Substitution

This is a geometric construction of the tiling.

These are the recursive substitution rules for a triangle define by ABC:

• Let φ be the golden ratio: (1 + √5) / 2
• Rule 0
  • Let P be a point on AB: P = A + (B - A) / φ
  • Add the triangles: CPB and PCA

• Rule 1
  • Let R be a point point on BC: R = B + (C - B) / φ
  • Add the triangle: RCA and BAR
  • Further apply Rule 0 to BAR

function subdivideStep (triangles) {
  return triangles.flatMap(function (triangle) {
    const { color, A, B, C } = triangle
    if (color === 0) {
      // P = A + (B - A) / goldenRatio;
      const P = goldenpoint(A, B)
      return [
        { color: 0, A: C, B: P, C: B },
        { color: 1, A: P, B: C, C: A }
      ]
    } else {
      // R = B + (C - B) / goldenRatio;
      const R = goldenpoint(B, C)
      return [
        { color: 1, A: R, B: C, C: A },
        // this triangle is further subdivided
        { color: 0, A: B, B: A, C: R }
      ]
    }
  })
}