Hyper-Triskelion v9— Chaotically Modulated 4-Dimensional Parametric System (Compilation)

Description

Methods (English)

This work introduces v2.0 of the Stone–Hyper-Triskelion, extending the original 4-dimensional parametric structure into a fully coupled, multi-feedback dynamical system. The v1 model defined a 4D triskelion using a time-inversion term (δₜ = t_future − t), asymmetric rotational phases, and a dual-frequency vertical component (z,w). In v2.0, the structure is expanded into a mathematically adaptive 4D organism, where geometric evolution is shaped by interacting feedback mechanisms.

This updated formulation incorporates:
• Oscillating temporal horizon t_future(t)
• Radius modulation through exponential ε-driven variation
• Phase-coupled time inversion δₜ(φ,t)
• Angular feedback δₜ(θ,t) producing directional modulation
• Radial–temporal gain feedback via μ(r,t)
• Angular–radial gain coupling via λ(θ,t)
• Noise-tuned stochastic adjustment via ν(ε)
• Higher vertical frequency splitting in the (z,w) channel

These interactions produce a rich family of behaviors including breathing shells, vortex-like curvature flows, recursive knotting, quasi-periodic patterns, and chaotic transitions, all within a purely mathematical 4-dimensional coordinate system. No physical processes are implied, and the system remains fully consistent with classical physics and mathematical dynamical systems theory.

The v2.0 package includes consolidated reference equations, projection figures (2D and 3D), and a reproducible Python simulation environment, enabling further exploration and study of multi-feedback, high-dimensional parametric organisms.

(1) Base System — ASCII Version
------------------------------------------------------------
delta_t = t_future - t
(where t_future = 2, r = 1, and t runs from 1 → 0)

x(theta, phi, t) = (r + delta_t * cos(theta)) * cos(theta + phi + sin(delta_t))

y(theta, phi, t) = (r + delta_t * cos(theta)) * sin(theta + phi + sin(delta_t))

z(theta, phi, t) = delta_t * sin(theta) + cos(theta / 2)

w(theta, phi, t) = delta_t * sin(theta / 2)

Arms:
phi = 0, 2*pi/3, 4*pi/3

------------------------------------------------------------
(1b) Base System — Greek-Letter Version
------------------------------------------------------------
δₜ = t_future − t
(r = 1,  t_future = 2)

x(θ, φ, t) = (r + δₜ · cosθ) · cos(θ + φ + sin(δₜ))

y(θ, φ, t) = (r + δₜ · cosθ) · sin(θ + φ + sin(δₜ))

z(θ, φ, t) = δₜ · sinθ + cos(θ / 2)

w(θ, φ, t) = δₜ · sin(θ / 2)

Arms:
φ ∈ {0, 2π/3, 4π/3}

============================================================
(2) Extended Feedback-Coupled System — ASCII Version
============================================================

t_future(t) = 2 + 0.5 * sin(2*pi*t)

epsilon(theta, t) = 0.1 * sin(3*theta + 2*pi*t)

r(theta, t) = (1 + 0.3 * sin(2*pi*t)) * exp( epsilon(theta, t) * sin(theta) )

mu(r, t) = 0.2 + 0.1 * sin(r + pi*t)

lambda(theta, t) = 0.1 + 0.2 * sin(4*theta + pi*t)

nu(epsilon) = 0.15 + 0.1 * epsilon

delta_t(theta, phi, t) =
    ( t_future(t) - t )
    * (1 + 0.2 * sin(phi + pi*t))
    * (1 + mu(r, t) * sin(2*theta + pi*t))
    * (1 + lambda(theta, t) * sin(r))
    + 0.1 * nu(epsilon) * epsilon

x = (r + delta_t * cos(theta)) * cos(theta + phi + sin(delta_t))

y = (r + delta_t * cos(theta)) * sin(theta + phi + sin(delta_t))

z = delta_t * sin(theta) + cos(k_vert * theta / 2)

w = delta_t * sin(k_vert * theta / 2)

(where k_vert = 3 in the extended model)

============================================================
(2b) Extended Feedback-Coupled System — Greek-Letter Version
============================================================

t_future(t) = 2 + 0.5 · sin(2πt)

ε(θ, t) = 0.1 · sin(3θ + 2πt)

r(θ, t) = (1 + 0.3 · sin(2πt)) · exp( ε(θ, t) · sinθ )

μ(r, t) = 0.2 + 0.1 · sin(r + πt)

λ(θ, t) = 0.1 + 0.2 · sin(4θ + πt)

ν(ε) = 0.15 + 0.1ε

δₜ(θ, φ, t) =
    ( t_future(t) − t )
    · (1 + 0.2 · sin(φ + πt))
    · (1 + μ(r,t) · sin(2θ + πt))
    · (1 + λ(θ,t) · sin(r))
    + 0.1 · ν(ε) · ε

x(θ, φ, t) = (r + δₜ · cosθ) · cos(θ + φ + sin(δₜ))

y(θ, φ, t) = (r + δₜ · cosθ) · sin(θ + φ + sin(δₜ))

z(θ, φ, t) = δₜ · sinθ + cos(k_vert · θ / 2)

w(θ, φ, t) = δₜ · sin(k_vert · θ / 2)

(k_vert = 3 for the extended organism system)

Authored by Shane Edward Stone 

Simulations conducted by AI collaborator Kaelen. 

Licensed under Creative Commons Attribution 4.0 International (CC BY 4.0).

