This page reports on my design process and preparations for this sculpture, which I call Gyrangle. The following text and images accumulated over the period of May-October 2012.  Pictures of the Oct 23-25 assembly are shown separately here.

1. The Sculpture

Here is a concept sketch for a sculpture I am designing. It is modular, being made of 490 hollow triangles.  Look carefully and you can see that many of the triangles have a fold in them. There are just these two geometric units---flat triangles and triangles with a 109.5 degree fold. And I plan to have four colors for each, making a total of eight types of part. What is most important to me is the interesting geometry and the fact that it can be built by many people who contribute by each attaching one or two triangles.  And what's especially interesting is the variety of patterns of tunnels that can be seen from various directions:
The triangles are made of laser-cut steel, with a six inch edge length. The total weight is about 120 pounds and the height is about 42 inches.  I've  named it Gyrangle, because it is based on a gyroid surface but is made of triangles.  (That is a title I made up, not a standard mathematical term.)

Note that I don't usually show my design ideas like this before completing a sculpture. I like to experiment, and the details of my plans usually continue to evolve up until the last moment.  So it is possible I will decide to make a rather different form from what is shown above, while still adhering to the mathematical ideas explained below. One advantage of these triangle modules is that they can be assembled into a wide variety of structures.  I will experiment with them after they are fabricated, before the assembly event, and may come up with a final design I like even better.

2. The Assembly Process

The sculpture will be assembled on the National Mall in Washington D.C. on Oct 23 and 24, 2010.  This will be the weekend of the USA Science and Engineering Festival. I will be leading the assembly process at the booth of the American Mathematical Society. The event is free and open to the public.  Come join us and participate in the construction!


The basic construction units are hollow metal triangles with bent connectors for joining to their neighbors via nuts and bolts. The hole's edge is exactly half the whole edge length. About half the parts are flat and half are partly folded. I will bring these laser-cut metal triangles with appropriate connectors to easily screw them together at the proper dihedral angles. Anyone who is able to use a screwdriver can visit the AMS booth and add a triangle piece to the growing sculpture.  The connectors will probably be much like the ones I used here, which worked very well, but with three dihedral angles: planar, tetrahedral, and octahedral.

I have led many previous sculpture barn raisings (see my web pages for examples) but I never can predict exactly about timing. We will have about seven and a half hours on Saturday and again on Sunday for the construction. I hope it will take under two minutes to select and add a piece to the growing sculpture, i.e., at least thirty per hour. So we can build something with roughly 7.5·2·30 = 450 triangles. The design above has 490 parts, so I expect it will take the two days to assemble, but might not be too ambitious if we can get some parallelism going.  But we'll see where we actually stand as saturday moves along. 

Afterward, the sculpture will be delivered to Towson University to be permanently displayed as a donation by the AMS.

3. The Underlying Mathematical Structure

The underlying mathematical ideas which make this sculpture possible are rather intricate, so I made some animations to help communicate the geometric concepts.  You don't need to understand the following to participate in the sculpture barn raising, but reading this may give some insight into what inspired me about this design and what you can look for in it.

The sculpture is based on what is called the (10,3)-a lattice, which is shown above. (To be more precise: the actual lattice extends infinitely in all directions and this is just a manageable cube-shaped chunk of it to display.) The blue spheres are the lattice points. They each have integer coordinates, so this is a subset of a standard cubic lattice, but it is a tricky subset to understand.  The green edges connect adjacent vertices; each is of length square-root-of-two. At each lattice point, three green edges meet with an equal angular spacing of 120 degrees. The thin yellow lines are 3-fold rotational axes of the entire structure.  They each are parallel to one of four directions---the directions of the long diagonals of a cube. The animation focuses on three important views: looking in the directions of a 4-fold axis, a 3-fold axis, and a 2-fold axis of the underlying cubic lattice.

