scale4d: Hyperdimensional Metamorphosis Engine

Constructing 4D Space-Time Geometries Through Homogeneous Coordinate Systems

The scale4d function family generates 4×4 transformation matrices that encode scaling relationships across four-dimensional homogeneous coordinate systems. These matrices represent linear transformations in projective space, enabling sophisticated geometric manipulations including perspective projections and hyperdimensional scaling.

Mathematical Foundation:

$$S_4(s) = \begin{pmatrix} s & 0 & 0 & 0 \\ 0 & s & 0 & 0 \\ 0 & 0 & s & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$

For anisotropic scaling with independent axis factors:

$$S_4(x,y,z,w) = \begin{pmatrix} x & 0 & 0 & 0 \\ 0 & y & 0 & 0 \\ 0 & 0 & z & 0 \\ 0 & 0 & 0 & w \end{pmatrix}$$

The fourth dimension serves dual purposes: in standard 3D graphics, $w=1$ maintains homogeneous coordinates for affine transformations, while $w \neq 1$ enables projective transformations and hyperdimensional scaling operations.

Relativistic Space-Time Compression

Demonstrates how scale4d transformations model relativistic effects by treating the fourth dimension as temporal, creating visual representations of Lorentz transformations and space-time curvature.

ライブエディター

const fragment = () => {
      const spacetime = uv.sub(0.5).mul(4)
      const cosmicTime = iTime.mul(0.3)

      // Relativistic scaling factors (Lorentz-like transformation)
      const velocity = cosmicTime.sin().mul(0.8)
      const gamma = float(1).sub(velocity.mul(velocity)).sqrt().reciprocal()

      // Space-time coordinates [x, y, z, t]
      const spatialZ = spacetime.length().add(cosmicTime).cos()
      const timeCoord = spacetime.x.add(spacetime.y).add(cosmicTime.mul(2)).sin()

      // Apply relativistic scaling transformation using scale4d
      const lorentzMatrix = scale4dXYZW(gamma, float(1), float(1), gamma)
      const spacetimePoint = vec4(spacetime.x, spacetime.y, spatialZ, timeCoord)

      // Transform space-time coordinates using scale4d matrix
      const transformedSpace = vec4(
              lorentzMatrix[0][0].mul(spacetimePoint.x),
              lorentzMatrix[1][1].mul(spacetimePoint.y),
              lorentzMatrix[2][2].mul(spacetimePoint.z),
              lorentzMatrix[3][3].mul(spacetimePoint.w)
      )

      // Calculate proper time and spatial intervals
      const properTime = transformedSpace.w.mul(transformedSpace.w).sub(
              transformedSpace.x.mul(transformedSpace.x).add(
              transformedSpace.y.mul(transformedSpace.y)).add(
              transformedSpace.z.mul(transformedSpace.z))).abs().sqrt()

      // Visualize curvature through metric tensor effects
      const curvature = properTime.mul(6).add(cosmicTime.mul(4)).sin()
      const distortion = smoothstep(-0.3, 0.3, curvature)

      // Geodesic field visualization
      const geodesic = transformedSpace.y.atan2(transformedSpace.x).add(
              properTime.mul(2)).sin()

      const field = distortion.mul(smoothstep(-0.5, 0.5, geodesic))

      // Color represents space-time compression effects
      const compression = gamma.div(2)
      const dilation = properTime.mul(0.5).add(0.5)

      return vec4(vec3(
              field.mul(compression),
              field.mul(dilation),
              field.mul(velocity.abs().add(0.2))
      ), 1)

}