dodecahedronSDF: Golden Ratio Dodecahedral Distance Field

Platonic Solid Geometry with Divine Proportion Mathematics

The dodecahedronSDF function generates signed distance fields for regular dodecahedra, one of the five Platonic solids. This geometry leverages the golden ratio (φ ≈ 1.618) to construct the characteristic twelve-sided polyhedron with pentagonal faces.

Mathematical Foundation

The dodecahedron distance calculation relies on the golden ratio-based normal vector:

$$\vec{n} = \frac{(\phi, 1, 0)}{|\vec{(\phi, 1, 0)}|}$$

where $\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618033988749895$ represents the golden ratio.

The distance function evaluates three symmetry planes:

$$d = \max(\max(a, b), c) - n_x$$

where:

• $a = |\vec{p}| \cdot (n_x, n_y, n_z)$
• $b = |\vec{p}| \cdot (n_z, n_x, n_y)$
• $c = |\vec{p}| \cdot (n_y, n_z, n_x)$

Function Variants

Function	Parameters	Description
dodecahedronSDF	p	Unit dodecahedron centered at origin
dodecahedronSDFRadius	p, radius	Scaled dodecahedron with specified radius

Implementation Demonstrations

ライブエディター

const fragment = () => {
      const up = vec3(0, 1, 0)
      const eps = vec3(0.01, 0, 0)
      const eye = rotate3dY(iTime).mul(vec3(5))
      const args = [1.5]
      const march = Fn(([eye, dir]: [Vec3, Vec3]) => {
              const p = eye.toVar()
              const d = dodecahedronSDFRadius(p, ...args).toVar()
              Loop(16, ({ i }) => {
                      If(d.lessThanEqual(eps.x), () => {
                              const dx = dodecahedronSDFRadius(p.add(eps.xyy), ...args).sub(d)
                              const dy = dodecahedronSDFRadius(p.add(eps.yxy), ...args).sub(d)
                              const dz = dodecahedronSDFRadius(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(dodecahedronSDFRadius(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)
}