Talk:Tetrahedral-octahedral honeycomb

Questions

Reading the article, I hardly figure out which of the following claims are true. Can someone help me (and maybe clarify the article so that people like me have no more such doubt)?

1. any face-to-face tilings by regular octahedra and tetrahedra is made up of stacked slabs (picture below)
2. slabs can be rotated by a multiple of 60°, yielding uncountably many different tilings
3. all these tilings are vertex-uniform with 8 tetrahedra and 6 octahedra around each vertex
4. some are edge-uniform with 2 tetrahedra and 2 octahedra alternating on each edge, but not all (e.g., not the gyrated tetrahedral-octahedral honeycomb)
5. the tetrahedral-octahedral honeycomb is one of these tilings, namely the one where slabs are identical
6. the gyrated tetrahedral-octahedral honeycomb is one of these tilings, namely the one where slabs are alternatively reflected (or, equivalently, rotated 180°)
7. putting spheres on each vertex yields a close-packing of equal spheres, with the HCP and the FCP packings respectively corresponding to the usual and the gyrated tetrahedral-octahedral honeycombs

Thank you very much 176.159.47.17 (talk) 09:23, 7 February 2019 (UTC)

There are certainly people much better qualified than I to answer, and I will probably be pointing out things that you already know. I hope some experts will see this and respond. To take your points one by one:

1. This sounds likely to be true, but maybe not trivial to prove. The article's references would be the best place to check whether this might have been done, but I don't have any of them at hand. Assuming that a face-to-face tiling is constructed from stacked slabs, you should maybe be clearer about what "stacked" means. If you lie a regular octahedron on one of its faces, the center of the top face will lie in a vertical line above the center of the bottom face, but the top triangle will point in the direction opposite to that of the bottom triangle. So if you look at the triangular tiling formed by the faces on the top of the slab in your drawing, the red triangles point out of the page. The red triangles on the bottom of the slab will be pointing into the page. Similarly, each of the yellow tetrahedron faces visible on the top of your slab will lie in a vertical line above a yellow face (of a different tetrahedron) pointing in the opposite direction on the bottom of the slab. This tells you that to get a face-to-face tiling by stacking slabs, you need to shift the top slab horizontally so that its bottom triangles align with the top triangles of the slab underneath. When you do this, you will be matching up red triangles with yellow triangles. To make the same point in a different way, the vertices of the triangular tiling on the top of the slab do not lie in vertical lines above the vertices on the bottom of the slab. When stacking slabs, you must perform a horizontal shift of one of the slabs to get vertices on the top of the lower slab to match up with vertices on the bottom of the upper slab.
2. If in addition to the horizontal shift, you rotate the top slab by 60° about the vertical line through one of its vertices, you will cause red faces on the bottom side of the top slab to match up with red faces on the top side of the bottom slab instead of red faces with yellow. Rotating by 120° produces the same result as not rotating at all. Rotating by 180° produces the same result as rotating by 60° and the same result as reflecting in a vertical plane which containing a line of triangle edges. Again, to make the same point differently, vertical lines through vertices on the bottom of the slab pass through centers of faces of yellow triangles on the top of the slab, while vertical lines through centers of red faces on the bottom of the slab pass through centers of red faces on the top of the slab. If you rotate the top slab by 60° so that yellow faces on the bottom of this slab match up with yellow faces on the top of the slab underneath, then vertices on the bottom of the bottom slab will be in a vertical line with vertices on the top of the top slab. If you don't rotate, then vertices on the bottom of the bottom slab will be in a vertical line with centers of red faces on the top of the top slab; if you then place a third slab on top of that, again unrotated, you will have vertices on the bottom of the bottommost slab in vertical alignment with vertices on the top of the topmost slab. This will be important to Point 7. And you indeed do get uncountably many different tilings by specifying whether each stacked slab is rotated or unrotated.
3. In all of these tilings there are eight tetrahedra and six octahedra meeting at each vertex. Is this the same as vertex uniform though? If some slabs are rotated and others are not, you will have octahedra face-to-face with octahedra between some slabs but not between others.
4. All edges in the interior of a slab have this alternation. If you don't rotate any slabs, the same is true of edges between two slabs, but if you do rotate you get octahedron-octahedron-tetrahedron-tetrahedron instead of alternation.
5. The tetrahedral-octahedral honeycomb is the one where none of the slabs is rotated.
6. The gyrated tetrahedral-octahedral honeycomb is the one where every slab is rotated 60° relative to the one underneath. Since rotating 120° is equivalent to not rotating, you can say that every other slab is rotated.
7. I think it's reversed. The usual honeycomb corresponds to face-centered cubic, the gyrated one to hexagonal close packing. See Point 2.

Let me know if any if this is unclear or seems incorrect.Will Orrick (talk) 10:57, 8 February 2019 (UTC)

Thank you. Yes, I did not mention the shift from slab to slab. I agree with all in particular with point 3 (hence with point 7) : in the usual the vertex figure is a Cuboctahedron (as the spheres adjacent to a given sphere in a FCC close-packing), in the gyrated version the vertex figure is a Triangular orthobicupola (as the spheres adjacent to a given sphere in a HCP close-packing). the other mixe the two figures, thus are not vertex-uniform. I would be interested if someone can point out a references (or a simple proof?) for point 1. 176.159.47.17 (talk) 14:14, 8 February 2019 (UTC)

I eventually convinced myself for 1 (a case study shows that there is a unique way to "grow" one of the vertex figure to get a slab), but if anyone find a reference it could improve the article.176.159.47.17 (talk) 09:12, 9 February 2019 (UTC)