quatIdentity: Quaternion Identity Element

Mathematical Foundation of Rotational Null State

The quatIdentity function returns the quaternion identity element $(0, 0, 0, 1)$, representing the null rotation state in 4D space. This element serves as the multiplicative identity for quaternion operations, providing the mathematical foundation for all rotational transformations.

Mathematical Definition: The identity quaternion is defined as:

$$q_{\text{identity}} = (0, 0, 0, 1) = 0\mathbf{i} + 0\mathbf{j} + 0\mathbf{k} + 1$$

Properties:

• Multiplicative identity: $q \cdot q_{\text{identity}} = q$ for any quaternion $q$
• Represents zero rotation in 3D space
• Provides reference coordinate system for all transformations
• Serves as the neutral element in quaternion algebra

This fundamental constant enables initialization of rotational systems and serves as the baseline for quaternion interpolation algorithms.

Quantum Vacuum State Visualization

This demonstration explores the quaternion identity as a quantum vacuum state, where minute perturbations create observable fluctuations around the null rotation baseline. The identity element generates crystalline symmetry patterns that reveal the mathematical structure of rotational neutrality.

Live Editor

const fragment = () => {
      const cellPos = uv.mul(5)
      const cell = cellPos.floor()
      const local = cellPos.fract().sub(0.5)

      const identity = quatIdentity()
      const hash = cell.x.mul(73.1).add(cell.y.mul(157.3)).sin().mul(8134.7).fract()

      const rotationalEnergy = hash.mul(iTime.mul(0.3)).sin().abs()
      const disturbance = vec4(
              local.x.mul(rotationalEnergy),
              local.y.mul(rotationalEnergy),
              local.length().mul(rotationalEnergy).mul(0.5),
              identity.w
      ).normalize()

      const identityDifference = quatAdd(disturbance, quatConj(identity))
      const nullDeviation = identityDifference.xyz.length()

      const crystallineField = nullDeviation.mul(15).sin().abs()
      const vacuumDensity = crystallineField.mul(identity.w.add(nullDeviation))

      const neutralState = vec3(1, 1, 1).mul(vacuumDensity.mul(0.8))
      const perturbationColor = vec3(0.2, 0.5, 1.0).mul(crystallineField.mul(0.6))

      return vec4(neutralState.add(perturbationColor), 1)

}

The quatIdentity function provides the essential null rotation reference, enabling quantum vacuum state visualizations and serving as the mathematical foundation for all quaternion-based rotational transformations.