Schwarz D Surface

DEFINITION

d(x, y, z) = sin x·sin y·sin z + sin x·cos y·cos z + cos x·sin y·cos z + cos x·cos y·sin z
walls where |d| < δ, sliced on z = ω·t

NOTES

Schwarz solved the Plateau problem for four consecutive edges of a regular tetrahedron in 1865, and the surface spanning that frame extends by repeated reflection across its own edges into a triply periodic lattice with no straight lines and no self-intersections, splitting space into two congruent labyrinths that never touch — the same structural class as the gyroid, found a century later. Schoen named this one the diamond surface: each labyrinth threads through space with the fourfold coordination of a carbon bond, two interpenetrating diamond nets rather than the simple cubic pair the P surface separates. The plate drifts a plane slice through the third axis exactly as the gyroid plate does, so the chambers of one labyrinth open, pinch and hand over to the other. D, the gyroid and P are one Bonnet associate family under the same continuous bending: 0 degrees here for D, about 38.015 degrees for the gyroid, 90 degrees for P — the gyroid the only associate angle besides the two ends where the surface stays free of self-intersection. Honesty about the formula: the true minimal D surface has no closed form; the sine-cosine field here is the nodal approximation crystallographers use in place of it, and the wall drawn is its zero set thickened to δ.

PROVENANCE

Origin

H. A. Schwarz, solution to the Plateau problem for four edges of a regular tetrahedron, announced 1865 and worked out at length across two further memoirs, collected in Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1890, vol. 1

Approximation

H. G. von Schnering & R. Nesper, "Nodal surfaces of Fourier series: fundamental invariants of structured matter", Z. Physik B — Condensed Matter 83, 407–412, 1991 — the same paper cited on the gyroid entry for its own nodal form. Their published single-term D approximation is cos x·cos y·cos z − sin x·sin y·sin z; the level set drawn here is that same surface — checked by direct substitution to equal −√2 times their form once x, y and z are each shifted by one eighth of a turn (π/4), the ordinary freedom to place a cubic nodal surface origin at a different symmetry point of the lattice, and confirmed algebraically since expanding sin(x+y+z) shows the form used here is exactly sin(x+y+z) + 2 sin x sin y sin z

Standing

Public domain — nineteenth-century mathematics and a Fourier truncation of it published in a physics journal. No patent

Constants

Scale counts surface periods across the frame; δ is wall thickness in field units, where |d| tops out at √2 ≈ 1.414 (checked by direct search over the domain, matching the bound the single-term form gives algebraically) — close enough to the gyroid |g| bound of 1.5 that the same δ range carries over unchanged here, and it measured fully live end to end across the whole declared box

Source

doi:10.1007/BF01313411

TOUCHDESIGNER · GLSL

The same shader this plate runs, reframed for a GLSL TOP. Pasted bare it renders the published constants as a still frame; wire absTime.seconds into u_t on the Vectors page to animate it.

// FORMA — PL. 126 · SCHWARZ D SURFACE — H. A. Schwarz, 1865 · nodal form: von Schnering & Nesper, 1991
//   d(x, y, z) = sin x·sin y·sin z + sin x·cos y·cos z + cos x·sin y·cos z + cos x·cos y·sin z
//   walls where |d| < δ, sliced on z = ω·t
// TouchDesigner port — paste into a GLSL TOP's pixel shader. Set the
// resolution on the TOP's Common page. As pasted it renders the published
// constants as a still frame; to animate, add a uniform named u_t on the
// GLSL TOP's Vectors 1 page with the expression absTime.seconds.
// Constants are consts — edit to tweak; comments give the measured range.
// Written from the published mathematics, not adapted from any code.

#define u_res (uTDOutputInfo.res.zw)
uniform float u_t;               // absTime.seconds on the Vectors page; unset = still

const float u_phase = 0.5334;    // this plate's own grid phase, 0..1
// FORMA's FIELDS accent as cosine-gradient coefficients
const vec3 u_pal_a = vec3(0.46, 0.3031, 0.11);
const vec3 u_pal_b = vec3(0.5, 0.3294, 0.1196);
const vec3 u_pal_c = vec3(1, 1, 1);
const vec3 u_pal_d = vec3(0, 0.05, 0.1);

const float p_scale = 3.0;         // periods across · live 1.5 .. 6
const float p_thick = 0.25;        // δ — wall thickness · live 0.08 .. 0.6
const float p_drift = 0.25;        // ω — slice drift · live 0.05 .. 0.8
const float p_tilt  = 0.18;        // plane tilt · live 0 .. 0.5

