fbm: Multi-Scale Pattern Generation

Hierarchical Noise Composition Through Octave Layering

Fractional Brownian Motion constructs complex organic patterns by accumulating multiple octaves of noise functions at varying frequencies and amplitudes. Each successive layer contributes finer detail while maintaining coherent structure across different scales.

Mathematical Foundation

FBM accumulates $n$ octaves of noise function $f(x)$ with geometric progression parameters:

$$\text{FBM}(x) = \sum_{i=0}^{n-1} A_i \cdot f(F_i \cdot x)$$

Where:

• $A_i = A_0 \cdot r^i$ (amplitude progression)
• $F_i = F_0 \cdot s^i$ (frequency progression)
• $r = 0.5$ (amplitude reduction factor, persistence)
• $s = 2$ (frequency scaling factor, lacunarity)

Parameter Configuration

Parameter	Default	Description
Octaves	4	Number of noise layers
Initial Amplitude	0.5	Starting amplitude value
Amplitude Scalar	0.5	Amplitude reduction per octave
Scale Scalar	2	Frequency increase per octave

Function Variants

Function	Input Type	Purpose
fbmVec2	vec2	2D FBM using simplex noise
fbmVec3	vec3	3D FBM using simplex noise
fbmVec3Tiled	vec3, float	Tileable 3D FBM using gradient noise

Implementation

Live Editor

const fragment = () => {
      const p = uv.mul(3)
      const base = fbmVec2(p.add(iTime))
      const warped = fbmVec2(p.add(vec2(base)))
      const detail = base.add(warped)
      return vec4(detail.mul(0.5).add(0.5), base.mul(warped), detail.pow(2), 1)
}

Live Editor

const fragment = () => {
      const p = vec3(uv.mul(2), iTime.mul(0.1))
      const layers = fbmVec3(p)
      const ridged = layers.abs().mul(-1).add(1)
      const turbulence = fbmVec3(p.mul(2)).mul(0.3)
      const final = ridged.add(turbulence)
      return vec4(vec3(final), 1)
}