icosahedronSDF: Regular Icosahedron Distance Field

Twenty-Faced Platonic Solid with Golden Ratio Geometry

The icosahedronSDF function generates a signed distance field for a regular icosahedron, one of the five Platonic solids. This polyhedron features 20 equilateral triangular faces and utilizes the golden ratio in its geometric construction.

Mathematical Foundation

The icosahedron distance field leverages the golden ratio φ ≈ 1.618 and its relationship:

$$\phi + 1 = 2.618...$$

The distance calculation involves projecting the point onto characteristic planes:

$$d_{\text{icosa}} = \max\left(\max(a, b, c) - \phi, d\right) \cdot \text{radius}$$

where:

• $a, b, c$ represent dot products with rotated normal vectors
• $d$ represents the distance to the circumscribed sphere
• $\vec{n_1} = \text{normalize}(\phi + 1, 1, 0)$ defines the primary orientation
• $\vec{n_2} = \frac{1}{\sqrt{3}}$ represents the uniform sphere distance

Function Signature

Parameter	Type	Description
p	vec3	Sample point position
radius	float	Icosahedron circumradius

Implementation Demonstrations

Live Editor

const fragment = () => {
      const up = vec3(0, 1, 0)
      const eps = vec3(0.01, 0, 0)
      const eye = rotate3dY(iTime).mul(vec3(5))
      const args = [1.2]
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = icosahedronSDF(p, ...args).toVar()
              Loop(16, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = icosahedronSDF(p.add(eps.xyy), ...args).sub(d)
                              const dy = icosahedronSDF(p.add(eps.yxy), ...args).sub(d)
                              const dz = icosahedronSDF(p.add(eps.yyx), ...args).sub(d)
                              return vec4(vec3(dx, dy, dz).normalize().mul(0.5).add(0.5), 1)
                      })
                      p.addAssign(d.mul(dir))
                      d.assign(icosahedronSDF(p, ...args))
              })
              return vec4(0)
      })
      const z = eye.negate().normalize()
      const x = z.cross(up)
      const y = x.cross(z)
      const scr = vec3(uv.sub(0.5), 2)
      const dir = mat3(x, y, z).mul(scr).normalize()
      return march(eye, dir)
}