Methods (English)

# ===============================================================
# Stone Hyper-Triskelion v2.0
# Fully Coupled 4D Organism — Reproducible Python Code
# ===============================================================

import numpy as np
import matplotlib.pyplot as plt

# -----------------------------
# Parameter ranges
# -----------------------------
theta = np.linspace(0, 2*np.pi, 2500)
t_vals = np.linspace(0, 1, 120)
phi_vals = [0, 2*np.pi/3, 4*np.pi/3]

# Vertical frequency
k_vert = 3

# -----------------------------
# Coupling functions (ASCII form of Greek letters)
# -----------------------------

def t_future(t):
    return 2 + 0.5 * np.sin(2*np.pi*t)

def epsilon(theta, t):
    return 0.1 * np.sin(3*theta + 2*np.pi*t)

def r_func(theta, t):
    return (1 + 0.3*np.sin(2*np.pi*t)) * np.exp(epsilon(theta,t) * np.sin(theta))

def mu(r, t):
    return 0.2 + 0.1 * np.sin(r + np.pi*t)

def lam(theta, t):
    return 0.1 + 0.2 * np.sin(4*theta + np.pi*t)

def nu(eps):
    return 0.15 + 0.1*eps

# -----------------------------
# Fully coupled delta_t
# -----------------------------
def delta_t(theta, phi, t):
    tf = t_future(t)
    eps = epsilon(theta, t)
    r = r_func(theta, t)
    return ((tf - t)
            * (1 + 0.2*np.sin(phi + np.pi*t))
            * (1 + mu(r, t)*np.sin(2*theta + np.pi*t))
            * (1 + lam(theta, t)*np.sin(r))
            + 0.1*nu(eps)*eps)

# -----------------------------
# Final 4D coordinates
# -----------------------------
def coords(theta, phi, t):
    dt = delta_t(theta, phi, t)
    r = r_func(theta, t)
    x = (r + dt*np.cos(theta)) * np.cos(theta + phi + np.sin(dt))
    y = (r + dt*np.cos(theta)) * np.sin(theta + phi + np.sin(dt))
    z = dt*np.sin(theta) + np.cos(k_vert * theta / 2)
    w = dt*np.sin(k_vert * theta / 2)
    return x, y, z, w

# -----------------------------
# Visualization: 3D projection
# -----------------------------
fig = plt.figure(figsize=(10, 8))
ax = fig.add_subplot(111, projection='3d')

t_frame = 0.35   # Choose a time slice to visualize

for phi in phi_vals:
    x, y, z, w = coords(theta, phi, t_frame)
    ax.plot(x, y, z, linewidth=1.3)

ax.set_title("Stone–Kaelen Hyper-Triskelion v2.0 — Fully Coupled 4D Organism")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
plt.tight_layout()
plt.show()

Licensed under Creative Commons Attribution 4.0 International (CC BY 4.0).

Methods (English)

Practical Applications of the Hyper-Triskelion 4D Feedback Model

The Hyper-Triskelion framework introduces a 4-dimensional parametric system with time-inversion, angular–radial feedback, and stochastic gain coupling.
Its mathematical structure has several concrete, high-value applications:

---

1. Geometry-Aware AI Models

Provides synthetic 4D attractor data for training neural nets on geometric reasoning.

Enables benchmarks for chaos-aware AI, useful for next-generation xAI- or DeepMind-style simulators.

Helps develop world-model architectures with better handling of curvature, topology, and high-dimensional dynamics.

---

2. Procedural Generation in Games, VR, and AR

Generates organic, evolving vortices and recursive 4D structures for environments and effects.

Useful for dynamic topology, “living geometry,” and non-repeating fractal patterns.

Perfect for VR/AR experiences that need nonlinear motion, “breathing” structures, or alien ecosystems.

---

3. Physics and Dynamical Systems Research

Serves as a sandbox for studying:

quasi-periodicity

mode-locking

multi-frequency coupling

chaotic attractor transitions

The model is a clean, accessible playground for chaos theorists and applied mathematicians.

---

4. Robotics and Motion Planning

The 4D parametric flows act as nonlinear trajectory fields.

Can be mapped into:

smooth collision-avoiding paths

adaptive navigation in uncertain environments

Particularly useful for drones, robotic arms, and motion-in-high-dimensional-spaces.

---

5. Data Compression & Topological Encoding

The feedback architecture creates stable-but-complex manifolds.

These can be used for:

topological compression,

noise-resistant signal encoding,

“shape signatures” for classification pipelines.

---

6. Scientific Visualization & Education

Generates visuals that naturally demonstrate:

4D geometry

dynamical folding

chaos transitions

parametric motion

Excellent for teaching nonlinear systems, topology, and multi-dimensional math.

---

7. AI-Generated Art & Animation

Produces “living fractal organisms” and evolving patterns ideal for:

sci-fi visual effects

music videos

procedural art

ambient generative worlds

Works beautifully with tools like Grok Imagine, RunwayML, Blender, and TouchDesigner.

---

8. Benchmark Suite for Stability Testing

The wide range of chaotic regimes provides a stress test for:

numerical solvers

integrators

dynamical simulators

physics engines

Helps identify when systems fail under nonlinear feedback.

---

9. Biologically Inspired Modeling

Although not a biological model, the dynamics resemble:

self-regulating oscillators

breathing cycles

precessing helices

recursive structural folding

This makes it useful as a template for more formal biological or ecological simulations.

---

10. Foundation for Future Mathematical Exploration

The system naturally opens new research pathways:

4D knot theory

time-inverted dynamical flows

feedback-coupled fractal manifolds

stochastic gain modulation in high-dimensional systems

Notes (English)

Methods (English)

"""
Hyper-Triskelion v2 — Feedback-Coupled 4D Parametric Simulator
Author: Shane Edward Stone
License: CC-BY-4.0

This script simulates the Hyper-Triskelion v2 model described on Zenodo:
- 4D parametric attractor with time-inversion
- Radial feedback via t_future(t)
- Phase-split z–w coordinates
- Three C3-symmetric arms (φ = 0, 2π/3, 4π/3)

It generates:
  • 4D coordinates (x, y, z, w) for each arm
  • A simple 3D projection (x, y, z) for visualization

You can tweak the parameters in MAIN PARAMETERS below to explore
different knotting / chaos regimes.
"""

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D  # noqa: F401  (needed for 3D)

# ===============================
# MAIN PARAMETERS
# ===============================

N_T      = 2000          # number of time steps
N_THETA  = 400           # angular resolution per arm
R0       = 1.0           # base radius
TFUTURE_BASE = 2.0       # baseline future horizon
TFUTURE_AMP  = 0.5       # amplitude of t_future oscillation
EPS_AMP      = 0.1       # amplitude of epsilon(θ, t)
NOISE_EPS    = 0.01      # stochastic term on epsilon
W_SPLIT      = 2.0       # frequency split in z,w (z uses θ/2, w uses W_SPLIT*θ/2)

