Simmetrized Pattern*

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Preview
Performance Cost 🟡 Medium
Space 2D UV Space
Implementation Native Compiled C++
Devices (Cycles) CPU / GPU (1:1)

Symmetrized Pattern is a cutting-edge geometric generator, capable of creating breathtakingly complex, quasi-regular structures. Based on cutting-edge research (2024) into resonant Hamiltonian functions and weak chaos theory, it visualizes smooth 2-manifold surfaces constrained by precise symmetry groups.

Unlike standard tilings which repeat a single image, this node mathematically "folds" space using Dihedral, Rotational, or Reflection symmetries to generate seamless, infinite Mandalas, webs, and quasicrystals. It includes 18 distinct mathematical models, ranging from simple trigonometric waves to multi-scale fractal compositions.

Usage & Behavior

Pattern 01
Pattern 02
Pattern 03
Pattern 04

This node is a "pattern engine." By combining a Mathematical Model (the shape source) with a Symmetry Type (the folding logic) and a Color Mode, you can generate everything from ancient religious geometry to sci-fi force fields. The Resonance Count parameter adds layers of complexity, turning simple shapes into intricate, self-weaving knots.

Basically, it constructs a periodic tiling of regular polygons (uv cells), applies canonical centering and rotation, and symmetrizes the base quasi-regular field using invariant mappings (dihedral, rotational, or reflectional). The result is a smooth, symmetric procedural pattern suitable for material texturing, decorative shaders, or analytic pattern synthesis. Colors are also fully tweakable and will create new pattern themselves. Infinite variations at your finger tips. They also can be tiled or not (see last pattern below).

Best Used For: Mandalas, sci-fi energy fields, intricate wallpaper, quasicrystals, Persian rugs, and complex motion graphics backgrounds.

Key Features

  • 18 Math Models: Cubic, Hyperbolic, Fractal, Ripple, Twisted, and more.
  • 3 Symmetry Groups: Dihedral (Rotation+Reflection), Rotational, and Reflection.
  • Advanced Coloring: Built-in palettes with Discrete, Smooth, Contour, and Edge modes.

Deep Dive: The Math of Symmetry

The Symmetry parameter determines how the mathematical domain is "folded" before the pattern is drawn. It transforms the underlying chaotic function into structured geometry.

1. Dihedral (Kaleidoscope)

Rotation + Reflection.
This is the most complex symmetry ($D_n$). It mimics a kaleidoscope: the image is mirrored and rotated. It creates perfectly balanced, static structures like snowflakes, starfishes, or Gothic rose windows. If you fold it in half, the sides always match.

2. Rotational (Pinwheel)

Rotation Only.
This symmetry ($C_n$) repeats the pattern around the center without mirroring it. This preserves "handedness" or direction, creating patterns that look like they are spinning, such as turbines, galaxies, or sawblades.

3. Reflection (Mirror)

Mirror Axis Only.
A simple reflection across the axes. This creates bilateral symmetry, similar to a Rorschach test, a butterfly, or a human face. It is less "circular" than the others and more architectural.

Cheat Sheet: Controlling the Color Modes

Each Color Mode changes how the pattern is rendered and activates a specific control slider:

  • 1. DISCRETE (Cel-Shaded)
    Creates flat, stepped bands of color.
    Control: Contours Threshold
  • 2. CONTOURS (Topographic)
    Draws thin isolines (like a map).
    Control: Contours Threshold
  • 3. CYCLIC (Psychedelic)
    Repeats the palette over and over.
    Control: Cyclic Threshold
  • 4. EDGES (Wireframe)
    Detects boundaries between regions.
    Control: Edge Threshold

Pro Tip: The N-Fold parameter controls the density of these operations. For example, Dihedral + 6-Fold creates exactly 12 segments (6 rotations × 2 mirrors).

Parameters

Coords Vector
The input texture coordinates.
Model Enum
The underlying mathematical equation used to generate the field:
  • BASIC / CUBIC / POWER: Standard trigonometric bases.
  • SINCOS / PRODUCT / TAN COMP: Composite functions for richer detail.
  • HYPERBOLIC / EXP DECAY: Bounded, smooth transitions.
  • FRACTAL / RIPPLE / SPIRAL: High-complexity, organic forms.
  • CELLULAR / TWISTED: Distance-field and non-linear distorted patterns.
Symmetry Enum
How the domain is folded:
  • DIHEDRAL: Full kaleidescope (Rotation + Mirroring). Perfect for Mandalas.
  • ROTATIONAL: Rotation only (Swirls/Pinwheels).
  • REFLECTION: Mirror axes only.
N-Fold Enum
The order of symmetry (3, 4, 6, 8, 12). E.g., "6 Folds" creates hexagonal symmetry (snowflakes).
Resonance Count (default: 2.0)
Controls the harmonic complexity. Higher values add more overlapping wave layers, creating intricate interference patterns.
Pattern Scale (default: 2.0)
Scales the internal math function relative to the symmetry frame.
Use Tiling (bool, default: true)
When enabled, the pattern repeats infinitely across the UV space. When disabled, it generates a single centered rosette.

Colorization

Color Palette Enum
Selects a pre-configured color scheme (Warm, Cool, Neon, Earth, etc.).
Color Mode Enum
Determines how the math values are mapped to the palette:
  • DISCRETE: Stepped, flat regions (Cel-shaded look).
  • SMOOTH: Continuous gradient interpolation.
  • CONTOURS: Distinct banded isolines.
  • EDGES: Draws sharp lines at boundary transitions (Technical drawing look).
  • CYCLIC: Wraps the palette repeatedly.
Value Min / Max (Manual Range)
Defines the expected range of the mathematical output to normalize it into 0-1 for coloring.
Contours / Cyclic Threshold
Controls the density of bands (for Contours mode) or repetitions (for Cyclic mode).
Edge Threshold (default: 0.15)
Controls the thickness of the black lines in Edge color mode.

Quick Recipes

Start with these setups.

Sacred Geometry

  • Model BASIC
  • Symmetry DIHEDRAL
  • N-Fold 12
  • Color Mode EDGES
Classic 12-fold rosette with sharp outline strokes.

Alien Energy

  • Model TWISTED
  • Symmetry ROTATIONAL
  • Palette NEON
  • Color Mode CYCLIC
Twisted model combined with cyclic neon colors creates a glowing, pulsing vortex.

Quasicrystal Floor

  • Model CUBIC
  • N-Fold 6
  • Use Tiling True
  • Resonance 4.0
High resonance tiling creates aperiodic, interlocking floor tiles.
Technical Note: This shader implements "Symmetrization of Quasi-Regular Patterns with Periodic Tilting" (2024). It evaluates a Hamiltonian energy function H(q, p) across the 2D plane, then applies symmetry operators to the resulting phase space to generate the final tiling.
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