It is a wonderful fact that we can wrap triangles around the lattice as shown above. The hollow triangles allow you to see some of the lattice inside. A pair of triangles make a sandwich around each lattice point, but they point in opposite directions, like the bases of a triangular antiprism. The four families of yellow 3-fold axes pass orthogonally through the holes of the triangles.  As there are four families of yellow lines, so the triangles are in four different parallel families, which are distinguished here by differing colors. Notice that the triangles do not share a full edge with any other triangle.
This animation shows how the triangles meet. The blue sphere here is a triangle vertex, not a lattice point. Every vertex of every triangle has this same configuration of four triangles around to it. It is a vertex of two opposite facing triangles and is the midpoint of two neighbors' edges.  Triangle edges half overlap, yet (ignoring the hole shown in each triangle) everything seals up to make a 2D manifold of triangles. All the triangles are equivalent in the structure, i.e., you can rotate the whole infinite structure to position any given triangle in the location of any other given triangle, and the entire structure will look unchanged. I used red and yellow to color the two sides of the surface, which turns out to be equivalent to a gyroid surface.

In the above image, I have removed the lattice to show just the pattern of the triangles, and again focus on a 4-fold, 3-fold, and 2-fold direction of the surrounding cube. The different shapes of tunnels in different directions are spectacular.  Because you are mainly seeing the exterior of the cube, it may be difficult to realize that in the interior everything seals up nicely.  The unhappy gaps and sharp corners along the outer boundaries of this cube-shaped chunk of the lattice will be addressed below.

The above animation shows another direction from which one can come to understand this structure.  Perhaps you know that regular octahedra and regular tetrahedra can pack space without any gaps when put together so each octahedron is surrounded by eight tetrahedra and each tetrahedron is surrounded by four octahedra. If you had magnetic octahedra and tetrahedra building blocks, you could stick them together to make many cool things. M.C. Escher's Planaria is all about exploring this idea. It turns out that by removing cells from the infinite "oct-tet" space packing, we can make the triangle construction. The large hollow triangles of the earlier images are now divided into four half-size triangles---one octahedron face surrounded by three tetrahedra faces. A (10,3)-a lattice can be intricately linked with a mirror-image copy of itself, as illustrated here. So the various tunnels in our sculpture have room to hold a mirror image copy of the same sculpture!

In this tetrahedron/octahedron derivation of the structure, some odd-shaped polyhedral blobs at the boundaries are the result of our cutting this finite chunk from an infinite pattern. And recall that sharp corners and open gaps were the analogous boundary effects in the earlier triangle-sandwich-around-a-lattice construction. Neither of these consequences is satisfactory to me, but with some work, we can terminate the infinite construction in a very nice manner:

The above chunk was derived by repeatedly pruning all leaves from the cube-shaped chunk of the lattice, until each remaining lattice point has two or three neighbors. It turns out that by applying this algorithm, the only gaps are adjacent pairs of triangles and we can partially fold one pair of the sharp triangle corners to close each gap. When a triangle is folded, its paired triangle around a lattice point (the matching "slice of bread" from one "sandwich") is also folded to meet it. So by using modules which are triangles and bent triangles, I have the freedom to choose a volume of any shape, as long as every (10,3)-a lattice vertex is of degree at least two. (Of course, there are other constraints, such as complexity, cost, stability, and engineering strength to consider when designing a sculpture.) The quasi-randomness in the pattern of colors that face the viewer is a nice feature of this method that I hadn't expected.

4. An Alternate Design

As an alternate to the tetrahedral frame shown at the top of this page, another possible design is this simpler head-and-shoulders form, consisting of 446 triangles.  The red base derives from a cube standing up on a corner, from which we slice the top and bottom vertex. Think of the polyhedron made famous in Durer's Melancholia, but shorter.  Walking around it, the viewer may be surprised to observe three of the four hexagonal tunnel directions. The fourth set of hexagon tunnels can be seen from directly above, because a 3-fold axis is vertical. The yellow orb that sits on it is based on a truncated cube also, but with different proportions. I like the subtle allusion to Durer.  And it is wonderful to realize that we could also make a mirror image structure which fits in the exterior holes that pass through it. I think this would be interesting to assemble, but I am still working out all the assembly issues.