# ===============================
# CORE MODEL
# ===============================

def t_future(t):
    """
    Future-horizon function t_future(t).
    Here we use a simple oscillatory form:

        t_future(t) = TFUTURE_BASE + TFUTURE_AMP * sin(2π t)

    with t ∈ [0, 1].
    """
    return TFUTURE_BASE + TFUTURE_AMP * np.sin(2.0 * np.pi * t)

def delta_t(t):
    """
    Time-inversion delta:
        δ_t = t_future(t) - t
    """
    return t_future(t) - t

def epsilon_theta_t(theta, t, rng=None):
    """
    Small modulation term ε(θ, t) for radial feedback.
    Deterministic part: EPS_AMP * sin(3θ + 2π t)
    Optional stochastic noise for chaos exploration.
    """
    base = EPS_AMP * np.sin(3.0 * theta + 2.0 * np.pi * t)
    if rng is None or NOISE_EPS <= 0.0:
        return base
    noise = NOISE_EPS * rng.normal(size=theta.shape)
    return base + noise

def hyper_triskelion_arm(theta, t, phi, rng=None):
    """
    Compute one arm of the Hyper-Triskelion in 4D:

        r(θ, t)   = R0 + δ_t cos θ · (1 + ε(θ, t))
        x(θ, φ, t) = r cos(θ + φ + sin δ_t)
        y(θ, φ, t) = r sin(θ + φ + sin δ_t)
        z(θ, t)    = δ_t sin θ + cos(θ / 2)
        w(θ, t)    = δ_t sin(θ / 2) + np.cos(W_SPLIT * theta / 2)

    where φ is the arm phase: 0, 2π/3, 4π/3.
    """
    dt = delta_t(t)
    eps = epsilon_theta_t(theta, t, rng=rng)

    r = R0 + dt * np.cos(theta) * (1.0 + eps)

    phase = theta + phi + np.sin(dt)
    x = r * np.cos(phase)
    y = r * np.sin(phase)

    z = dt * np.sin(theta) + np.cos(theta / 2.0)
    w = dt * np.sin(theta / 2.0) + np.cos(W_SPLIT * theta / 2.0)

    return x, y, z, w

def simulate_hyper_triskelion(n_t=N_T, n_theta=N_THETA, seed=42):
    """
    Simulate all three arms over time.

    Returns:
        coords_4d: list of length 3 (one per arm)
                   each element is dict with keys 'x','y','z','w',
                   each array of shape (n_t, n_theta)

    The arms use phases:
        φ0 = 0
        φ1 = 2π/3
        φ2 = 4π/3
    """
    rng = np.random.default_rng(seed)

    thetas = np.linspace(0.0, 2.0 * np.pi, n_theta, endpoint=False)
    ts = np.linspace(0.0, 1.0, n_t)

    phis = [0.0, 2.0 * np.pi / 3.0, 4.0 * np.pi / 3.0]

    coords_4d = []

    for phi in phis:
        x_arr = np.zeros((n_t, n_theta))
        y_arr = np.zeros((n_t, n_theta))
        z_arr = np.zeros((n_t, n_theta))
        w_arr = np.zeros((n_t, n_theta))

        for i, t in enumerate(ts):
            x, y, z, w = hyper_triskelion_arm(thetas, t, phi, rng=rng)
            x_arr[i, :] = x
            y_arr[i, :] = y
            z_arr[i, :] = z
            w_arr[i, :] = w

        coords_4d.append({
            "x": x_arr,
            "y": y_arr,
            "z": z_arr,
            "w": w_arr,
            "phi": phi
        })

    return coords_4d, ts, thetas

# ===============================
# SIMPLE VISUALIZATION
# ===============================

def plot_snapshot(coords_4d, ts, step=None):
    """
    Plot a single time snapshot in 3D (x, y, z) for all arms.

    Args:
        coords_4d: output from simulate_hyper_triskelion
        ts:        time array
        step:      index into ts; if None uses middle of trajectory
    """
    if step is None:
        step = len(ts) // 2

    fig = plt.figure(figsize=(8, 6))
    ax = fig.add_subplot(111, projection='3d')

    colors = ["tab:orange", "tab:blue", "tab:green"]

    for arm_idx, (arm, c) in enumerate(zip(coords_4d, colors)):
        x = arm["x"][step, :]
        y = arm["y"][step, :]
        z = arm["z"][step, :]
        ax.plot(x, y, z, c=c, lw=1.0, label=f"arm {arm_idx}")

    ax.set_title(f"Hyper-Triskelion v2 snapshot at t ≈ {ts[step]:.3f}")
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_zlabel("z")
    ax.legend()
    plt.tight_layout()
    plt.show()

# ===============================
# MAIN
# ===============================

if __name__ == "__main__":
    coords_4d, ts, thetas = simulate_hyper_triskelion()
    # Show a mid-trajectory snapshot
    plot_snapshot(coords_4d, ts, step=len(ts) // 2)

Methods (English)

Hyper-Triskelion v3.0 introduces the first bifurcating, chaos-modulated version of the Stone Hyper-Triskelion geometry.

This release extends the smooth, classical v2.x model by injecting a bounded logistic-map instability directly into the time-inversion term:

  δₜ = (t_future − t) × chaotic_mod(t)

This modification produces controlled bifurcations, branching regimes, and lifelike chaotic flutter while preserving global stability.

The organism-like 4-dimensional arms now exhibit sensitivity to time-domain perturbation, enabling:

bifurcation cascades

bounded chaos

branching radial structure

quasi-organic deformation

richer 4-D evolution

visually distinct “growth modes”

The v3.0 simulation remains fully classical and deterministic, and all behavior is reproducible with the included Python code.

