Technique · Fields

Lattice shell

A translucent crystal made from a gyroid sheet cut by an octahedron, tinted red to violet, with contour lines and a slow turn.

Lattice shell is a field drawn for every pixel by the graphics chip. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.

Good to know

  • A minimal surface has zero mean curvature everywhere: a soap film spanning a wire frame takes this shape because it makes the least area for the boundary it has.
  • Gyroid structures have been used in 3D printing as an infill, because the surface is strong, uses little material and has no flat overhangs that need support.
  • The Brill models of plaster and string, made in Germany from the 1870s, let students hold algebraic surfaces in their hands.
  • The gyroid was found in 1970 by the NASA scientist Alan Schoen: a surface that repeats in three directions, divides space into two interlocking labyrinths, and contains no straight line.
  • The wing scales of the green hairstreak butterfly are built as a gyroid, a structure on the scale of light itself that makes their green colour without any pigment.

The mathematics

  • The sheet is the set where g = sin x cos y + sin y cos z + sin z cos x is zero, a close approximation to Schoen's gyroid; the picture shows the part where |g| is below 0.28, a thickened skin, repeating every 2 pi / 7.50 = 0.84 units in each direction.
  • The surface normal is the gradient of g, (cos x cos y - sin z sin x, cos y cos z - sin x sin y, cos z cos x - sin y sin z), so the shading needs no mesh.
  • The octahedron |x| + |y| + |z| <= R is the intersection of eight half-spaces, so a ray enters and leaves it where it crosses the planes: the slab method, a few comparisons per pixel.

In art

  • Contour lines drawn on a surface, as they are here, are a way of showing a form by its slices, the same trick as the bands on a topographic map.
  • Architects and designers now borrow gyroids for lightweight facades and furniture, where the same surface gives strength, airflow and a pattern with no seams.

More ways to use it

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