5. History

The (10,3)-a lattice has a long history of independent rediscoveries. It has also been called the Laves graph (of degree three or of girth ten), and the triamond lattice. There is a list of references about it here.

But I don't know anything about the history of this triangle construction. I discovered it around 1980 and made this paper model.  Only later did I learn about the gyroid surface and realize that this is a discrete version of it.  I assumed that, as with the (10,3)-a lattice, my discovery was a rediscovery and others had known of it previously, but I don't know of any previous description of it. Because the triangles do not meet edge-to-edge, it doesn't fit into the standard categories of infinite polyhedra. You can think of the triangles as irregular hexagons which then do meet edge-to-edge, but I like triangles. Or you can divide the triangles into four smaller triangles to have two orbits of triangles in the pattern (a³.b.a³.b). The closest relatives I know that are described in print are what John H. Conway calls the "propeller-hedron" and the (6.3.4.3.3) on pp. 326-327 and pp. 338-339 in The Symmetries of Things, by Conway, Burgiel and Goodman-Strauss.  Please let me know if you can provide me with additional references.

Since 1980, this paper model has been sitting in my office and I've thought off and on about making a large sculpture of some kind based it. I am very happy that this commission by the AMS has given me an opportunity to do so. I feel the deep geometric ideas which make this construction possible are very worthy of an artwork for a professional mathematical organization like the AMS, and the modularity is what is needed for a public sculpture barn-raising assembly.

Incidentally, the blue spiral strips in the paper model above provide a continuous set of connectors to join the triangles together.  Each blue strip forms a triangulated helix in the direction of a 4-fold axis. They are really quite cool and deserve to be the subject of a future sculpture.  Let me know if you want to commission me...

6. Make Your Own Model

A good way to learn about any topic in 3D geometry is to make your own physical models.  If you want to understand more about this sculpture, I suggest you make a simple paper model.  Cut out a pile of equilateral triangles and tape them together so the edges half overlap like this:


It is quite fun so spin this model around in your hands, looking for all the various shapes of tunnels. In this image, there is a place at the top right where you can see an opening the size of two small (half-edge size) triangles and the points of two triangles above and below the gap.  Those points can be folded over to close the gap. An example of a gap covered in this way is shown just left of center, where two of the paper triangles are folded. Inside each of those spaces is a lattice point with just two neighbors instead of three.  The third connection would have been in the direction of the gap.


The layout of the triangles in each of the four orientations of planes is that they meet tip-to-tip as shown above. They alternately face inside and outside the enclosed volume, as suggested by the two shades of green.  Other families of planes cross this plane at the tetrahedral dihedral angle of roughly 109.5 degrees.  Along each of the triangle-boundary lines, analogous lines from two other intersecting planes meet, but those lines are slid over half a triangle edge length relative to these triangles. At each vertex, there are two triangle vertices meeting and two edge midpoints (from one triangle on each side of this plane). Remember to maintain handedness. You can make either of the mirror image equivalents, but be consistent throughout. It all becomes clearer as you build it, but it takes a good number of triangles before you close up around some tunnels.


To fully understand the sculpture, you may also want to study the underlying (10,3)-a lattice. If you have Zometool, it provides parts with the appropriate lengths and angles to make a hands-on model. The green lattice edges and yellow 3-fold axes in the animations above are color coded to match the colors of the corresponding Zometool struts. The blue struts here outline the underlying cubic lattice.  Replicating the cube above as many times as you wish allows you to create a very informative model of this intricate and beautiful lattice. Within a 4x4x4 cube, the lattice points to connect are at coordinates: (0,0,4), (1,0,3), (2,1,3), (3,1,4), (4,0,4), then (4,0,0), (3,1,0), (3,2,1), (4,3,1), (4,4,0), then (0,4,0), (0,3,1), (1,3,2), (1,4,3), (0,4,4), then connect (2,2,2) to its three neighbors. Pairs separated by a distance of square-root-of-two get connected with a green edge.

7. Thinking Bigger