The structure of a diamond crystal is fascinating. But there’s an equally fascinating form of carbon, called the triamond, that’s theoretically possible but never yet seen in nature. Here it is:
In the triamond, each carbon atom is bonded to three others at 120° angles, with one double bond and two single bonds. Its bonds lie in a plane, so we get a plane for each atom.
But here’s the tricky part: for any two neighboring atoms, these planes are different. In fact, if we draw the bond planes for all the atoms in the triamond, they come in four kinds, parallel to the faces of a regular tetrahedron!
If we discount the difference between single and double bonds, the triamond is highly symmetrical. There’s a symmetry carrying any atom and any of its bonds to any other atom and any of its bonds. However, the triamond has an inherent handedness, or chirality. It comes in two mirror-image forms.
A rather surprising thing about the triamond is that the smallest rings of atoms are 10-sided. Each atom lies in 15 of these 10-sided rings.
Some chemists have argued that the triamond should be ‘metastable’ at room temperature and pressure: that is, it should last for a while but eventually turn to graphite. Diamonds are also considered metastable, though I’ve never seen anyone pull an old diamond ring from their jewelry cabinet and discover to their shock that it’s turned to graphite. The big difference is that diamonds are formed naturally under high pressure—while triamonds, it seems, are not.
Nonetheless, the mathematics behind the triamond does find its way into nature. A while back I told you about a minimal surface called the ‘gyroid’, which is found in many places:
• The physics of butterfly wings.
It turns out that the pattern of a gyroid is closely connected to the triamond! So, if you’re looking for a triamond-like pattern in nature, certain butterfly wings are your best bet:
• Matthias Weber, The gyroids: algorithmic geometry III, The Inner Frame, 23 October 2015.
Instead of trying to explain it here, I’ll refer you to the wonderful pictures at Weber’s blog.
Building the triamond
I want to tell you a way to build the triamond. I saw it here:
• Toshikazu Sunada, Crystals that nature might miss creating, Notices of the American Mathematical Society 55 (2008), 208–215.
This is the paper that got people excited about the triamond, though it was discovered much earlier by the crystallographer Fritz Laves back in 1932, and Coxeter named it the Laves graph.
To build the triamond, we can start with this graph:
It’s called since it’s the complete graph on four vertices, meaning there’s one edge between each pair of vertices. The vertices correspond to four different kinds of atoms in the triamond: let’s call them red, green, yellow and blue. The edges of this graph have arrows on them, labelled with certain vectors
Let’s not worry yet about what these vectors are. What really matters is this: to move from any atom in the triamond to any of its neighbors, you move along the vector labeling the edge between them… or its negative, if you’re moving against the arrow.
For example, suppose you’re at any red atom. It has 3 nearest neighbors, which are blue, green and yellow. To move to the blue neighbor you add to your position. To move to the green one you subtract since you’re moving against the arrow on the edge connecting blue and green. Similarly, to go to the yellow neighbor you subtract the vector from your position.
Thus, any path along the bonds of the triamond determines a path in the graph
Conversely, if you pick an atom of some color in the triamond, any path in starting from the vertex of that color determines a path in the triamond! However, going around a loop in may not get you back to the atom you started with in the triamond.
Mathematicians summarize these facts by saying the triamond is a ‘covering space’ of the graph
Now let’s see if you can figure out those vectors.
Puzzle 1. Find vectors such that:
A) All these vectors have the same length.
B) The three vectors coming out of any vertex lie in a plane at 120° angles to each other:
For example, and lie in a plane at 120° angles to each other. We put in two minus signs because two arrows are pointing into the red vertex.
C) The four planes we get this way, one for each vertex, are parallel to the faces of a regular tetrahedron.
If you want, you can even add another constraint:
D) All the components of the vectors are integers.
Diamonds and hyperdiamonds
That’s the triamond. Compare the diamond:
Here each atom of carbon is connected to four others. This pattern is found not just in carbon but also other elements in the same column of the periodic table: silicon, germanium, and tin. They all like to hook up with four neighbors.
