Saturday, February 19, 2011

Filling Three Dimensional Space

How does one go about filling Euclidean Three-dimensional Space?

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 Honeycomb.svg Cantitruncated Cubic Honeycomb.svg Truncated Alternated Cubic Honeycomb.svg
The cell-transitive bitruncated cubic honeycomb can also be seen as the Voronoi tessellation of the body-centred cubic lattice. The truncated octahedron is one of five three-dimensional primary parallelohedron.

Bitruncated cubic:

This honeycomb has three uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions:
  1. 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.
  2. B~3 or [4,31,1] group - Three types of truncated octahedra in 2:1:1 ratios.
  3. A~3 or [3[4]] group - Four types of truncated octahedra in 1:1:1:1 ratios.
These uniform symmetries can be represented by coloring differently the cells in each construction.
Bitruncated Cubic Honeycomb.svg
A 2-colored honeycomb with C~3 symmetry.
Bitruncated cubic honeycomb3.png
A 3-colored honeycomb with B~3 symmetry.
Bitruncated cubic honeycomb2.png
A 4-colored honeycomb with A~3 symmetry.

Here is the net:

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


Rhombo-hexagonal dodecahedron

If you find the subject interesting, you may further explore at Zonohedron and Convex uniform honeycomb.

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