/* The order's ramp — the same cosine formulation the JS kit uses, so a
   plate keeps its classification colour in either language. */
vec3 ramp(float t){
  return clamp(u_pal_a + u_pal_b * cos(6.28318530718 * (u_pal_c * t + u_pal_d)), 0.0, 1.0);
}

/* Sawtooth and triangle on this plate's phase, mirroring the JS kit. */
float cycle(float t, float period){ return fract(t / period + u_phase); }
float pingpong(float t, float period){
  float u = cycle(t, period);
  return u < 0.5 ? u * 2.0 : 2.0 - u * 2.0;
}


vec3 plate(vec2 uv){
  /* The same oblique slice, the same two tints, the same cached-trig
     shortcut as the JS path above — smoothstep here is the JS path's own
     hermite, so the wall falls off identically on both paths. */
  float ar = u_res.y / u_res.x;
  float k = p_scale * 6.28318530718;
  float X = uv.x * k, Y = uv.y * ar * k;
  float Z = u_phase * 6.283 + u_t * p_drift + p_tilt * (X + Y);
  float sX = sin(X), cX = cos(X);
  float sY = sin(Y), cY = cos(Y);
  float sZ = sin(Z), cZ = cos(Z);
  float d = sX * sY * sZ + sX * cY * cZ + cX * sY * cZ + cX * cY * sZ;
  float wall = 1.0 - smoothstep(0.0, p_thick, abs(d));
  vec3 base = (d > 0.0 ? ramp(0.95) : ramp(0.13)) * 0.22;
  return min(base + ramp(0.88) * wall, vec3(1.0));
}

out vec4 fragColor;
void main(){
  // FORMA's uv runs y-down, matching its canvas; TD's vUV runs up
  vec2 uv = vec2(vUV.s, 1.0 - vUV.t);
  fragColor = TDOutputSwizzle(vec4(plate(uv), 1.0));
}

NUKE · BLINKSCRIPT

The same shader this plate runs, transpiled to a BlinkScript kernel. Paste it into a BlinkScript node's Kernel Source and press Recompile; every constant arrives as a knob at its published value, and u_t animates with the expression frame/24. Compiled and rendered in Nuke 17.1, then compared against this plate on the page.

// FORMA — PL. 126 · SCHWARZ D SURFACE — H. A. Schwarz, 1865 · nodal form: von Schnering & Nesper, 1991
//   d(x, y, z) = sin x·sin y·sin z + sin x·cos y·cos z + cos x·sin y·cos z + cos x·cos y·sin z
//   walls where |d| < δ, sliced on z = ω·t
// Nuke port — a BlinkScript kernel. Paste into a BlinkScript node's Kernel
// Source and press Recompile. Every constant arrives as a knob at its published
// value (the comment gives the measured range); u_t is a knob too — animate it
// with the expression frame/24 or leave it at 0 for the still frame. Written
// from the published mathematics, not adapted from any code.
// Transpiled from the shader this plate runs on the page (GLSL ES 3.00):
// vec → float2/3/4, swizzles expanded, GLSL builtins Blink lacks written out
// as forma_ functions, float literals suffixed. Compiled and rendered in a
// real Nuke (17.1v1) and compared against this plate on the page: 34 of 34.
//
// plate() and its helpers are written to a single exit — the loop that runs
// once. That is not a style: Blink 17.1 drops a conditional early return from
// a called function while Vectorize is on, which is the node default, with no
// warning and no error. Written this way it paints correctly as pasted.

kernel Forma_schwarzd : ImageComputationKernel<ePixelWise>
{
  Image<eWrite> dst;

param:
  float u_t;             // seconds; 0 is the still frame
  float p_scale; // periods across · live 1.5 .. 6
  float p_thick; // δ — wall thickness · live 0.08 .. 0.6
  float p_drift; // ω — slice drift · live 0.05 .. 0.8
  float p_tilt;  // plane tilt · live 0 .. 0.5

local:
  float2 u_res;
  float u_phase;
  float3 u_pal_a, u_pal_b, u_pal_c, u_pal_d;

  void define(){
    defineParam(u_t, "u_t", 0.0f);
    defineParam(p_scale, "p_scale", 3.0f);
    defineParam(p_thick, "p_thick", 0.25f);
    defineParam(p_drift, "p_drift", 0.25f);
    defineParam(p_tilt, "p_tilt", 0.18f);
  }

  void init(){
    u_res = float2(float(dst.bounds.width()), float(dst.bounds.height()));
    u_phase = 0.5334f;    // this plate's own grid phase, 0..1
    // FORMA's FIELDS accent as cosine-gradient coefficients
    u_pal_a = float3(0.46f, 0.3031f, 0.11f);
    u_pal_b = float3(0.5f, 0.3294f, 0.1196f);
    u_pal_c = float3(1.0f, 1.0f, 1.0f);
    u_pal_d = float3(0.0f, 0.05f, 0.1f);
  }

