Mathematical Constants for Creative Coding

Mathematical Foundation

Mathematical constants form the foundation of algorithmic visual art. These universal values appear repeatedly in natural patterns, geometric relationships, and visual effects. TSL provides immediate access to essential constants without manual calculation.

The fundamental mathematical constants are defined as:

$\pi = 3.14159265358979323846... = \lim_{n \to \infty} \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots$

$\tau = 2\pi = 6.28318530717958647692... \quad \text{(full circle in radians)}$

$\phi = \frac{1 + \sqrt{5}}{2} = 1.61803398874989484820... \quad \text{(golden ratio)}$

$\varepsilon = 10^{-7} = 0.0000001 \quad \text{(machine epsilon for comparisons)}$

Additional derived constants include:

• Golden angle: $\theta_g = 2\pi - \frac{2\pi}{\phi} \approx 2.39996$ radians
• Half-pi: $\frac{\pi}{2} = 1.5707963...$
• Square root of two: $\sqrt{2} = 1.41421356...$

Core Constants Library

Constant	Value	Application
PI	3.14159...	Circle geometry, wave functions
TAU	6.28318...	Full circle rotations
PHI	1.61803...	Golden ratio proportions
EPSILON	0.0000001	Floating point comparisons

Circle and Wave Mathematics

PI and TAU constants enable precise circular motion and wave calculations. TAU represents a complete circle rotation, eliminating the need for 2 * PI calculations.

ライブエディター

const fragment = () => {
      const angle = iTime.mul(TAU) // Complete rotation per time unit
      const wave = angle.mul(4).sin() // Four waves per rotation

      const radius = wave.mul(0.3).add(0.5)
      const x = angle.cos().mul(radius)
      const y = angle.sin().mul(radius)

      const color = vec3(x.add(0.5), y.add(0.5), wave.add(1).mul(0.5))
      return vec4(color, 1)

}

Golden Ratio Aesthetics

PHI and GOLDEN_RATIO constants create naturally pleasing proportions. The golden angle creates spiral patterns found in nature.

ライブエディター

const fragment = () => {
      const center = vec2(0.5)
      const pos = uv.sub(center)
      const radius = pos.length()

      const angle = pos.y.atan2(pos.x).add(iTime.mul(0.2))
      const spiral = angle.div(GOLDEN_ANGLE).fract()

      const color1 = vec3(spiral, radius.mul(2), 1)
      const color2 = vec3(1, spiral.mul(PHI.fract()), radius)

      const finalColor = color1.mix(color2, iTime.sin().mul(0.5).add(0.5))
      return vec4(finalColor, 1)

}