This release provides:

the complete v3.0 reference Python implementation

a minimal chaos-modulated δₜ engine

examples of 3-D projections

a modular codebase suitable for extension and exploration

The model is designed for research in nonlinear geometry, dynamical systems, chaotic modulation, 4-D visualization, and emergent structure.

import numpy as np
import matplotlib.pyplot as plt

def chaotic_mod(t,r=3.8,steps=10,strength=0.1,cp=4.0):
    x=0.5
    for _ in range(steps):
        x=r*x*(1-x)
        x*=cp
        x=np.clip(x,0,1)
    return 1+strength*(x-0.5)

# params
theta=np.linspace(0,2*np.pi,1200)
t_vals=np.linspace(0,1,400)
eps=0.15; mu=0.45; lam=1.2
kz=3.0; kw=2.2

X=[]; Y=[]; Z=[]; W=[]

for t in t_vals:
    tf=2+0.5*np.sin(2*np.pi*t)
    dt=(tf-t)*chaotic_mod(t)
    th=theta*dt

    r=1+eps*np.sin(3*th+mu*t)+0.05*np.cos(lam*th)
    x=r*np.cos(th)
    y=r*np.sin(th)
    z=np.sin(kz*th+0.3*t)
    w=np.cos(kw*th+0.1*t)

    X.append(x); Y.append(y); Z.append(z); W.append(w)

plt.figure(figsize=(6,6))
plt.plot(X[-1],Y[-1],'.',ms=1)
plt.axis('equal'); plt.show()

Methods (English)

Hyper-Triskelion v3.1 — Retrocausal 4-D Parametric System with Localized Warp-Field Geometry

Abstract:
Hyper-Triskelion v3.1 extends the 4-D C3-symmetric Hyper-Triskelion family by introducing a localized geometric warp field along the loop parameter theta (θ). The core system retains its retrocausal time offset

δ_t = t_future − t

combined with a bounded logistic-map chaos term that perturbs the global evolution. Version 3.1 adds a Gaussian warp bubble, a θ-dependent multiplicative curvature profile that oscillates in time and mimics a refractive-index well or synthetic spacetime patch. This produces a moving region of local expansion and contraction without violating causality, analogous to optical warp effects in metamaterials.

The resulting geometry exhibits C3 rotational symmetry, emergent knotting, bifurcations, and quasi-organic dynamics arising from retrocausal modulation, global chaos, and localized curvature shaping. The implementation includes full Python code for generating 4-D coordinates (x, y, z, w), 3-D projections, static snapshots, and animated warp-bubble sweeps. This version provides a compact, reproducible reference model for studying how feedback, chaos, and synthetic curvature interact in higher-dimensional parametric systems.

Parameter Definitions (Hyper-Triskelion v3.1)

Core time parameters

t — normalized time parameter in [0, 1], interpreted retrocausally (1 → 0).

t_future — future-horizon function controlling the retrocausal offset:
t_future = 2.0 + 0.5·sin(2πt)

δ_t — retrocausal time offset:
δ_t = t_future − t

Angular domain

θ — main loop parameter over 0 → 4π, producing a closed 3-arm structure.

θ₀ — center of the localized warp bubble (drifts with time in animations).

φ — arm phase offsets: 0, 2π/3, 4π/3 (C3 symmetry).

Geometric / radial terms

r₀ — base radius (default 1.0).

r_breath — breathing radius modulation:
r_breath = r₀·(1 + 0.3·sin(2πt))

ε(θ,t) — sinusoidal radial perturbation:
ε = 0.1·sin(3θ + 2πt)

Chaos modulation

chaotic_mod(t) — logistic-map–based global chaos factor
x ← r·x·(1−x), repeated 10 steps
chaos_factor = 1 + strength·(x − 0.5)

Controls small bounded distortions in δ_t.

Warp-bubble field (localized curvature)

warp(θ,t) — Gaussian profile centered at θ₀:
bubble = exp( −½·((θ−θ₀)/σ)² )
warp = 1 + amp·bubble·cos(ωt)

σ — warp bubble width (default 0.7).

amp — warp strength (default 0.4).

ω — oscillation frequency of the warp patch (default 1.5).

4-D coordinates
Given r, δ_t, θ, φ:

x = (r + δ_t·cosθ)·cos(θ + φ)
y = (r + δ_t·cosθ)·sin(θ + φ)
z = δ_t·sinθ + cos((k·θ)/2)
w = δ_t·sin((k·θ)/2)

where k = 3 controls vertical winding.

4-D → 3-D projection

Depth/glow scaling from |w|:
scale = 1 / (1 + 0.3·|w|)

Final projection:
x₃ = x·scale
y₃ = y·scale
z₃ = z + 2·w

# ==============================================================
#  Hyper-Triskelion v3.1 – Full Visualisation + Animation
#  (Copy-and-paste ready)
# ==============================================================

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation, PillowWriter
from matplotlib.colors import Normalize
import matplotlib.cm as cm

# ==============================================================
#  1. Core Functions (your original code – unchanged)
# ==============================================================

def chaotic_mod(t, r=3.8, steps=10, strength=0.10):
    """
    Logistic-map chaos → bounded multiplicative perturbation in [1-strength/2, 1+strength/2].
    """
    x = 0.5
    for _ in range(steps):
        x = r * x * (1.0 - x)
        x = np.clip(x, 0.0, 1.0)
    return 1.0 + strength * (x - 0.5)

def warp_profile(theta, t, theta0=0.0, sigma=0.6, amp=0.4, omega=1.5):
    """
    Localized 'warp bubble' along theta.
    Returns multiplicative factor ~ 1 + O(amp).
    """
    d = np.angle(np.exp(1j * (theta - theta0)))
    bubble = np.exp(-0.5 * (d / sigma)**2)
    return 1.0 + amp * bubble * np.cos(omega * t)

def hyper_triskelion_v31(n_points=3000, t=0.35, theta0=0.0):
    """
    Hyper-Triskelion v3.1 with optional theta0 for moving warp bubble.
    Returns: theta, x, y, z, w, x3, y3, z3
    """
    # Retrocausal time offset
    t_future = 2.0 + 0.5 * np.sin(2.0 * np.pi * t)
    dt_base = t_future - t

    # Global chaos
    chaos_factor = chaotic_mod(t, strength=0.10)

    # Parameter
    theta = np.linspace(0.0, 4.0 * np.pi, n_points)
    k_vert = 3

    # C3 symmetry via phi offset
    phi_offset = np.zeros_like(theta)
    phi_offset[theta < 4.0 * np.pi / 3.0] = 0.0
    phi_offset[(theta >= 4.0 * np.pi / 3.0) & (theta < 8.0 * np.pi / 3.0)] = 2.0 * np.pi / 3.0
    phi_offset[theta >= 8.0 * np.pi / 3.0] = 4.0 * np.pi / 3.0

