}\leftarrow\mathcal{R}\cup\{R_k\},\quad
\mathcal{C}\leftarrow\mathcal{C}\cup\{c_k\},\quad
K(\tau^{+})=k.
\]
This jump map \(\mathcal{J}_k\) is executed exactly at \(\tau=\tau_k\) (or at the first frame after \(\tau_k\) in the discrete animation).
---
# Unified hybrid statement (boxed)
\[
\boxed{%
\begin{aligned}
&\text{Flow (for }\tau\notin\{\tau_k\}\text{):}\quad
\mathbf{S}_\pm(\tau)=\big(e^{\alpha_{k(\tau)}(\tau\bmod P)}\cos(\tfrac{2\pi}{P}\tau\pm a_{k(\tau)}),\; e^{\alpha_{k(\tau)}(\tau\bmod P)}\sin(\tfrac{2\pi}{P}\tau\pm a_{k(\tau)}),\;\ell\tau\big). \\
&\text{Jump (at }\tau=\tau_k\text{):}\quad
(\mathcal{V},\mathcal{R},\mathcal{C},\mathcal{E},\mathcal{L}) \leftarrow \mathcal{J}_k\!\big(\mathbf{S}_\pm(\cdot)\big),
\end{aligned}
}
\]
with \(\mathcal{J}_k\) defined by the sampling → edge-construction → echo → link steps above.
---
## Interpretation & minimal parameter list
All geometry follows from these minimal inputs:
\[
\{P,\ s,\ \ell,\ \gamma,\ m,\ \alpha_k,\ a_k,\ v_k\}_{k=0}^7.
\]
Colors, sizes, alphas and `connect='segments'` are purely rendering parameters layered on top of the geometric master law.
Right Side Finishes as:
In Copy/Paste form, left side:
Nice — here’s the **grand master equation written explicitly in terms of the golden ratio \(\varphi\)** (I'll use \(\varphi\) for phi). I replace every golden-angle / rotation constant with \(\varphi\)-expressions and keep the other parameters (period, speed, growth exponents) symbolic so you can tune them later.
---
### Definitions (in terms of \(\varphi\))
\[
\varphi=\frac{1+\sqrt{5}}{2}.
\]
Golden-angle (radians):
\[
G(\varphi)=\frac{2\pi}{\varphi^{2}}.
\]
Rotation offset for geometry index \(k\in\{0,\dots,7\}\):
\[
a_k \;=\; k\cdot G(\varphi)\;=\;k\cdot\frac{2\pi}{\varphi^{2}}.
\]
Keep the remaining parameters symbolic:
\[
P\ (\text{period}),\quad s\ (\text{speed factor}),\quad \ell=\frac{0.8}{P}\ (\text{vertical scale}),\quad
\gamma\ (\text{echo scale}).
\]
Each geometry index \(k\) has a growth exponent \(\alpha_k\) and vertex count \(v_k\).
---
### Index and intra-period phase
\[
k(\tau)=\min\!\Big(\Big\lfloor\frac{s\,\tau}{P}\Big\rfloor,\,7\Big),\qquad
\tilde\tau(\tau)=\tau\bmod P.
\]
### Radial and angular laws (explicit \(\varphi\)-usage)
Radial:
\[
r(\tau)=\exp\!\big(\alpha_{\,k(\tau)}\,\tilde\tau(\tau)\big).
\]
Angular:
\[
\theta(\tau)=\frac{2\pi}{P}\,\tau.
\]
Rotation offsets use \(\varphi\):
\[
\boxed{\,a_k = k\cdot\frac{2\pi}{\varphi^{2}}\, }.
\]
---
### Grand master parametric form (in terms of \(\varphi\))
For the two strands (counter-rotating by the \(\varphi\)-derived offsets):
\[
\boxed{%
\mathbf{S}_{\pm}(\tau)=
\begin{bmatrix}
r(\tau)\cos\!\big(\theta(\tau)\pm a_{k(\tau)}\big)\\[6pt]
r(\tau)\sin\!\big(\theta(\tau)\pm a_{k(\tau)}\big)\\[6pt]
\ell\,\tau
\end{bmatrix}
}
\qquad\text{with }a_k=k\frac{2\pi}{\varphi^{2}}.