The pattern of atoms in a diamond is called the diamond cubic. It’s elegant but a bit tricky. Look at it carefully!
To build it, we start by putting an atom at each corner of a cube. Then we put an atom in the middle of each face of the cube. If we stopped there, we would have a face-centered cubic. But there are also four more carbons inside the cube—one at the center of each tetrahedron we’ve created.
If you look really carefully, you can see that the full pattern consists of two interpenetrating face-centered cubic lattices, one offset relative to the other along the cube’s main diagonal.
The face-centered cubic is the 3-dimensional version of a pattern that exists in any dimension: the Dn lattice. To build this, take an n-dimensional checkerboard and alternately color the hypercubes red and black. Then, put a point in the center of each black hypercube!
You can also get the Dn lattice by taking all n-tuples of integers that sum to an even integer. Requiring that they sum to something even is a way to pick out the black hypercubes.
The diamond is also an example of a pattern that exists in any dimension! I’ll call this the hyperdiamond, but mathematicians call it Dn+, because it’s the union of two copies of the Dn lattice. To build it, first take all n-tuples of integers that sum to an even integer. Then take all those points shifted by the vector (1/2, …, 1/2).
In any dimension, the volume of the unit cell of the hyperdiamond is 1, so mathematicians say it’s unimodular. But only in even dimensions is the sum or difference of any two points in the hyperdiamond again a point in the hyperdiamond. Mathematicians call a discrete set of points with this property a lattice.
If even dimensions are better than odd ones, how about dimensions that are multiples of 4? Then the hyperdiamond is better still: it’s an integral lattice, meaning that the dot product of any two vectors in the lattice is again an integer.
And in dimensions that are multiples of 8, the hyperdiamond is even better. It’s even, meaning that the dot product of any vector with itself is even.
In fact, even unimodular lattices are only possible in Euclidean space when the dimension is a multiple of 8. In 8 dimensions, the only even unimodular lattice is the 8-dimensional hyperdiamond, which is usually called the E8 lattice. The E8 lattice is one of my favorite entities, and I’ve written a lot about it in this series:
• Integral octonions.
To me, the glittering beauty of diamonds is just a tiny hint of the overwhelming beauty of E8.
But let’s go back down to 3 dimensions. I’d like to describe the diamond rather explicitly, so we can see how a slight change produces the triamond.
It will be less stressful if we double the size of our diamond. So, let’s start with a face-centered cubic consisting of points whose coordinates are even integers summing to a multiple of 4. That consists of these points:
(0,0,0) (2,2,0) (2,0,2) (0,2,2)
and all points obtained from these by adding multiples of 4 to any of the coordinates. To get the diamond, we take all these together with another face-centered cubic that’s been shifted by (1,1,1). That consists of these points:
(1,1,1) (3,3,1) (3,1,3) (1,3,3)
and all points obtained by adding multiples of 4 to any of the coordinates.
The triamond is similar! Now we start with these points
(0,0,0) (1,2,3) (2,3,1) (3,1,2)
and all the points obtain from these by adding multiples of 4 to any of the coordinates. To get the triamond, we take all these together with another copy of these points that’s been shifted by (2,2,2). That other copy consists of these points:
(2,2,2) (3,0,1) (0,1,3) (1,3,0)
and all points obtained by adding multiples of 4 to any of the coordinates.
Unlike the diamond, the triamond has an inherent handedness, or chirality. You’ll note how we used the point (1,2,3) and took cyclic permutations of its coordinates to get more points. If we’d started with (3,2,1) we would have gotten the other, mirror-image version of the triamond.
I mentioned that the triamond is a ‘covering space’ of the graph More precisely, there’s a graph whose vertices are the atoms of the triamond, and whose edges are the bonds of the triamond. There’s a map of graphs
This automatically means that every path in is mapped to a path in But what makes a covering space of is that any path in comes from a path in which is unique after we choose its starting point.
