The pictured object is very likely a gyroid—a triply periodic minimal surface—shown with a honeycomb-like perforated shell or mesh treatment. It is not usually an exact “single closed blob” from the formula alone: the equation defines an infinite repeating surface, and artists/manufacturers crop it to a ball or cube and often thicken it for fabrication. The book’s coverage includes such math-art minimal-surface work, and descriptions associated with it specifically identify Bathsheba Grossman’s Gyroid as based on triply periodic minimal surfaces. artpublikamag
Standard gyroid equation
A common implicit level-set equation is
[ \sin x\cos y+\sin y\cos z+\sin z\cos x=c. ]
- Setting (c=0) gives the classical nodal gyroid approximation.
- The geometry repeats with period (2\pi) along each coordinate direction.
- Rescale spatial frequency with (k=2\pi/a), where (a) is the desired unit-cell width:
[ \sin(kx)\cos(ky)+\sin(ky)\cos(kz)+\sin(kz)\cos(kx)=c. ]
The zero level set has mean curvature approximately zero and separates space into two interwoven, non-intersecting labyrinths. Unlike the Schwarz P or D surfaces, the gyroid has no straight lines or mirror planes, which gives it that flowing, chiral appearance.
How to obtain the pictured form
The cover object appears to combine three operations:
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Create the gyroid level surface [ f(x,y,z)=\sin(kx)\cos(ky)+\sin(ky)\cos(kz)+\sin(kz)\cos(kx). ]
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Give it thickness for a printable or manufacturable solid: [ |f(x,y,z)-c|\le t. ] Here (t) controls wall thickness; changing (c) also changes the relative volume of the two labyrinthine phases.
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Clip it to a rounded bounding volume, plausibly a sphere: [ x^2+y^2+z^2\le R^2. ] A smooth fade near the boundary can make the surface appear to dissolve naturally into the spherical outline rather than terminate abruptly.
A practical volumetric definition is therefore:
[ \left| \sin(kx)\cos(ky)+\sin(ky)\cos(kz)+\sin(kz)\cos(kx)-c \right| \le t, \qquad x^2+y^2+z^2\le R^2. ]
Use marching cubes / a signed-distance-field workflow to turn that inequality into a triangular mesh, then apply mesh offset/remeshing if needed.
The visible holes
The regular hexagon-like openings are consistent with a thickened gyroid sheet rendered or fabricated as a porous surface. They are not an independently imposed hexagonal lattice in the usual gyroid construction; they arise visually from the local topology and from viewing a finite-thickness surface.
However, the cover could also include an artistic lattice modifier—such as a wireframe/remesh, Voronoi-type perforation, or image/mesh processing—on top of the underlying gyroid. The smooth, wide ribbon-like regions make it particularly likely that the geometry is a gyroid-derived mesh that was offset, cropped, and artistically refined rather than a raw (f=0) plot.
Useful variants
The following related implicit surfaces may produce nearby aesthetics:
| Surface | Implicit equation, level (=c) | Visual character |
|---|---|---|
| Gyroid | (\sin x\cos y+\sin y\cos z+\sin z\cos x=c) | Smooth, swirling, no straight channels |
| Schwarz P | (\cos x+\cos y+\cos z=c) | More orthogonal, sponge-like pores |
| Schwarz D | (\sin x\sin y\sin z+\sin x\cos y\cos z+\cos x\sin y\cos z+\cos x\cos y\sin z=c) | More diamond-like connectivity |
| Neovius | (3(\cos x+\cos y+\cos z)+4\cos x\cos y\cos z=c) | Rounder bulb-and-window pattern |
For matching this cover, start with the gyroid at roughly 2–3 periods across a sphere, choose a modest thickness (t), and crop the result with a spherical boundary. That produces the closest underlying mathematical family. The book itself is a collection of mathematical art and explicitly discusses artworks based on surfaces that minimize area, including related sculptural forms. ppl-ai-file-upload.s3.amazonaws
