Perfect Space: Zero-Gap 3D Tilings

Cell Shape

Lattice

Grid radius 2

Cell gap 0%

Live Readouts

Cells shown

Gap between cells

Volume / cell

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Surface / cell

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Isoperimetric quotient (sphere = 1.000, lower needs more surface per volume)

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How it works

A "perfect" partition of space is a set of equal-volume convex cells that fill all of it with zero gaps and zero overlaps — a honeycomb. Only a handful of convex solids can do this on their own. This simulator renders three of them at their exact space-filling lattice:

Cube — simple cubic lattice, position (2i, 2j, 2k). The trivial perfect tiling.
Rhombic dodecahedron — the Voronoi cell of the face-centred cubic (FCC) lattice: 14 vertices (8 cube corners at (±1,±1,±1) plus 6 octahedron tips at (±2,0,0) etc.), placed at every FCC lattice point.
Truncated octahedron — the Voronoi cell of the body-centred cubic (BCC) lattice: 24 vertices, every permutation of (0,±1,±2). This is Kelvin's cell, Lord Kelvin's 1887 candidate for the partition of space into equal volumes with the least surface area — unbeaten by a straight-edged solid until the Weaire–Phelan foam structure in 1993.

Isoperimetric quotient:  IQ = 36πV² / A³   (sphere: IQ = 1)

Cube:                V = 8    A = 24        IQ ≈ 0.524
Rhombic dodecahedron: V = 16   A = 24√2   IQ ≈ 0.740
Truncated octahedron: V = 32   A = 6a²(1+2√3), a=√2  IQ ≈ 0.753

Kelvin's cell wins: for the same enclosed volume it needs the least surface area of the three, which is why soap-froth structures and crystal grain boundaries tend toward truncated-octahedron-like cells rather than cubes. The gap slider shrinks every cell toward its own center — at 0% the cells share faces exactly and space is perfectly filled; above 0% gaps open up between neighbours, showing what an imperfect tiling looks like by contrast.

Space-filling cell Lattice point

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