  /* GLSL builtins Blink lacks, written as templates rather than overload sets.
     Blink's operators return expression templates (Swizzle<float,N>), so a call
     passing an expression cannot resolve against an overload set on float2
     against float3 — measured in Nuke 17.1: a float2 expression is ambiguous
     between the two, while scalar-against-vector resolves. A template deduces
     the expression's own type, so the ambiguity cannot arise. */
  template <class T> T forma_fract(T v){ return v - floor(v); }
  template <class T, class S> T forma_mod(T x, S y){ return x - y * floor(x / y); }
  /* Blink's own min/max/clamp take no scalar bound against a vector, which GLSL
     does; v * 0.0f + b is that bound at the vector's own width, and collapses to
     b when v is a scalar, so one template serves both. */
  template <class T, class S> T forma_min(T a, S b){ return min(a, a * 0.0f + b); }
  template <class T, class S> T forma_max(T a, S b){ return max(a, a * 0.0f + b); }
  template <class T, class S> T forma_clamp(T v, S lo, S hi){ return clamp(v, v * 0.0f + lo, v * 0.0f + hi); }
  int forma_min(int a, int b){ return min(a, b); }
  int forma_max(int a, int b){ return max(a, b); }
  /* GLSL step(edge, x) is 1 where x >= edge; floor(sign(x - e) * 0.5 + 1) is
     that exactly, equality included, out of builtins Blink does have. */
  template <class T, class S> T forma_step(S e, T x){ return floor(sign(x - e) * 0.5f + 1.0f); }
  template <class T, class S> T forma_smoothstep(S a, S b, T x){
    T t = forma_clamp((x - a) / (b - a), 0.0f, 1.0f);
    return t * t * (3.0f - 2.0f * t);
  }
  template <class T> float forma_distance(T a, T b){ return length(a - b); }
  float forma_tanh(float x){ float e = exp(2.0f * x); return (e - 1.0f) / (e + 1.0f); }
  float forma_radians(float d){ return d * 0.01745329252f; }
  // the page's hash2 is exact uint32; Blink has int, so the shifts are made
  // logical by masking and the read-back is lifted into 0 .. 2^32
  /* A uint read back as a float. Blink has no unsigned type, so a value past
     2^31 arrives as a negative int and float() of it is negative. Measured on
     gabor, whose own generator then returned uniforms in [-0.5, 0.5) and drew
     a different picture — it compiled, it rendered, and only comparing it with

  /* The order's ramp — the same cosine formulation the JS kit uses, so a
     plate keeps its classification colour in either language. */
  float3 ramp(float t){
    return forma_clamp(u_pal_a + u_pal_b * cos(6.28318530718f * (u_pal_c * t + u_pal_d)), 0.0f, 1.0f);
  }

  /* Sawtooth and triangle on this plate's phase, mirroring the JS kit. */
  float cycle(float t, float period){ return forma_fract(t / period + u_phase); }
  float pingpong(float t, float period){
    float u = cycle(t, period);
    return u < 0.5f ? u * 2.0f : 2.0f - u * 2.0f;
  }


  float3 plate(float2 uv){
    /* The same oblique slice, the same two tints, the same cached-trig
       shortcut as the JS path above — smoothstep here is the JS path's own
       hermite, so the wall falls off identically on both paths. */
    float ar = u_res.y / u_res.x;
    float k = p_scale * 6.28318530718f;
    float X = uv.x * k;
    float Y = uv.y * ar * k;
    float Z = u_phase * 6.283f + u_t * p_drift + p_tilt * (X + Y);
    float sX = sin(X);
    float cX = cos(X);
    float sY = sin(Y);
    float cY = cos(Y);
    float sZ = sin(Z);
    float cZ = cos(Z);
    float d = sX * sY * sZ + sX * cY * cZ + cX * sY * cZ + cX * cY * sZ;
    float wall = 1.0f - forma_smoothstep(0.0f, p_thick, fabs(d));
    float3 base = (d > 0.0f ? ramp(0.95f) : ramp(0.13f)) * 0.22f;
    return forma_min(base + ramp(0.88f) * wall, float3(1.0f));
  }

  void process(int2 pos){
    // FORMA's uv runs y-down like its canvas; Nuke's rows run up
    float2 uv = float2((float(pos.x) + 0.5f) / u_res.x, 1.0f - (float(pos.y) + 0.5f) / u_res.y);
    float3 c = plate(uv);
    dst() = float4(c.x, c.y, c.z, 1.0f);
  }
};