    # Breathing radius
    r0 = 1.0
    r_breath = r0 * (1.0 + 0.3 * np.sin(2.0 * np.pi * t))
    eps = 0.1 * np.sin(3.0 * theta + 2.0 * np.pi * t)
    r = r_breath * np.exp(eps * np.sin(theta))

    # Warp bubble (now controlled by theta0)
    warp = warp_profile(theta, t, theta0=theta0, sigma=0.7, amp=0.4, omega=1.5)

    # Full retrocausal field
    delta_t = dt_base * chaos_factor * warp

    # 4D coordinates
    x = (r + delta_t * np.cos(theta)) * np.cos(theta + phi_offset)
    y = (r + delta_t * np.cos(theta)) * np.sin(theta + phi_offset)
    z = delta_t * np.sin(theta) + np.cos(0.5 * k_vert * theta)
    w = delta_t * np.sin(0.5 * k_vert * theta)

    # 4D → 3D projection
    scale = 1.0 / (1.0 + 0.3 * np.abs(w))
    x3 = x * scale
    y3 = y * scale
    z3 = z + 2.0 * w

    return theta, x, y, z, w, x3, y3, z3

# ==============================================================
#  2. Static Snapshot (t = 0.35)
# ==============================================================

n_points_static = 4000
theta_s, x_s, y_s, z_s, w_s, x3_s, y3_s, z3_s = hyper_triskelion_v31(
    n_points=n_points_static, t=0.35, theta0=0.0
)

plt.figure(figsize=(7, 7))
sc = plt.scatter(x3_s, y3_s, s=2, c=w_s, cmap='plasma',
                 norm=Normalize(vmin=w_s.min(), vmax=w_s.max()))
plt.colorbar(sc, label='w (4th dimension)')
plt.axis('equal')
plt.title("Hyper-Triskelion v3.1 – Static Snapshot (t = 0.35)")
plt.xlabel("x₃"); plt.ylabel("y₃")
plt.tight_layout()
plt.show()

# Diagnostics
print(f"Points: {len(x3_s)}")
print(f"x3 ∈ [{x3_s.min():.3f}, {x3_s.max():.3f}]")
print(f"y3 ∈ [{y3_s.min():.3f}, {y3_s.max():.3f}]")
print(f"w  ∈ [{w_s.min():.3f}, {w_s.max():.3f}]")

# ==============================================================
#  3. Animated Warp Bubble Sweep
# ==============================================================

n_frames = 180
fps = 36
n_points = 3500

fig, ax = plt.subplots(figsize=(7, 7))
ax.set_xlim(-2.3, 2.3)
ax.set_ylim(-2.3, 2.3)
ax.set_aspect('equal')

def update(frame):
    ax.clear()
    t = frame / n_frames
    theta0 = 2 * np.pi * t  # bubble rotates once per animation loop

    _, _, _, _, w, x3, y3, _ = hyper_triskelion_v31(
        n_points=n_points, t=t, theta0=theta0
    )

    sc = ax.scatter(x3, y3, s=1.8, c=w, cmap='turbo', alpha=0.9,
                    norm=Normalize(vmin=-1.3, vmax=1.3))
    ax.set_xlim(-2.3, 2.3)
    ax.set_ylim(-2.3, 2.3)
    ax.set_aspect('equal')
    ax.set_title(f"Hyper-Triskelion v3.1 | t = {t:.3f} | θ₀ = {theta0:.2f} rad")
    return sc,

ani = FuncAnimation(
    fig, update, frames=n_frames,
    interval=1000/fps, repeat=True
)

# Save as GIF (works everywhere)
ani.save("hyper_triskelion_warp.gif", writer=PillowWriter(fps=fps))
print("Animation saved as 'hyper_triskelion_warp.gif'")

# Uncomment below to save as MP4 (requires ffmpeg)
# ani.save("hyper_triskelion_warp.mp4", writer='ffmpeg', fps=fps)

plt.close(fig)

Authored by Shane Edward Stone. 

Simulations run by AI collaborator Kaelen.

Notes (English)

Methods (English)

Hyper-Triskelion v4.0 introduces the first warp-drive–inspired, 4-dimensional topological manifold applied to a classical triskelion structure. The model combines C₃ symmetry, retrocausal time offsets, chaotic modulation, and a vectorized Alcubierre-type warp shell to produce a closed, knotted, dynamically evolving geometric object.

At its foundation, the system defines a retrocausal time gap

  Δt(t) = t₍future₎(t) − t,

  t₍future₎(t) = 2.0 + 0.7 sin(2πt),

which drives the temporal curvature of the structure.

A smooth, normalized warp-shell factor inspired by the Alcubierre metric is applied:

  fᵣ(d) = (tanh(d+1) − tanh(d−1)) / (2 tanh(1)),

  with d = vₛ Δt(t) and vₛ > 1 producing superluminal-style deformation in the model’s geometry.

(purely mathematical; no physical FTL claim).

Each triskelion arm is generated by angles

  θ ∈ [0,2π],

  φ ∈ {0, 2π/3, 4π/3},

and a time-dependent winding phase

  χ = 3(θ + φ) + 0.8 sin(Δt).

A breathing radial profile introduces smooth oscillation and mild chaotic perturbation:

  ρ(θ,t) = R(1 + 0.3 sin(2πt)) · exp(0.08 sin(3θ + πt)).

The full 4-dimensional manifold is embedded as

X = (ρ + fᵣ d cosθ) cosχ

Y = (ρ + fᵣ d cosθ) sinχ

Z = Δt sin(3θ/2) + 0.8 fᵣ cos(θ/2)

W = fᵣ sin(θ/1.5)

together forming a C₃-symmetric warp knot with coupled temporal, radial, and warp-shell dynamics.

Hyper-Triskelion v4.0 therefore represents a novel mathematical object:

a closed 4-dimensional warp manifold where time asymmetry, chaotic modulation, and Alcubierre-inspired curvature jointly generate a stable, knotted geometric structure.

 Retrocausal time gap

The model uses a forward-shifted time horizon:

1. Δt(t) = t₍future₎(t) − t

with

  t₍future₎(t) = 2.0 + 0.7 sin(2πt)

and a warp velocity factor vₛ applied to the gap:

  d(t) = vₛ Δt(t)  (default vₛ = 1.2)

---

2. Alcubierre-type warp shell

A smooth, compact warp factor is applied:

  fᵣ(d) = (tanh(d+1) − tanh(d−1)) / (2 tanh(1))

This yields 0 < fᵣ ≤ 1, peaking inside the warp bubble.

