How about cubes? Yes that will do it, Cubes "tessellate", that is to say "fill" space, with no gaps. Spheres certainly don't. A cube is one of the five Regular polyhedrons that so fascinated the ancients, along with the tetrahedron, the octahedron, the dodecahedron, and the icosahedron.

I once made Archimedes' mistake, which is that tetrahedrons fill space too. But they don't. However, a combination of tetrahedrons and octahedron do, which I figured out for myself, only to find out through further research that so had many others, long ago.

So, hy did Archimedes and I make this mistake? Possibly because of the octa- and tetra- thing, and possibly because triangles tessellate well in 2-space.

So the cube (aka the hexahedron), is the only one of the 5 regular polyhedra to fill space. Anything else?

Yes, the Johnson solid known as gyrobifastgium, pictured above.

Anything else?

Yes, several more things: hexagonal, square and triangular prisms, and the Archimedean solid known as the Truncated octahedron

As so:

The truncated octahedron exists in three different convex uniform honeycombs (space-filling tessellations):

Bitruncated cubic | Cantitruncated cubic | Truncated alternated cubic |
---|---|---|

Bitruncated cubic:

This honeycomb has three uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions:

- C
^{~}_{3}or [4,3,4] group - Two types of truncated octahedra in 1:1 ratio. Half from the original cells of a cubic honeycomb, and half are centered on the vertices of the original honeycomb. - B
^{~}_{3}or [4,3^{1,1}] group - Three types of truncated octahedra in 2:1:1 ratios. - A
^{~}_{3}or [3^{[4]}] group - Four types of truncated octahedra in 1:1:1:1 ratios.

A 2-colored honeycomb with C ^{~}_{3} symmetry. | A 3-colored honeycomb with B ^{~}_{3} symmetry. | A 4-colored honeycomb with A ^{~}_{3} symmetry. |

Here is the net:

And in fact, there are many more, such as this:

Rhombic dodecahedron |

and

Rhombo-hexagonal dodecahedron |

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