This picture by Greg Egan shows the Laves graph, a structure discovered by the crystallographer Fritz Laves in 1932. It is also called the ‘\(\mathrm{K}_4\) crystal’, since is an embedding of the maximal abelian cover of the complete graph on 4 vertices in 3-dimensional Euclidean space. It is also called the ‘triamond’, since it is a theoretically possible — but never yet seen — crystal structure for carbon.

In the Laves graph, each vertex is connected to three others with edges at 120° angles. These edges lie in a plane, so we get a plane for each atom. However, for any two neighboring vertices, these planes are different. In fact, these planes come in four equally spaced families, parallel to the four faces of a regular tetrahedron.

The Laves graph is highly symmetrical. There is a symmetry carrying any vertex and any of its edges to any other vertex and any of its edges. However, the Laves graph has an inherent handedness, or chirality: it comes in two different mirror-image forms.

The smallest cycles in the Laves graph have 10 edges. Each vertex lies in 15 of these 10-cycles.

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. However, diamonds are formed naturally under high pressure — while triamonds, it seems, are not.

Nonetheless, the mathematics behind the Laves graph does find its way into nature. The minimal surface called a gyroid is topologically the boundary of a tubular neighborhood of the Laves graph, and this surface shows up in the structure of certain butterfly wings. For details, see:

• John Baez, The physics of butterfly wings, *Azimuth*, August 11, 2015.

• S. T. Hyde, M. O’Keeffe, and D. M. Proserpio, A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics, *Angew. Chem. Int. Ed.* **47** (2008), 7996–8000.

Mathematically, the most interesting way to construct the Laves graph is as a ‘topological crystal’. This method was introduced by Sunada:

• Toshikazu Sunada, Crystals that nature might miss creating, *Notices of the American Mathematical Society* **55** (2008), 208–215.

• Toshikazu Sunada, *Topological Crystallography*, Springer, Berlin, 2012.

Just as the universal cover of a connected graph \(X\) has the fundamental group \(\pi_1(X)\) as its group of deck transformations, its **maximal abelian cover**, denoted \(\overline{X}\), has the abelianization of \(\pi_1(X)\) as its group of deck transformations. It thus covers every other connected cover of \(X\) whose group of deck transformations is abelian. Since the abelianization of \(\pi_1(X)\) is the first homology group \(H_1(X,\mathbb{Z})\), there is a close connection between the maximal abelian cover and homology theory.

For a large class of graphs there is a systematic way to embed the maximal abelian cover in the vector space \(H_1(X,\mathbb{R})\), the first homology group with real coefficients. We call this embedded copy of \(\overline{X}\) a ‘topological crystal’. When \(X\) is the complete graph with 4 vertices, \(H_1(X,\mathbb{R})\) is 3-dimensional Euclidean space, and the topological crystal is the Laves graph!

In more detail, the construction of topological crystals proceeds as follows. Any graph \(X\) has a space \(C_0(X,\mathbb{R})\) of **0-chains**, which are formal linear combinations of vertices, and a space \(C_1(X,\mathbb{R})\) of **1-chains**, which are formal linear combinations of edges. There is a boundary operator

$$ \partial \colon C_1(X,\mathbb{R}) \to C_0(X,\mathbb{R}) , $$

the linear operator sending any edge to the formal difference of its two endpoints. The kernel of this operator is the space of **1-cycles**, \(Z_1(X,\mathbb{R})\). There is an inner product on the space of 1-chains such that edges form an orthonormal basis. This determines an orthogonal projection

$$ \pi \colon C_1(X,\mathbb{R}) \to Z_1(X,\mathbb{R}) . $$

For a graph we have \(H_1(X,\mathbb{R}) \cong Z_1(X,\mathbb{R})\). Thus, to build the topological crystal of \(X\), we only need to embed its maximal abelian cover \(\overline{X}\) in \(Z_1(X,\mathbb{R})\). We do this by embedding \(\overline{X}\) in \(C_1(X,\mathbb{R})\) and then projecting it down via \(\pi\).

To accomplish this, we need to fix a basepoint for \(X\). Each path \(\gamma\) in \(X\) starting at this basepoint determines a 1-chain \(c_\gamma\). It is easy to show that these 1-chains correspond to the vertices of \(\overline{X}\). Furthermore, the graph \(\overline{X}\) has an edge from \(c_\gamma\) to \(c_{\gamma’}\) whenever the path \(\gamma’\) is obtained by adding an extra edge to \(\gamma\). We can think of this edge as a straight line segment from \(c_\gamma\) to \(c_{\gamma’}\).

The hard part is checking that the projection \(\pi\) maps this copy of \(\overline{X}\) into \(Z_1(X,\mathbb{R})\) in a one-to-one manner. This happens precisely when the graph \(X\) has no **bridges** that is, edges whose removal would disconnect \(X\). The resulting copy of \(\overline{X}\) embedded in \(Z_1(X,\mathbb{R})\) is called a **topological crystal**.

For details, see:

• John Baez, Topological crystals.

This paper arose from a long discussion here:

• John Baez, Diamonds and triamonds, *Azimuth*, April 11, 2016.

The image of the unit cell of sphalerite was created by Benjah-bmm27 and put into the public domain on Wikicommons.

*Visual Insight* is a place to share striking images that help explain advanced topics in mathematics. I’m always looking for truly beautiful images, so if you know about one, please drop a comment here and let me know!