[Someone]

How do you plan to fill 3D space with ‘triangles’?

You can’t do that with tetrahedrons. You can use triangular prisms, but IMO, that’s cheating. You can also split a cube into six pyramids with a square base and height ½, but I don’t think that’s a natural way to fill space.

Looking at https://mathworld.wolfram.com/Space-FillingPolyhedron.html, the Truncated Octahedron probably is the nicest alternative to the cube.

[kragen]

I think this question is sort of another way to phrase, "What are the crystal systems [or possibly lattice systems] other than primitive cubic?" And the answer is in https://en.wikipedia.org/wiki/Crystal_system; there are 6 or 31 answers depending on how you look at it. Or maybe the question is "What are the 3-honeycombs other than the cubic honeycomb?" which is discussed in https://en.wikipedia.org/wiki/Honeycomb_%28geometry%29 and to which there are an infinite number of answers.

Crystal systems are usually described in terms of point lattices, while you're talking about polyhedra, but the Voronoi polyhedra of the points in the lattice are the polyhedra you're looking for. (This is mentioned at the end of the Wolfram™Ⓡ MathWorld™Ⓡ article you linked.) One of my favorites is the cuboctahedral honeycomb corresponding to hexagonal close-packed crystals.

Even within cubic crystals, you could reasonably argue that face-centered cubic crystals "fill[] 3-D space with ‘triangles’".

Honeycombs do not, as I understand it, have to be periodic. In particular, any 3-D rep-tile can be used to tile space in a manner similar to the Penrose tiling, and usually the result is aperiodic. I wrote a 2-D demonstration of this process is at http://canonical.org/~kragen/sw/dev3/skitch#!ffffrrrfffffrrr....

[jedharris]

From Wikipedia: The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells.

https://en.wikipedia.org/wiki/Tetragonal_disphenoid_honeycom...

[lstodd]

I once did a 3d-space representation with https://en.wikipedia.org/wiki/Rhombic_dodecahedron

but in the end it turned out to be easier to just supersample everything down to cube grid, then put some n-vectors on top than to explain everyone how to calculate stuff in that grid.

Still have couple dozen ~3" cardboard ones somewhere from when I tried to understand how exactly do they stack. Made a nice lamp when stuffed with LEDs.

[smegsicle]

> Truncated Octahedron

looks to me like they fill space pretty nicely as an offset of cubes, in the same way as hexgrid is sort of an offset of squares

https://en.wikipedia.org/wiki/Bitruncated_cubic_honeycomb

[scotty79]

https://mathworld.wolfram.com/RhombicDodecahedron.html

Rhombic dodecahedron looks quite nice.

And you could mix it with cube grid for artificial structures in your world.