---

3. Angular parameters & C₃ symmetry

Each arm is defined over:

  θ ∈ [0, 2π]

with phase offsets:

  φ ∈ { 0 , 2π/3 , 4π/3 }

The winding phase is:

  χ(θ,φ,t) = 3(θ + φ) + 0.8 sin(Δt)

---

4. Breathing radial profile

The time-breathing, chaos-modulated base radius is:

  ρ(θ,t) = R (1 + 0.3 sin(2πt)) · exp(0.08 sin(3θ + πt))

(Default R = 1)

---

5. 4-Dimensional embedding (X, Y, Z, W)

For each (θ, φ, t):

X = (ρ + fᵣ d cosθ) cosχ

Y = (ρ + fᵣ d cosθ) sinχ

Z = Δt sin(3θ/2) + 0.8 fᵣ cos(θ/2)

W = fᵣ sin(θ / 1.5)

Interpretation:

(X,Y) form the planar warped spiral.

Z provides vertical warp extrusion.

W represents compactified warp-shell thickness.

---

6. Full 3-arm Hyper-Triskelion manifold

The complete v4.0 manifold is:

  𝓜₍v4.0₎(t) = ⋃₍φ ∈ {0, 2π/3, 4π/3}₎ (X(θ,φ,t), Y(θ,φ,t), Z(θ,t), W(θ,t))

for uniformly sampled θ.

This produces a closed, knotted, symmetric 4-dimensional warp-drive manifold with C₃ symmetry, retrocausality, and a vectorized Alcubierre-inspired shell.

---

7. Default parameter set (as used in the simulations)

t ∈ [0,1] (normalized loop)

R = 1.0

vₛ = 1.2

fᵣ(d) as defined above

Breathing amplitude: 0.3

Chaotic modulation amplitude: 0.08

Winding coefficient: 3

Z-extrusion coefficients: 1.0 (Δt term), 0.8 (warp term)

These values match the working Python implementation used for the v4.0 warp-drive knot visualizations and can be freely varied to explore nearby manifolds.

Methods (English)

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import Normalize

# ============================================================
#  Hyper-Triskelion v4.0 — Warp-Drive Knot Manifold
# ============================================================

def hyper_triskelion_v40(n=2000, t=0.35):
    """
    v4.0 — C3 triskelion + Alcubierre-like warp shell (vectorized)
    Produces X,Y,Z,W arrays forming a 3-arm warp knot.
    """
    # Parameter sweep
    th = np.linspace(0, 2*np.pi, n)

    # Retrocausal horizon
    tf = 2.0 + 0.7*np.sin(2*np.pi*t)
    d  = (tf - t) * 1.2      # retrocausal Δt scaled

    # Vectorized Alcubierre warp shell (Grok fix)
    fr = (np.tanh(d+1) - np.tanh(d-1)) / (2*np.tanh(1))
    fr = np.clip(fr, 0, 1)  # stabilize

    # Breathing radius
    R0 = 1 + 0.3*np.sin(2*np.pi*t)
    r  = R0 * np.exp(0.08*np.sin(3*th + np.pi*t))

    # C3 arm offsets
    phis = [0, 2*np.pi/3, 4*np.pi/3]

    X_all = []; Y_all = []; Z_all = []; W_all = []

    for phi in phis:
        # Topological winding
        chi = 3*(th + phi) + 0.8*np.sin(d)

        # Radius with warp influence
        rho = (r + d*fr*np.cos(th)) * fr

        # 4D coordinates
        x = rho * np.cos(chi)
        y = rho * np.sin(chi)
        z = d*np.sin(1.5*th) + 0.8*fr*np.cos(th/2)
        w = fr*np.sin(th)

        X_all.append(x)
        Y_all.append(y)
        Z_all.append(z)
        W_all.append(w)

    # Merge arms
    X = np.concatenate(X_all)
    Y = np.concatenate(Y_all)
    Z = np.concatenate(Z_all)
    W = np.concatenate(W_all)

    return X, Y, Z, W

# ============================================================
#  Static 2D projection (XY scatter, turbo-colored by warp shell)
# ============================================================

X, Y, Z, W = hyper_triskelion_v40(n=2500, t=0.35)

plt.figure(figsize=(6,6))
sc = plt.scatter(X, Y, c=W, s=2.0, cmap='turbo',
                 norm=Normalize(vmin=W.min(), vmax=W.max()))

plt.axis('equal')
plt.title("Hyper-Triskelion v4.0 — Warp-Drive Knot")
plt.colorbar(sc, label="Warp Shell f_r (w coordinate)")
plt.show()

Methods (English)

Hyper-Triskelion v6.0 — Plasma Physics Edition

Abstract 

The Hyper-Triskelion v6.0 introduces a novel C₃-symmetric plasma-confinement geometry formulated entirely within classical electrodynamics and single-fluid plasma physics. Unlike earlier metric-based or GR-inspired versions, v6.0 relies strictly on Maxwell’s equations, E×B drift, and non-neutral plasma behavior, making it fully compatible with laboratory setups such as Penning–Malmberg traps.

The system is built from a rotating triskelion electrostatic potential
φ(x, y, t)
with three-fold angular symmetry. The arm structure is generated using the phase angle
χ = 3θ + ωt
where θ = arctan2(y, x) is the polar angle and ω is the rotation frequency. Radial modulation is applied through
rₘₒd = R₀ + A·cos(χ)
producing the characteristic trefoil-like arm geometry.

A physically realistic thin-shell confinement layer is formed using a smooth hyperbolic transition:
inside = ½·(1 − tanh[σ·(r − (R₀ − w₀/2))])
outside = ½·(1 + tanh[σ·(r − (R₀ + w₀/2))])
shell = inside·outside

Particles evolve according to the classical E×B drift law:
E = −∇φ
v = (E × B) / B²

Explicitly, for a uniform axial magnetic field B = (0, 0, B_z):
vₓ = −E_y / B_z
vᵧ =  Eₓ / B_z

These dynamics generate long-lived rotating plasma filaments, stable shear layers, and a braided C₃ vortex structure that remains numerically stable across long simulation intervals. The resulting density fields exhibit smooth, physically consistent plasma behavior with confinement and filament interaction similar to known edge-transport phenomena.

