2 dimensions

each shape covers the same area
13% of the maximum possible gain
100% = ∞ dimensions
square: 4 sides  ·  hexagon: 6 sides

3 dimensions

each shape has the same volume
19% of the maximum possible gain
100% = ∞ dimensions
cube: 6 faces  ·  truncated octahedron: 14 faces
…

8 dimensions

we can't draw these cells, but we can measure them
47% of the maximum possible gain
100% = ∞ dimensions
8D cube: 16 facets  ·  E8 cell: 240 facets
cube-shaped cells (rounding each value separately) best lattice cells: as round as possible while still filling space with no gaps a ball of the same size