Frail soap films inspire ultra-strong+light materials. Iridescent colors in 🐚 🦋🪲🦜 - same structures: GYROIDS. Minimal surfaces can naturally form to minimize area while balancing tension and pressure — like soap films stretched across frames. The gyroid, discovered in 1970 by Alan Schoen (NASA crystallographer), holds no straight lines or reflective symmetry. Its intricate, repeating geometry bridges mathematics, modeling, nature, engineering.
Gyroids manipulate light in butterfly wings, bird feathers, and even photonic crystals. They create “structural color” via interference as different from pigment color. Engineers use them for lightweight 3D-printed structures in aerospace, scaffolds for medical tissue growth. Gyroid structures optimize strength and weight, distributing stress efficiently for diverse engineering uses.
Surface and equation in the video are a simple widely-used approximation of a GYROID, which in reality has a complex description. I confine it within a spherical region and send 3D waves through the structure -- a visual clue that often these structures are in motion in the real world.
Start from creating a boundary outlining the intersection of a sphere and gyroid:
tb=ContourPlot3D[​​Sin[x]Cos[y]+Sin[y]Cos[z]+Sin[z]Cos[x],​​{x,-2Pi,2Pi},{y,-2Pi,2Pi},{z,-2Pi,2Pi},​​Contours->{0},ContourStyle->None,​​MeshFunctions->{Function[{x,y,z,f},x^2+y^2+z^2]},​​Mesh->{{(2Pi)^2}},MeshStyle->Directive[Red,Tube[0.08]],​​Axes->False,Boxed->False,BoundaryStyle->None,​​PlotRange->All,Lighting->"ThreePoint"]/.Line[pts_]:>{MaterialShading[{"Glazed",Red}],Tube[pts,0.4]}
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Now plot gyroid within the sphere and add the boundary:
imgFiN=Show[tb,​​RegionPlot3D[​​(Sin[x]Cos[y]+Sin[y]Cos[z]+Sin[z]Cos[x])^2<.1&&x^2+y^2+z^2<=(2π)^2,​​{x,-2π,2π},{y,-2π,2π},{z,-2π,2π},​​Mesh->None,Axes->False,Boxed->False,PlotStyle->MaterialShading["Gold"],​​PlotPoints->70,PlotRange->All,Lighting->"ThreePoint"​​],Lighting->"ThreePoint",Background->GrayLevel[.1],ImageSize->500{1,1}]
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Extract geometric primitives from this Graphics:
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cpGeometry=Cases[imgFiN,_GraphicsComplex,Infinity];
Transform or replace the primitives with the ones that move according to periodic in time wave motion:
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gr[t_]=Graphics3D[cpGeometry/.GraphicsComplex[pts_,data__]:>GraphicsComplex[pts/.{x_,y_,z_}:>(#+0.15Sin[t+Times@@#/4.5]&@{x,y,z}),data],​​PlotRange->All,Axes->False,Boxed->False,Lighting->"ThreePoint",Background->GrayLevel[.0],BoxRatios->{1,1,1},ImageSize->1080{1,1}];
Create a video parallelizing over multiple CPUs:
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Parallelize@AnimationVideo[gr[t],{t,0,2Pi},RasterSize->1080];//AbsoluteTiming
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{439.812,Null}
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CITE THIS NOTEBOOK

Approximate gyroid visualization: minimal surface in motion with 3D waves and spherical constraints​
by Vitaliy Kaurov​
Wolfram Community, STAFF PICKS, January 7, 2025
​https://community.wolfram.com/groups/-/m/t/3352303