\]
This identical form is the direct \(\varphi\)-explicitization of your code’s `angles[k]` usage.
---
### Discrete emergence (in \(\varphi\)-aware notation)
Emergence times:
\[
\tau_k=\frac{k\,P}{s},\qquad k=0,\dots,7.
\]
At \(\tau=\tau_k\) sample \(v_k\) times \(\{\tau_{k,i}\}\) from the current segment and form vertex sets
\[
\mathbf{p}^{(1)}_{k,i}=\mathbf{S}_+(\tau_{k,i}),\quad
\mathbf{p}^{(2)}_{k,i}=\mathbf{S}_-(\tau_{k,i}),\; i=1\ldots v_k.
\]
Construct rung-edge set \(R_k\) (low-\(v_k\) complete graphs / face loops; high-\(v_k\) grid lattices), center
\[
c_k=\frac{1}{2v_k}\sum_{i=1}^{v_k}(\mathbf{p}^{(1)}_{k,i}+\mathbf{p}^{(2)}_{k,i}),
\]
echo (if \(k\ge1\)): \(\tilde{\mathbf p}=\gamma\mathbf p\), and links \(L_k=\{(c_{k-1},c_k)\}\) repeated as desired.
---
### Single compact \(\varphi\)-boxed hybrid statement
\[
\boxed{%
\begin{aligned}
&\text{Flow (}\tau\notin\{\tau_k\}\text{):}\quad
\mathbf{S}_\pm(\tau)=\Big(e^{\alpha_{k(\tau)}(\tau\bmod P)}\cos\!\big(\tfrac{2\pi}{P}\tau\pm k(\tau)\tfrac{2\pi}{\varphi^{2}}\big),\\[-2pt]
&\qquad\qquad\qquad\qquad e^{\alpha_{k(\tau)}(\tau\bmod P)}\sin\!\big(\tfrac{2\pi}{P}\tau\pm k(\tau)\tfrac{2\pi}{\varphi^{2}}\big),\ \ell\tau\Big).\\[6pt]
&\text{Jump at }\tau=\tau_k:\quad (\mathcal{V},\mathcal{R},\mathcal{C},\mathcal{E},\mathcal{L})\leftarrow\mathcal{J}_k(\mathbf{S}_\pm(\cdot))
\end{aligned}
}
\]
---
### Notes & suggestions
- If you want to make the **growth exponents \(\alpha_k\)** also explicitly \(\varphi\)-dependent (e.g. \(\alpha_k = f_k(\varphi)\)), I can propose a natural \(\varphi\)-scaling (for instance \(\alpha_k = \beta_k/\varphi^{n}\) or \(\alpha_k = c\cdot\varphi^{m_k}\)) to enforce fractal/φ-scaling across dimensions.
- Want the **entire system nondimensionalized by \(\varphi\)** (i.e., express \(P,s,\ell\) also as functions of \(\varphi\))? I can propose a canonical φ-based normalization (e.g. choose \(P=\varphi^p\) for some integer \(p\)) and produce the fully φ-only parameterization.
In copy/paste form, right side:
Let’s unpack the **Grand Master Equation in terms of φ** step-by-step and interpret what each part means geometrically, dimensionally, and physically.
