scale3d: Hyperbolic Metamorphosis Engine

Constructing Dimensional Reality Through Multiplicative Topology

The scale3d function generates a 3×3 transformation matrix that encodes scale relationships across three-dimensional space. This matrix represents a linear transformation that preserves the origin while multiplying coordinate values by specified scaling factors.

Mathematical Definition:

$$S(\mathbf{s}) = \begin{pmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & s_z \end{pmatrix}$$

For uniform scaling with factor $s$:

$$S(s) = s \cdot I_3 = \begin{pmatrix} s & 0 & 0 \\ 0 & s & 0 \\ 0 & 0 & s \end{pmatrix}$$

Where $I_3$ represents the 3×3 identity matrix. The transformation applies as: $\mathbf{p'} = S(\mathbf{s}) \cdot \mathbf{p}$, where $\mathbf{p}$ is the original position vector.

Volumetric Resonance Chambers

Shows how scale3d transformations affect wave frequencies in 3D space, demonstrating the relationship between geometric scaling and wave propagation characteristics.

Live Editor

const fragment = () => {
      const p = uv.mul(5).sub(2.5)
      const time = iTime.mul(0.5)

      // Scale factor affects wave frequencies
      const waveScale = time.mul(0.8).sin().mul(0.4).add(1.1)
      const scaleMatrix = scale3d(waveScale)

      // 3D wave coordinates
      const z = p.length().mul(2).add(time).cos()
      const wavePoint = vec3(p, z)

      // Apply scaling to wave coordinates
      const scaledWave = vec3(
              scaleMatrix[0][0].mul(wavePoint.x),
              scaleMatrix[1][1].mul(wavePoint.y),
              scaleMatrix[2][2].mul(wavePoint.z)
      )

      // Multiple wave frequencies affected by scaling
      const wave1 = scaledWave.x.mul(4).add(time.mul(2)).sin()
      const wave2 = scaledWave.y.mul(3).add(time.mul(1.5)).cos()
      const wave3 = scaledWave.z.mul(5).add(time.mul(2.5)).sin()

      // Wave interference patterns
      const interference = wave1.mul(wave2).add(wave2.mul(wave3)).add(wave1.mul(wave3))
      const resonance = smoothstep(-0.6, 0.6, interference)

      // Visualize scaling effect on wave amplitude
      const amplitude = scaledWave.x.abs().add(scaledWave.y.abs()).add(scaledWave.z.abs()).div(6)
      const frequency = waveScale.mul(0.5)

      const color = vec3(
              resonance.mul(amplitude.add(0.3)),
              resonance.mul(frequency),
              resonance.mul(float(2).sub(amplitude).mul(0.7))
      )

      return vec4(color, 1)

}