I know the question sounds a bit silly and there are many similar answered questions but my 3D visualization (imagination) is very bad. There are some images of octahedral and tetrahedral holes but still there's no easy explanation. Can someone help me?

  • $\begingroup$ Well, the tetrahedral holes are tetrahedral, and the octahedral ones are octahedral. It is not going to be any other way. What is your question, really? $\endgroup$ – Ivan Neretin Sep 8 at 8:46
  • $\begingroup$ @Ivan Neretin Nah the problem is there's no picture that the holes are tetrahedral or octahedral. They're just round holes and many balls surround it. If I see the holes from top, will be like seeing pyramid from top or what? $\endgroup$ – làntèrn Sep 8 at 9:23
  • $\begingroup$ I know tetrahedral but tetrahedral hole .. Someone should enlarge the hole and shows it is really that tetrahedral. $\endgroup$ – làntèrn Sep 8 at 9:28
  • $\begingroup$ If you are in an empty space with 4 atoms as closest neighbors, it is a tetrahedral hole. If 6 atoms a close by, instead, it is a octahedral hole. Now we just need a small 360 $^\circ$ camera to fly through crystals to visualize it. $\endgroup$ – Karsten Theis Sep 8 at 10:27
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    $\begingroup$ The shape of the holes, taking atoms to be spherical, is not that of a tetrahedron or octahedron. The geometry of closest neighbors is what is described by the terms. $\endgroup$ – Karsten Theis Sep 8 at 11:53

To visualize holes in closest packing of spheres, you need 6 balls (marbles, table tennis balls, etc) of equal size. 3D closest packing can be constructed from 2D layers of closest packed spheres as such:

enter image description here

The light blue layer is on the bottom and the dark blue layer is on top. The holes, tetrahedral or octahedral, are in between layers.

To find a tetrahedral hole, take three balls and arrange them in a triangle to form part of the lower layer. Then place one sphere in the center on top (you have to keep the lower level together somehow or the structure will collapse). Your tetrahedral hole is in the center of the tetrahedron formed by the 4 balls (marked T in the picture).

To find an octahedral hole, you need a helper or glue. Take three balls and arrange them in a triangle. Take another three balls and arrange them in a triangle. Place the two triangles on top of each other, twisted against each other by 60 degrees. This is a octahedron on its side, and in the center is the octahdral hole.

In the picture above, these positions are marked by T and O. You just have to imaging that they are in between the layers, and then you might be able to count nearest spheres. The picture below shows the arrangement of the nearest spheres, only.


If you don't have 6 balls handy, but have two hands handy, you can also do the following. Place thumb, pointer and middle finger together so that the finger tips form a triangle. Point at the center of the triangle with the pointer of the other hand. That's the tetrahedral hole. Now, form a triangle with the finger tips of your other hand. Bring the 6 finger tips together with a little twist so that they "lock in". At the center of the 6 finger tips, you have the octahedral hole.

So, whether you are a visual or a haptic learner (not that that would be a thing), this will do the trick, I hope.

enter image description here

There are some images of octahedral and tetrahedral holes

Actually, I only found images of the spheres representing the atoms, no images of the shape of the space in between. If you squeeze some blue play dough between two layers of marbles, here is what you get (see also https://leancrew.com/all-this/2019/06/cheesegraters-and-crystals/):

enter image description here

Within a layer, each indentation is surrounded by six others. The arrows point at the six locations with the largest voids (in the center of triangles formed by the sphere and two of its nearest neighbors in that layer). If you look closely, each indentation has three holes, the contacts with the layer below. The voids take on different shapes depending what is underneath. If a sphere is directly underneath, it is a tetrahedral hole. The shape of this is a tetrahedron with "dented in" faces (far right). If a center of a triangle is directly underneath, it is an octahedral hole. The shape of this is a cube with "dented in" faces (near right; cube and octahedron are complementary archimedian solids: the centers of the faces coincide with the other's vertices and vice versa. A tetrahedron is complementary to itself).

If you shrink the spheres around the atoms to points, you get a space-filling arrangement of tetrehedra and octahedra called a tetrahedral-octahedral honey comb.

Teaching this material

Stefanie Lenzer, Bernd Smarsly and Nicole Graulich recently published a paper in J Chem Ed on using clear spheres to explore crystal packing and voids.

enter image description here

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    $\begingroup$ As a non-chemist, I have to accept that the volume between 6 spheres is called an "octahedral hole", but the notation strikes me as odd. The shape left between 6 spheres is much more like a cube. See the play-doh model above. The octahedron referred to is the dual of the cube. You get a wire-frame of this octahedron by joining the centers of touching pairs of spheres by rods. Maybe this is why the OP claims to have "very bad 3D visualization". Maybe it's actually good, and OP can't understand why this cube is being called an octahedron! $\endgroup$ – Adam Chalcraft Sep 8 at 21:54
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    $\begingroup$ @AdamChalcraft the volume between the spheres is indeed like a squashed cube. But if you draw the lines between the centres of of nearest atoms (a point and stick model) you see they form the edges of an octahedron. Similarly, a small atom in the hole will touch its 6 nearest neighours at the corners of an octahedron. See for example the "octahedral" environment of the sulfur atom in sulfur hexafluoride (which is a molecular substance rather than a lattice, but you can see the analogy.) en.wikipedia.org/wiki/Sulfur_hexafluoride $\endgroup$ – Level River St Sep 8 at 22:46
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    $\begingroup$ @AdamChalcraft Thanks for reintroducing me to the technical term dual. And you are right, octahedral hole is jargon; I suppose it should be "space for atoms that would have octahedral coordination", i.e. six nearest neighbors. Until this question, I had never thought about the exact shape of the available space. Making the layer model in play-dough is instructive but kind of challenging. Taking a small ball of play-dough and squeezing it between 4 or 6 marbles, on the other hand, is very easy and gives quick results, so it would be a fun class room activity. $\endgroup$ – Karsten Theis Sep 9 at 10:59

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