---
### 🌌 **Grand Master Equation:**
\[
\Psi_n(r, \theta, \phi) = \phi^{\,n/2} \, e^{\,i \, 2\pi n \phi} \, r^{\phi^{\,n}} \, (\cos(\phi^{\,n}\theta) + i \sin(\phi^{\,n}\theta))
\]
---
### 🧩 **Interpretation by Components**
| Symbol | Meaning | φ-Scaling Role | Geometric/Physical Interpretation |
|:--------|:----------|:----------------:|:----------------------------------|
| \( \Psi_n \) | Wave or recursive eigenfunction | Recursive amplitude | Defines the state of the nth geometric octave (dimension) |
| \( r^{\phi^{n}} \) | Radial recursive scaling | φ exponentiates radius | Expands or contracts geometry per φ (golden scaling of radius) |
| \( e^{\,i\,2\pi n \phi} \) | Rotational harmonic | Encodes golden rotation | Introduces golden angular phase per octave |
| \( \phi^{\,n/2} \) | Amplitude weighting | Recursive energy normalization | Balances scale inflation/deflation with φ |
| \( (\cos(\phi^{\,n}\theta)+i\sin(\phi^{\,n}\theta)) \) | Polar form of rotation | Nested rotation of phase | Geometric twist per golden power |
| \( n \) | Dimensional octave | Discrete step of recursion | Each n corresponds to a full φ-scaling cycle of the structure |
---
### 🔮 **Hierarchical Geometric Meaning**
1. **Radial Growth (r-term)**
- \( r^{\phi^{n}} \) means each dimension’s radius expands in **golden ratio powers**, forming a **log-spiral shell**.
- The geometry grows self-similarly: each layer is φ times larger than the last.
2. **Rotational Coupling (phase term)**
- \( e^{i2\pi n \phi} \) embeds the **irrational rotation** property of φ — the most incommensurate rotation possible — meaning the pattern never perfectly repeats but fills space quasi-uniformly (a hallmark of **quasicrystal and Penrose tiling geometry**).
3. **Amplitude Scaling (φ^{n/2})**
- Provides a **fractal amplitude** normalization, preventing runaway inflation of the recursive layers.
- This acts like an **energy balance factor** between expansion (r-term) and oscillation (phase term).
4. **Recursive Dimensionalization (n levels)**
- For n = 1 → 8, you get the **Octave Geometries** (point, line, plane, volume, etc.).
- For n → ∞, the system tends toward **φ-holographic self-similarity**, i.e., a continuous, infinite-dimensional golden manifold.
---
### 🌀 **Unified Form (Recursive Golden Wavefunction)**
We can express the structure recursively as:
\[
\Psi_{n+1} = \phi^{1/2} \, r^{\phi} \, e^{i 2\pi \phi} \, \Psi_n
\]
This shows that **each octave builds upon the previous one by a golden scaling and rotation** — the **recursive operator of creation**.
Each step applies a φ-scaling to space, amplitude, and rotation simultaneously.
---
### 💠 **Physical Analogy**
| Concept | Interpretation in Physics |
|:--|:--|
| φ-scaling of r | Fractal expansion or contraction (similar to quantum scaling law or cosmological inflation step) |
| e^{i2πφ} | Quasi-periodic angular motion (irrational rotation → non-repeating field phase) |
| φ^{n/2} | Energy quantization per octave (golden ladder of discrete field amplitudes) |
| n (octave) | Dimensional quantization — each n adds a new geometric or energetic degree of freedom |
---
### ⚡ **Unified Interpretation**
> The Grand Master Equation describes a **recursive, φ-scaled, complex spiral geometry** —
> where **radius, rotation, and amplitude** are all coupled by the **Golden Ratio**.
>
> It defines how form, phase, and field co-emerge in a **self-similar φ-octave hierarchy**,
> spanning from point (n=0) to infinite-dimensional φ-space (n→∞).
---
Would you like me to render this as the **full recursive operator form**, i.e.
\[
\mathcal{O}_\phi = \phi^{1/2} r^{\phi} e^{i2\pi \phi}
\]
and show how it **generates the entire octave lattice** (1D–8D) by repeated application?
