July 2, 2019

# Sections of Triply Periodic Surfaces- WXML Spring 2019

Taking Cross Sections of Triply Periodic Polyhedral Surfaces

This is a guest post by Benjamin Zielinski, based on work of his and Aadya Bhatnagar, mentored by Gabriel Dorfsman-Hopkins, Charles Camacho, Dami Lee, and Jayadev Athreya.

Perhaps the easiest way to visualize a triply periodic surface is to build one from scratch. The building blocks for polyhedral surfaces are polyhedra, typically platonic solids or archimedean solids. The constituent polyhedra can serve two roles in the resulting surface, either center or handle. Handles bridge between two centers, while centers can be connected to many handles.

As an example, we will build up the surface called cube-6. It receives this name because the centers are cubes and there are six handles connected to each center. To begin, we simply take a cube. We will call it a center and color it blue.

Next, we glue a handle to each of the faces of the center. For cube-6, handles are cubes as well, and they will be colored pink. Notice that every face of the center will be covered in this process, so no blue will show through.

Then we add centers. We must follow the rule that handles touch exactly two centers. So, we add a blue center to the opposite face of the handle, as illustrated.

Now we cover all exposed blue faces with a handle. To build the entire surface, this process is repeated infinitely. This surface has the property that it is triply periodic. This means that cube-6 repeats itself in three distinct directions.

Notice that cube-6 does not fill all of space. It has gaps, and looks like an infinite jungle gym, or perhaps an infinite sponge. So, the question of cross sections naturally arises. In the animation below, we see a small portion of cube-6, and a plane with normal vector (1,1,1) is moving back and forth, so that we can visualize the intersections with cube-6. The intersection of a plane with a cube will be a polygon. When the plane intersects a center, it is drawn blue. Intersections with handles are pink.

Because cube-6 is triply periodic, and our normal vector is rational, moving our plane forward far enough will eventually get us a cross section exactly the same as the starting position. So, we can create closed loops of animation.

Now that we have looked at cube-6, we will investigate a surface called octa-8. It received this name because the centers are octahedron, and there are eight handles connected to every center.

This surface is built entirely of octahedron, and has small gaps between the handles. Taking cross sections using a plane with normal vector (1,1,1) we get the following animation.

If we change our plane so that the normal vector is (1,0,0) we will be looking through a vertex of the centers. Thus, taking these cross sections, moving through the octahedron centers will give us square slices that grow and then shrink.

The next surface we will examine is called octa-4. We can obtain octa-4 from octa-8 by removing half of the centers and handles. This leaves a less densely packed surface which has almost a 3D honeycomb pattern. For more images from different angles, look here.

We will take the same intersecting planes as with octa-8. Notice that shapes that occur look similar to the octa-8 animations, but with portions removed. Since the handles of octa-4 do not attach to every face of the centers, when we take intersecting planes with normal vector (1,0,0) we will see diagonal lines that alternate direction.

These animations are beautiful representations of fascinating mathematical objects, but they are certainly not exhaustive. There are infinite normal vectors we could look at, and many more triply periodic surfaces we could explore. There are no restrictions, and hopefully there will be many more exciting animations to share in the future!