modi: Discrete Periodicity Mathematics

Integer Domain Periodicity

The modi function creates discrete periodic structures by constraining integer values within specified boundaries. This operation establishes mathematical periodicity through remainder calculation, enabling precise control over discrete pattern generation.

Unlike continuous modulo operations, modi maintains integer precision throughout the calculation process, making it essential for algorithmic pattern creation and discrete mathematical visualizations.

ライブエディター

const fragment = () => {
      const uv = position.xy.div(iResolution).sub(0.5).mul(20)

      // Create discrete cellular automata-like patterns
      const cellX = uv.x.toInt()
      const cellY = uv.y.toInt()

      const pattern = modi(cellX.mul(cellY), int(7))
      const intensity = pattern.toFloat().div(6)

      const color = vec3(
              intensity.mul(pattern.toFloat().sin()),
              intensity.mul(pattern.toFloat().mul(2).cos()),
              intensity.mul(pattern.toFloat().mul(3).sin())
      )

      return vec4(color, 1)

}

Prime Number Visualization

Mathematical prime number theory visualization through modulo arithmetic creates complex interference patterns that reveal number-theoretic structures invisible to continuous mathematics.

ライブエディター

const fragment = () => {
      const uv = position.xy.div(iResolution).sub(0.5).mul(12)

      // Prime modulo visualization
      const x = uv.x.mul(3).toInt()
      const y = uv.y.mul(3).toInt()

      const mod3 = modi(x, int(3))
      const mod5 = modi(y, int(5))

      const r = mod3.toFloat().div(2)
      const g = mod5.toFloat().div(4)
      const b = r.mul(g)

      return vec4(r, g, b, 1)

}

Mathematical Foundation

Discrete Modulo Arithmetic

The modi function implements integer remainder calculation maintaining discrete mathematical properties:

$$\text{modi}(x, y) = x - y \cdot \lfloor\frac{x}{y}\rfloor$$

This ensures the result $r$ satisfies: $0 \leq r < |y|$ for positive divisors.

For integer domain operations, the mathematical relationship can be expressed as: $\text{modi}(x, y) \in \{0, 1, 2, \ldots, y-1\}$

Periodicity Properties

The modulo operation creates periodic boundary conditions where adding multiples of the divisor maintains the same remainder value. This periodicity enables construction of discrete tiling patterns and algorithmic visualizations.

Computational Architecture

Phase	Mathematical Operation	Implementation Logic
Division	Integer division with floor	x.div(y).floor()
Multiplication	Quotient scaling	quotient.mul(y)
Subtraction	Remainder extraction	x.sub(scaled_quotient)

The algorithm maintains integer precision throughout the computation, ensuring discrete mathematical correctness essential for algorithmic pattern generation and number-theoretic visualizations.