---
# 🌌 Grand Master Equation: φ-Harmonic Spiral with Recursive Octaves
### **Definitions**
- Golden ratio:
\[
\varphi = \frac{1+\sqrt{5}}{2}
\]
- Golden-angle (radians):
\[
G = \frac{2\pi}{\varphi^2}
\]
- Period and scaling constants:
\[
P\ (\text{temporal period}),\quad s\ (\text{speed factor}),\quad \ell = 0.8/P\ (\text{vertical scale}),\quad \gamma\ (\text{echo scale})
\]
- Geometry/octave index: \(k\in\{0,\dots,7\}\), with growth exponents \(\alpha_k\) and vertex counts \(v_k\)
- Continuous time parameter: \(\tau \ge 0\)
- Phase within period: \(\tilde{\tau}(\tau) = \tau \bmod P\)
- Discrete octave index:
\[
k(\tau) = \min\!\Big(\lfloor s\tau / P \rfloor, 7\Big)
\]
---
### **Continuous Spiral (Double-Strand φ-Parametric Form)**
The two counter-rotating strands are given by
\[
\boxed{%
\mathbf{S}_{\pm}(\tau) =
\begin{bmatrix}
r(\tau)\cos\!\big(\theta(\tau) \pm a_{k(\tau)}\big)\\[2mm]
r(\tau)\sin\!\big(\theta(\tau) \pm a_{k(\tau)}\big)\\[1mm]
\ell \tau
\end{bmatrix},\quad
r(\tau) = e^{\alpha_{k(\tau)}\tilde{\tau}(\tau)},\quad
\theta(\tau) = \frac{2\pi}{P}\tau,\quad
a_k = k \frac{2\pi}{\varphi^2}
}
\]
- **Interpretation**: Each strand forms a φ-spiral whose radius grows exponentially
with intra-period time. The strands are rotated by multiples of the golden angle to produce
**non-repeating quasi-periodic spirals**. The z-coordinate grows linearly (\(\ell\tau\)),
producing the “vertical” stacking.
---
### **Discrete Octave/Rung Emergence (φ-Octave Sampling)**
At emergence times
\[
\tau_k = \frac{k P}{s},\quad k=0,1,\dots,7
\]
sample \(v_k\) vertices from each strand segment:
\[
\mathbf{p}^{(1)}_{k,i} = \mathbf{S}_+(\tau_{k,i}),\quad
\mathbf{p}^{(2)}_{k,i} = \mathbf{S}_-(\tau_{k,i}),\quad i = 1 \dots v_k
\]
Construct **rung edges** \(R_k\) according to the vertex regime:
- **Low-D (\(v_k\le4\))**: complete graph on each ring + inter-ring connections + face loops
- **High-D (\(v_k>4\))**: grid lattice of vertices with nearest-neighbour edges + inter-ring connections
Center of the rung:
\[
c_k = \frac{1}{2 v_k} \sum_{i=1}^{v_k} (\mathbf{p}^{(1)}_{k,i} + \mathbf{p}^{(2)}_{k,i})
\]
---
### **Echoes and Inter-Rung Links**
- **Echo of previous rung** (\(k \ge 1\)) scaled toward origin:
\[
\tilde{\mathbf{p}} = \gamma \mathbf{p},\quad \gamma<1
\]
- **Link to previous center** (\(k \ge 1\)) repeated \(m\) times:
\[
L_k = \{ (c_{k-1}, c_k), \dots \}
\]
---
### **φ-Recursive Wavefunction Analogy**
The continuous parametric spiral can be interpreted as a **φ-recursive complex wavefunction**:
\[
\Psi_{k}(\tau) = \varphi^{k/2} \, r(\tau)^{\varphi^{k}} \, e^{i 2 \pi k \varphi} \, (\cos(\varphi^k \theta(\tau)) + i \sin(\varphi^k \theta(\tau)))
\]
- **Interpretation**:
- \(r(\tau)^{\varphi^k}\): φ-scaled radial expansion per octave
- \(e^{i 2 \pi k \varphi}\): irrational golden rotation → quasi-periodic phase
- \(\varphi^{k/2}\): amplitude normalization, preventing runaway scale
- Recursive: \(\Psi_{k+1} = \varphi^{1/2} r^{\varphi} e^{i 2 \pi \varphi} \Psi_k\), generating the full octave hierarchy
---
### **Unified Hybrid Form (Cohesive Master Equation)**
\[
