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Show[Graphics3D[{Cuboid[]}], ContourPlot3D[{x==1/3,y==1/4,z==1/5},{x,-0.2,1.2},{y,-0.2,1.2},{z,-0.2,1.2}]]

enter image description here

Three planes divide the unit cube into 8 small cubes, what's the easiest way to get these 8 little cuboids? I've tried the regional boolean operation
RegionIntersection[Cuboid[], ImplicitRegion[x < 1/3 && y < 1/4 && z < 1/5, {x, y, z}]] gives

BooleanRegion[#1&&#2&,{Cuboid[{0,0,0}],ImplicitRegion[x<1/3&&y<1/4&&z<1/5,{x,y,z}]}]

but I want

Cuboid[{0, 0, 0}, {1/3, 1/4, 1/5}]

The expected final output is

{Cuboid[{1/3,1/4,1/5},{1,1,1}], Cuboid[{1/3,1/4,0},{1,1,1/5}], Cuboid[{1/3,0,1/5},{1,1/4,1}], Cuboid[{1/3,0,0},{1,1/4,1/5}], Cuboid[{0,1/4,1/5},{1/3,1,1}], Cuboid[{0,1/4,0},{1/3,1,1/5}], Cuboid[{0,0,1/5},{1/3,1/4,1}], Cuboid[{0,0,0},{1/3,1/4,1/5}]}

Show[Graphics3D[{Cuboid[]}], ContourPlot3D[{x==1/3,y==1/4,z==1/5,x==3/4,y==5/6,z==6/7}, {x,-0.2,1.2},{y,-0.2,1.2},{z,-0.2,1.2}]]

If can be extended to the case of six planes, it will be better. enter image description here

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  • $\begingroup$ RegionIntersection[] is not sufficiently smart to figure out that the intersection of Cuboid[] and HalfSpace[] should be representable as a Cuboid[] as well, so you'll need to do it manually. $\endgroup$ Commented Sep 26, 2020 at 10:09
  • $\begingroup$ @J.M. RegionIntersection[Cuboid[],HalfSpace[{0,1,0},1/3]] The returned nor cuboid. $\endgroup$ Commented Sep 26, 2020 at 10:25
  • $\begingroup$ That's what I meant by "not sufficiently smart", yes. $\endgroup$ Commented Sep 26, 2020 at 10:27

2 Answers 2

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xmesh = {0, 1/3, 2/3, 1}; ymesh = {0, 1/4, 2/4, 3/4, 1}; zmesh = {0, 1/5, 3/5, 4/5, 1}; cuboids = Cuboid@@@(Transpose /@ Tuples[Partition[#, 2, 1]& /@ {xmesh, ymesh, zmesh}]); Graphics3D[{Opacity[.3, RandomColor[]], #} & /@ cuboids, Boxed -> False]

enter image description here

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  • $\begingroup$ Once again... bested by the mysterious @kglr (+1). $\endgroup$ Commented Sep 26, 2020 at 18:35
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This can be made far more elegant and terse using Outer, but it works:

xplanes = {1/3, 2/3}; yplanes = {1/4, 2/4, 3/4}; zplanes = {1/5, 3/5, 4/5}; ends = {0, 1}; myPlanes = Union[ends, #] & /@ {xplanes, yplanes, zplanes}; theCuboidlets=Flatten[Table[ Cuboid[{myPlanes[[1, i]], myPlanes[[2, j]], myPlanes[[3, k]]}, {myPlanes[[1, i + 1]], myPlanes[[2, j + 1]], myPlanes[[3, k + 1]]}], {i, Length[myPlanes[[1]]] - 1}, {j, Length[myPlanes[[2]]] - 1}, {k, Length[myPlanes[[3]]] - 1}]]; Graphics3D[theCuboidlets]
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  • $\begingroup$ Good job, Thank you. $\endgroup$ Commented Sep 26, 2020 at 12:30

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