The (10, 3)-a Network

George W. Hart

Above is a sculptural concept based on the crystal structure (10,3)-a, described in the 1950's by the chemist A.F. Wells. I have taken that arrangement of vertices and edges and wrapped a smooth organic volume around it using a general-purpose algorithm described here.  The result is a "slimmed down" gyroid surface. Above is a 4-by-4-by-4 block of the lattice structure made as a 3.5 inch nylon model using selective laser sintering. Check out the various views below, but it is impossible to convey its 3D structure with 2D images. It would be better to look at the stl file in a 3D viewer or use it to make your own physical copy.

 The above rendering only hints at how cool the real physical model is when you hold it.
For one thing, my version stands up on one corner, balanced nicely on the three feet in the lower-left.

Looking down the X, Y, or Z axis drections, you see the above pattern of large and small tunnels.

The above rendering gives more perspective.

Looking down a 3-fold axis you see a hexagonal pattern of tunnels.

In any 110 direction, you get this pattern.

In other directions, you get other patterns.  It is a very rich object!

Wells points out that the structure can interpenetrate space with its mirror image.
This colored rendering clarifies that there are two separate components.

A physical model of the two interlocked structures is immensely complex.
The two parts in the above 3.5 inch nylon model are free to jiggle slightly.


Here is the view of the doubled structure along an X, Y, or Z direction.
This one is a 5-by-5-by-5 block of doubly covered unit cells.
(This doubled model also stands up nicely, balanced on a corner!)

The underlying structure of vertices and edges can be conveniently made with Zometool.
The (10, 3)-a network is just the green edges in the above unit cell. The yellow lines are its 3-fold axes.

The structure is described in A.F. Wells' 1956 book The Third Dimension in Chemistry.
It was recently rediscovered by Toshikazu Sunada and called "K4" in this AMS Notices paper
His subsequent letter updating that paper is online here and an update by Alan Schoen is here.
Also see H.S.M. Coxeter's "On Laves' Graph of Girth Ten" Canadian Journal of Math, 1955.
The first presentation of the structure was apparently in 1933, in this German publication:
H. Heesch, F. Laves, "Über dünne kugelpackungen," Z. Kristallogr., 1933, 85, 443-453.
A history with additional references is: Stephen T. Hyde, Michael O'Keeffe, Davide M. Proserpio,
"A short history of an elusive yet ubiquitous structure in chemistry, materials and mathematics,"
Angewandte Chemie, International Edition, 2008, 47 (42): 7996-8009. (abstract)

It leads to the above Heesch-Laves loose-packing of spheres.

Double Diamond

On a related topic, above is the analogous idea of the "double diamond lattice".

This is a small (1.6 inch) physical SLS model. It consists of two copies of the structure below.
They are free to wiggle slightly relative to each other.  It is said to occur in a sodium-thallium alloy,
where the atoms of each metal take the positions of one of the interlinked carbon lattices [reference]. 

This is a small (1.6 inch) SLS model of the diamond lattice.