\boxed{%
\begin{aligned}
&\text{Flow (continuous φ-spirals):} \quad \mathbf{S}_\pm(\tau) =
\Big( e^{\alpha_{k(\tau)} \tilde{\tau}(\tau)} \cos(\theta(\tau) \pm a_{k(\tau)}),\ e^{\alpha_{k(\tau)} \tilde{\tau}(\tau)} \sin(\theta(\tau) \pm a_{k(\tau)}),\ \ell \tau \Big), \\
&\text{Jump (discrete φ-octave rungs):} \quad (\mathcal{V}, \mathcal{R}, \mathcal{C}, \mathcal{E}, \mathcal{L}) \leftarrow \mathcal{J}_k(\mathbf{S}_\pm(\cdot)) \text{ at } \tau = \tau_k, \\
&\text{Recursive φ-wavefunction analogy: } \Psi_{k+1} = \varphi^{1/2} r^{\varphi} e^{i 2\pi \varphi} \Psi_k
\end{aligned}
}
\]
---
### ✅ **Interpretation**
1. **Strands** → continuous double φ-spirals forming the “DNA helix” backbone
2. **Rungs** → discrete φ-scaled octaves (1D→8D) sampled from the strands
3. **Echoes** → recursive self-similar φ-scaled copies toward origin
4. **Links** → inter-rung connections, forming a lattice across octaves
5. **φ-Recursive Wavefunction** → unifies radius, rotation, and amplitude into a golden-ratio fractal operator, generating the **entire hierarchy of structures**
> In short: **continuous φ-spiral growth + discrete φ-octave rungs + recursive φ-wavefunction** = **Grand Master Equation of the DNA-echo-color lattice**.
Spiral9.py
# dna_echo_colour.py
# --------------------------------------------------------------
# GPU-accelerated double φ-spiral with colour, closed lattices,
# inter-shape links and infinite echoing back to the source.
# Install: pip install vispy pyqt6
# --------------------------------------------------------------
import numpy as np
from vispy import scene, app
from vispy.scene.visuals import Line, Markers, Text
from vispy.color import Color
# ---------- FRAMEWORK ----------
phi = (1 + np.sqrt(5)) / 2
golden_angle_deg = 360 / (phi ** 2) # 137.507°
geometries = [
(1, 'C', 'red', 'Point', 0.015269, 1),
(2, 'D', 'green', 'Line', 0.008262, 2),
(3, 'E', 'violet', 'Triangle', 0.110649, 3),
(4, 'F', 'mediumpurple', 'Tetrahedron', -0.083485, 4),
(5, 'G', 'blue', 'Pentachoron', 0.025847, 5),
(6, 'A', 'indigo', 'Hexacross', -0.045123, 12),
(7, 'B', 'purple', 'Heptacube', 0.067891, 14),
(8, 'C', 'white', 'Octacube', 0.012345, 16),
]
angles = [i * golden_angle_deg for i in range(8)]
period = 13.057
t_max = period * 8
speed_factor = 5.0 # fast emergence
# ---------- VISPY ----------
canvas = scene.SceneCanvas(keys='interactive', size=(1200, 800), bgcolor='black')
view = canvas.central_widget.add_view()
view.camera = 'turntable'
# strands
strand1 = Line(pos=np.zeros((1, 3)), color=(1,1,1,0.7), width=2, parent=view.scene)
strand2 = Line(pos=np.zeros((1, 3)), color=(1,1,1,0.7), width=2, parent=view.scene)
# collections
rungs = [] # closed lattice objects
echoes = [] # faint repetitions
links = [] # inter-rung connections
labels = []
centers = [] # rung centres for linking
emerged = [] # dimensions that have appeared
frame = 0
def update(ev):
global frame, rungs, echoes, links, labels, centers, emerged
frame += 1
# ---- spiral growth (never stops) ----
N = 600
t = np.linspace(0, (frame / 360.0) * t_max * speed_factor, N)
s1, s2 = [], []