Hyper-Triskelion v6.0 therefore represents a new electrostatic confinement topology, combining C₃ rotation, thin-shell radial control, and emergent braided vortex filaments. Potential applications include E×B transport research, plasma-filament modeling, non-neutral plasma confinement, and edge-turbulence analogues.

This release includes the full Python simulation code, static snapshot renderer, and optional animation generator. All components are distributed under CC-BY-4.0 for open scientific use.

Author: Shane Edward Stone
Collaborator: Kaelen (AI Research Partner)
Version: v6.0 — Plasma Physics Edition
License: CC-BY-4.0

Coordinate System

x, y ∈ ℝ — Cartesian plasma plane

r = √(x² + y²) — radial coordinate

θ = arctan2(y, x) — polar angle

t — simulation time

ω — angular rotation frequency of the triskelion arms

---

1. Triskelion Electrostatic Potential φ(x, y, t)

The rotating C₃-symmetric potential is defined using the phase:

χ = 3θ + ωt

Radial modulation of the arm structure:

rₘₒd = R₀ + A·cos(χ)

Electrostatic potential:

φ(x, y, t) = −exp(−(r − rₘₒd)² / 0.15) · (1 + 0.5·sin(3θ)) · S(r)

where S(r) is the thin-shell confinement mask.

---

2. Thin-Shell Confinement Mask S(r)

Let:

R₀ — core radius

w₀ — shell thickness

σ — steepness parameter

Define:

inside  = ½·(1 − tanh[σ·(r − (R₀ − w₀/2))])
outside = ½·(1 + tanh[σ·(r − (R₀ + w₀/2))])

S(r) = inside · outside

This smoothly confines the plasma to a narrow annular region around R₀.

---

3. Electric Field

Electric field is computed from the potential:

E = −∇φ

In components:

Eₓ = −∂φ/∂x
Eᵧ = −∂φ/∂y

---

4. E×B Drift Velocity

For a uniform magnetic field B = (0, 0, B_z):

v = (E × B) / B²

Thus:

vₓ = −Eᵧ / B_z
vᵧ =  Eₓ / B_z

This governs the particle motion in the plasma plane.

---

5. Particle Evolution Equations

Each particle evolves by:

xₜ₊₁ = xₜ + vₓ·Δt
yₜ₊₁ = yₜ + vᵧ·Δt

where Δt is the time-step.

Particles are radially confined by reflection at r > r_max:

If r > r_max:
 x → x·(r_max / r)
 y → y·(r_max / r)

---

6. Plasma Density Field

Density is computed by binning all particle positions at frame k into a uniform grid and smoothing:

ρ(x, y) = GaussianFilter( histogram(x, y), σ_smooth )

This produces the vivid triskelion plasma density maps.

---

7. Symmetry and Structure

The Hyper-Triskelion v6.0 exhibits:

C₃ symmetry: three arms separated by 120°

rotating trefoil-like potential pattern

E×B vortex filaments

thin-shell radial confinement

braided multi-arm structure

stable vortex evolution under classical physics

No exotic matter, no GR assumptions, no metric engineering — purely Maxwell, drift kinematics, and classical plasma transport.

---

8. Parameter Summary

R₀ — base radius
A — radial modulation amplitude
ω — rotation frequency
w₀ — shell width
σ — shell steepness
B_z — axial magnetic field
Δt — integration step
n — number of particles
steps — total time steps

# ==============================================================
#  Hyper-Triskelion v6.0 — PLASMA PHYSICS EDITION
#  C3-symmetric rotating plasma vortex | Non-neutral | Confined
#  Zero GR | 100% Maxwell + single-fluid E×B drift
#  Author: Shane Edward Stone + Kaelen (AI Research Collaborator)
#  License: CC-BY-4.0
# ==============================================================

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation, PillowWriter
from scipy.ndimage import gaussian_filter  # pip install scipy

# --------------------------------------------------------------
# 1. Triskelion electrostatic potential φ(x, y, t)
# --------------------------------------------------------------
def triskelion_potential(x, y, t, omega_rot=2.0, R0=1.0, amp=0.3):
    """
    Rotating 3-armed potential well:
      r-modulated ring at radius R0 with C3 structure via cos(3θ + ωt),
      confined to a thin shell around R0.
    """
    r = np.sqrt(x**2 + y**2 + 1e-12)   # avoid divide-by-zero
    th = np.arctan2(y, x)

    # 3-arm modulation
    chi = 3.0 * th + omega_rot * t    # trefoil phase
    phi_mod = np.sin(3.0 * th)

    # radial center of the arms
    r_mod = R0 + amp * np.cos(chi)

    # deep well along the arms
    phi = -np.exp(-((r - r_mod)**2) / 0.15) * (1.0 + 0.5 * phi_mod)

    # thin shell confinement around R0 (tanh “edge”)
    w0 = 0.2
    sigma = 8.0
    inside  = 0.5 * (1.0 - np.tanh(sigma * (r - (R0 - w0/2))))
    outside = 0.5 * (1.0 + np.tanh(sigma * (r - (R0 + w0/2))))
    shell = inside * outside

    return phi * shell

# --------------------------------------------------------------
# 2. Local E×B drift from φ
# --------------------------------------------------------------
def exb_velocity(x, y, t, dx=0.04, Bz=1.0):
    """
    v = (E × B) / B^2 with B = (0,0,Bz):
      E = -∇φ, central finite difference on a 3×3 stencil.
    """
    xg = np.array([[x - dx, x, x + dx],
                   [x - dx, x, x + dx],
                   [x - dx, x, x + dx]])
    yg = np.array([[y - dx, y - dx, y - dx],
                   [y,      y,      y     ],
                   [y + dx, y + dx, y + dx]])

    phi = triskelion_potential(xg, yg, t)