If you’re a high-powered mathematician you might wonder if is the universal covering space of It’s not, but it’s the universal abelian covering space.
What does this mean? Any path in gives a sequence of vectors and their negatives. If we pick a starting point in the triamond, this sequence describes a unique path in the triamond. When does this path get you back where you started? The answer, I believe, is this: if and only if you can take your sequence, rewrite it using the commutative law, and cancel like terms to get zero. This is related to how adding vectors in is a commutative operation.
For example, there’s a loop in that goes “red, blue, green, red”. This gives the sequence of vectors
We can turn this into an expression
However, we can’t simplify this to zero using just the commutative law and cancelling like terms. So, if we start at some red atom in the triamond and take the unique path that goes “red, blue, green, red”, we do not get back where we started!
Note that in this simplification process, we’re not allowed to use what the vectors “really are”. It’s a purely formal manipulation.
Puzzle 2. Describe a loop of length 10 in the triamond using this method. Check that you can simplify the corresponding expression to zero using the rules I described.
A similar story works for the diamond, but starting with a different graph:
The graph formed by a diamond’s atoms and the edges between them is the universal abelian cover of this little graph! This graph has 2 vertices because there are 2 kinds of atom in the diamond. It has 4 edges because each atom has 4 nearest neighbors.
Puzzle 3. What vectors should we use to label the edges of this graph, so that the vectors coming out of any vertex describe how to move from that kind of atom in the diamond to its 4 nearest neighbors?
There’s also a similar story for graphene, which is hexagonal array of carbon atoms in a plane:
Puzzle 4. What graph with edges labelled by vectors in should we use to describe graphene?
I don’t know much about how this universal abelian cover trick generalizes to higher dimensions, though it’s easy to handle the case of a cubical lattice in any dimension.
Puzzle 5. I described higher-dimensional analogues of diamonds: are there higher-dimensional triamonds?
The Wikipedia article is good:
• Wikipedia, Laves graph.
They say this graph has many names: the K4 crystal, the (10,3)-a network, the srs net, the diamond twin, and of course the triamond. The name triamond is not very logical: while each carbon has 3 neighbors in the triamond, each carbon has not 2 but 4 neighbors in the diamond. So, perhaps the diamond should be called the ‘quadriamond’. In fact, the word ‘diamond’ has nothing to do with the prefix ‘di-‘ meaning ‘two’. It’s more closely related to the word ‘adamant’. Still, I like the word ‘triamond’.
This paper describes various attempts to find the Laves graph in chemistry:
• Stephen T. Hyde, Michael O’Keeffe, and Davide M. Proserpio, A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics, Angew. Chem. Int. Ed. 47 (2008), 7996–8000.
This paper does some calculations arguing that the triamond is a metastable form of carbon:
• Masahiro Itoh et al, New metallic carbon crystal, Phys. Rev. Lett. 102 (2009), 055703.
Abstract. Recently, mathematical analysis clarified that sp2 hybridized carbon should have a three-dimensional crystal structure () which can be regarded as a twin of the sp3 diamond crystal. In this study, various physical properties of the carbon crystal, especially for the electronic properties, were evaluated by first principles calculations. Although the crystal is in a metastable state, a possible pressure induced structural phase transition from graphite to was suggested. Twisted π states across the Fermi level result in metallic properties in a new carbon crystal.
The picture of the crystal was placed on Wikicommons by someone named ‘Workbit’, under a Creative Commons Attribution-Share Alike 4.0 International license. The picture of the tetrahedron was made using Robert Webb’s Stella software and placed on Wikicommons. The pictures of graphs come from Sunada’s paper, though I modified the picture of The moving image of the diamond cubic was created by H.K.D.H. Bhadeshia and put into the public domain on Wikicommons. The picture of graphene was drawn by Dr. Thomas Szkopek and put into the public domain on Wikicommons.