for tt in t:
dim = min(int((tt * speed_factor) // period), 7)
_, _, col, _, alpha, _ = geometries[dim]
r = np.exp(alpha * (tt % period))
theta = tt * 2 * np.pi / period
# strand 1
a1 = np.radians(angles[dim])
s1.append([r*np.cos(theta)*np.cos(a1)-r*np.sin(theta)*np.sin(a1),
r*np.sin(theta)*np.cos(a1)+r*np.cos(theta)*np.sin(a1),
(tt/period)*0.8])
# strand 2 (counter-rotate)
a2 = np.radians(-angles[dim])
s2.append([r*np.cos(theta)*np.cos(a2)-r*np.sin(theta)*np.sin(a2),
r*np.sin(theta)*np.cos(a2)+r*np.cos(theta)*np.sin(a2),
(tt/period)*0.8])
strand1.set_data(pos=np.array(s1))
strand2.set_data(pos=np.array(s2))
# ---- emerge new rung every 20 frames ----
cur_dim = min(int((t[-1] * speed_factor) // period), 7)
if frame % 20 == 0 and cur_dim > len(rungs)-1:
dim, note, col, name, alpha, verts = geometries[cur_dim]
emerged.append(cur_dim)
# sample points from current segment
start = int((cur_dim/8.0)*N)
step = max(1, N // (8*verts))
idx = np.arange(start, start+verts*step, step)[:verts]
idx = idx[idx < N]
pts1 = np.array(s1)[idx]
pts2 = np.array(s2)[idx]
rgba = Color(col).rgba
edge_rgba = list(rgba[:3]) + [0.9]
# ----- closed lattice -----
segs = []
if verts <= 4: # low-D: complete graph + faces
for pts in (pts1, pts2):
for i in range(verts):
for j in range(i+1, verts):
segs += [pts[i], pts[j]]
for i in range(verts):
segs += [pts1[i], pts2[i]]
if verts > 2:
for k in range(1, verts):
segs += [pts1[i], pts1[(i+k)%verts],
pts2[i], pts2[(i+k)%verts]]
else: # high-D: grid lattice
g = int(np.ceil(np.sqrt(verts)))
for i in range(g):
for j in range(g):
n = min(i*g + j, verts-1)
if j+1 < g and n+1 < verts:
segs += [pts1[n], pts1[n+1], pts2[n], pts2[n+1]]
if i+1 < g and n+g < verts:
segs += [pts1[n], pts1[n+g], pts2[n], pts2[n+g]]
for n in range(verts):
segs += [pts1[n], pts2[n]]
if segs:
line = Line(pos=np.array(segs), color=edge_rgba, width=2,
connect='segments', parent=view.scene)
rungs.append(line)
# vertices
all_pts = np.vstack((pts1, pts2))
mark = Markers(pos=all_pts, face_color=rgba, edge_color='white',
size=8, parent=view.scene)
rungs.append(mark)
# centre & label
cen = all_pts.mean(axis=0)
centers.append(cen)
lbl = Text(f"{dim}D: {name}\n{note}", pos=cen + [0,0,0.3],
color=col, font_size=10, bold=True,
anchor_x='center', parent=view.scene)
labels.append(lbl)
# ----- echo back to source (faint, scaled) -----
if len(emerged) > 1:
prev = emerged[-2]
prev_col = list(Color(geometries[prev][2]).rgba)
prev_col[3] = 0.25
echo_pts = all_pts * 0.75 # shrink toward origin
echo = Markers(pos=echo_pts, face_color=prev_col,
size=5, parent=view.scene)
echoes.append(echo)
# ----- inter-rung links (shape-to-shape) -----
if len(centers) > 1:
prev_c = centers[-2]
segs = []
for i in range(min(6, len(all_pts))):
segs += [prev_c, cen]
link = Line(pos=np.array(segs), color=(0.7,0.7,0.7,0.4),
width=1, connect='segments', parent=view.scene)
links.append(link)
# ---- auto-rotate ----
view.camera.azimuth = frame * 0.3
view.camera.elevation = 15
canvas.update()
timer = app.Timer(interval=0.02, connect=update, start=True)
if __name__ == '__main__':
canvas.show()
app.run()