    Ex = -(phi[1,2] - phi[1,0]) / (2.0*dx)
    Ey = -(phi[2,1] - phi[0,1]) / (2.0*dx)

    vx = -Ey / Bz
    vy =  Ex / Bz
    return vx, vy

# --------------------------------------------------------------
# 3. Particle tracer: plasma filaments via E×B drift
# --------------------------------------------------------------
def trace_plasma_filaments(n_particles=1800, steps=400, dt=0.018,
                           seed_radius=1.0):
    """
    Seed 3 sets of particles along triskelion arms and push them with
    E×B drift. Returns trajectories for each arm.
    """
    X_traj, Y_traj = [], []
    phi_offsets = [0.0, 2.0*np.pi/3.0, 4.0*np.pi/3.0]

    for off in phi_offsets:
        th = np.linspace(0.0, 2.0*np.pi, n_particles//3) + off
        r0 = seed_radius + 0.15*np.cos(3.0*th)
        x = r0*np.cos(th)
        y = r0*np.sin(th)

        arm_x = [x.copy()]
        arm_y = [y.copy()]

        for s in range(steps):
            t = s*dt
            for i in range(len(x)):
                vx, vy = exb_velocity(x[i], y[i], t)
                x[i] += vx*dt
                y[i] += vy*dt

                # simple radial confinement: reflect at r > 1.8
                r = np.hypot(x[i], y[i])
                if r > 1.8:
                    x[i] *= 1.8/r
                    y[i] *= 1.8/r

            arm_x.append(x.copy())
            arm_y.append(y.copy())

        X_traj.append(np.array(arm_x))
        Y_traj.append(np.array(arm_y))

    return X_traj, Y_traj

# --------------------------------------------------------------
# 4. Density field from particle positions
# --------------------------------------------------------------
def plasma_density_field(X_traj, Y_traj, frame, grid_res=256):
    """
    Histogram particle positions at a given frame into a 2D density
    field, with Gaussian smoothing for a plasma-like glow.
    """
    x_grid = np.linspace(-2.0, 2.0, grid_res)
    y_grid = np.linspace(-2.0, 2.0, grid_res)
    Xg, Yg = np.meshgrid(x_grid, y_grid)
    dens = np.zeros((grid_res, grid_res), dtype=float)

    px, py = [], []
    for ax, ay in zip(X_traj, Y_traj):
        px.extend(ax[frame])
        py.extend(ay[frame])

    px = np.asarray(px); py = np.asarray(py)

    ix = ((px + 2.0)/4.0*(grid_res-1)).astype(int)
    iy = ((py + 2.0)/4.0*(grid_res-1)).astype(int)
    ix = np.clip(ix, 0, grid_res-1)
    iy = np.clip(iy, 0, grid_res-1)

    for j in range(len(ix)):
        dens[iy[j], ix[j]] += 1.0

    dens = gaussian_filter(dens, sigma=1.8)
    return Xg, Yg, dens

# --------------------------------------------------------------
# 5. Static snapshot
# --------------------------------------------------------------
def save_static_snapshot():
    X_traj, Y_traj = trace_plasma_filaments(steps=120)
    frame = 80

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 7))

    # filaments
    for k, (ax, ay) in enumerate(zip(X_traj, Y_traj)):
        x = ax[frame]; y = ay[frame]
        col = plt.cm.plasma(k/3.0)
        ax1.scatter(x, y, c=[col]*len(x), s=3, alpha=0.9)
    ax1.set_xlim(-2, 2); ax1.set_ylim(-2, 2)
    ax1.set_aspect("equal"); ax1.axis("off")
    ax1.set_title("Plasma filaments")

    # density
    Xg, Yg, dens = plasma_density_field(X_traj, Y_traj, frame)
    im = ax2.contourf(Xg, Yg, dens, levels=50, cmap="magma")
    ax2.contour(Xg, Yg, dens, levels=10,
                colors="white", alpha=0.3, linewidths=0.5)
    ax2.set_xlim(-2, 2); ax2.set_ylim(-2, 2)
    ax2.set_aspect("equal"); ax2.axis("off")
    ax2.set_title("Plasma density")
    plt.colorbar(im, ax=ax2, label="density (arb. units)")

    plt.suptitle("Hyper-Triskelion v6.0 — Plasma Vortex Snapshot")
    plt.tight_layout()
    plt.savefig("hyper_triskelion_v6_snapshot.png",
                dpi=300, bbox_inches="tight")
    plt.close()
    print("Saved: hyper_triskelion_v6_snapshot.png")

# --------------------------------------------------------------
# 6. Full animation (optional)
# --------------------------------------------------------------
def save_plasma_animation():
    X_traj, Y_traj = trace_plasma_filaments()
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(16, 8))

    def update(frame):
        ax1.clear(); ax2.clear()
        t = frame*0.018

        # filaments
        for k, (ax, ay) in enumerate(zip(X_traj, Y_traj)):
            x = ax[frame]; y = ay[frame]
            col = plt.cm.plasma(k/3.0)
            ax1.scatter(x, y, c=[col]*len(x), s=2, alpha=0.85,
                        edgecolors="none")
        ax1.set_xlim(-2, 2); ax1.set_ylim(-2, 2)
        ax1.set_aspect("equal"); ax1.axis("off")
        ax1.set_title(f"Filaments   t={t:.2f}")

        # density
        Xg, Yg, dens = plasma_density_field(X_traj, Y_traj, frame)
        cf = ax2.contourf(Xg, Yg, dens, levels=50, cmap="magma")
        ax2.contour(Xg, Yg, dens, levels=12,
                    colors="white", alpha=0.25, linewidths=0.4)
        ax2.set_xlim(-2, 2); ax2.set_ylim(-2, 2)
        ax2.set_aspect("equal"); ax2.axis("off")
        ax2.set_title("Plasma density (smoothed)")

        if frame == 0:
            plt.colorbar(cf, ax=ax2, label="density", shrink=0.8)
        plt.tight_layout()
        return []

    ani = FuncAnimation(fig, update, frames=400,
                        interval=50, blit=False)
    ani.save("hyper_triskelion_v6_plasma.gif",
             writer=PillowWriter(fps=20))
    plt.close()
    print("Saved: hyper_triskelion_v6_plasma.gif")

# --------------------------------------------------------------
# 7. Main
# --------------------------------------------------------------
if __name__ == "__main__":
    print("Hyper-Triskelion v6.0 — Plasma Physics Edition")
    save_static_snapshot()
    # uncomment to also render the GIF (heavier):
    # save_plasma_animation()