(Symmetric Pipe-Cleaner Constructions)

In this exercise you will learn about polyhedral symmetry groups as you design and build symmetric balls from pipe cleaners. A tessellation drawing program will be used in the design phase. Rather than copying the exact design provided in the instructions below, you will create your own original form which most likely has never existed before. You need a computer, a printer, and pipe cleaners.

The picture above shows an icosahedral example made from 30 pipe-cleaners, but we will start in Part I with the simpler octahedral symmetry, requiring only 12 parts. Each part is a symmetric curved "worm" with both ends identical, which is centered on a 2-fold axis. Then in Part II we allow more general worm shapes in which the two ends are not identical and it is not centered on an axis. This requires 24 identical worms. Finally, Part III suggests variations for exploring other symmetry groups.

Part I.   Octahedral Symmetry --- 12 worms each having 2-fold "propeller" symmetry

1) Create a design using a tessellation program. Many programs exist for drawing planar tessellations. An easy one to use is the java applet Kali. (Others include Tess, Tessellation Exploration, Tesselmania, ...) Play with Kali for a while to see how whatever you draw is replicated with the symmetry of a 2D symmetry group.

To create a background lattice of triangles, choose the 632 Wallpaper Group in Kali and draw a black line from one 6-fold axis point to another. It will take you a few tries to find exactly where to start. But the result is that the triangle edges are each covered twice, as in the black lines of this screen shot:

Whatever you draw will be replicated with 3-fold symmetry in each triangle. Note that the midpoint of each triangle edge will be a 2-fold symmetry point, such as the green dot in the above image. Start at any 2-fold point and draw a curve which isn't too complex, such as the heavy red line above, ending it with a short backtrack to indicate a folded end of the pipe cleaner.  It will be replicated with the symmetry to make all the other copies. (On your screen there will be no green dot and the original line will not be heavier than the others. I modified this image for clarity.)

First try to replicate the above design, to learn how the program works, but then make your own design. A good design will not have any places where three or more lines cross simultaneously. You don't want your first project to be too complex, so don't let the line be too long or cross too many other lines. The line you draw from a 2-fold point to the end represents half a worm. The complete worm, from end to end, is symmetric like a propeller and has the 2-fold point at its center. Here are some other examples of reasonable complexity for a first time project: example 2 example 3 example 4.  These all have smooth curves, but sharp angles and zig-zags are OK also.

2) Make a paper net for an octahedron from eight triangles. To print out the pattern, first copy the screen to the clipboard by using the PrintScrn key. Then paste the screenshot from the clipboard into any paint program. Crop down the image to include just enough triangles to make an octahedron net. Save it to an image file and print it out. Cut out the paper net, which for the above example looks like this:

3) Use a marker to emphasize an over-and-under pattern on the paper. A good weaving pattern will respect the symmetry. Every triangle should have the same pattern and within each triangle the same one-third-of-a-triangle repeats three times. Draw the end as a heavy blob regardless of whether it would otherwise cross as an over or an under. There will be two overs or two unders in a row at the crossings closest to the 2-fold axis. This is a necessary consequence of the fact that it is a symmetry point. Here is a marked example:

(Ignore this fine point on first reading: Optionally, you  may be able to maintain over/under alternation along the entire length of the worm if four of the eight triangles have one pattern and the other four have the opposite pattern (with over/under reversed) and each triangle is surrounded by three triangles of the opposite type. But this may be trickier to assemble.)

4) Fold the pattern up to form an octahedron. A bit of tape will hold it together sufficiently. The triangle vertices, which are 6-fold rotation points in the 2D tessellation, become 4-fold rotation points in the octahedron.

5) Build a spherical version from pipe cleaners. An octahedron has 12 edges, so you need 12 worms. If the design is not too complex, you may prefer to cut the pipe cleaners in half and use a half of a pipe cleaner for each worm. The sequence of weaving operations is up to you and will depend on your design. It can be frustrating to keep track of the parts when it is only partially assembled, but follow the paper model systematically and be patient. I find it easiest to first fold the ends of each pipe cleaner back about a quarter inch from each end before weaving into the structure. Then squeeze the end tight after it is hooked over the place it is supposed to join. When the weaving is done, even up the spacings, smooth out irregular curves, round it out to a sphere, and make it as symmetrical as possible. Here is a wormball from the above pattern:

6) Observe the symmetry.  Be sure to notice the eight 3-fold points, the six 4-fold points, and the twelve 2-fold points. Below is the same wormball, as seen along a 4-fold axis:

Part II.   Octahedral Symmetry, 24 worms each not constrained by symmetry

The steps above are easily modified so you do not start at (or even pass through) a 2-fold point. In this case, twenty-four worms are required for octahedral symmetry. The two ends of the worm will be distinct, so think of them as ends A and B, perhaps marking them on your printout with two different colors. To mark the overs and unders of the weave, always start at end A and follow the same alternating pattern as you work towards B. During assembly you need to be conscious of whether you are holding end A or end B of any given worm. Below is a sample pattern to illustrate the symmetry, but make your own. It will be the only one in the universe of whatever it is!

Part III.  Variations you might try

A) Symmetric colorings.
The twelve midpoints of the octahedron's edges can be seen as the vertices of  three squares.With three colors of pipe cleaners you can make four worms of each color in a symmetric manner.

B) Symmetry with mirrors.
The above patterns are chiral---they come in left-hand and right-hand forms. If you use the *632 symmetry in Kali, you get mirrors as well. Design and build a symmetric wormball.  Below is a sample pattern to illustrate the symmetry:

(Fine point to ignore at first: Although the pattern of lines in the plane will be exactly mirror symmetric, the over/under weaving will have to depart slightly from this: Where a worm crosses a mirror at an angle, it and its reflection cross, but in the physical construction only one can be the over and the other must be the under. You can still maintain the rotational symmetry at the mirror crossings in the over/under pattern you choose, unless the worm crosses the mirror at a  2-fold axis point.)

C) Icosahedral symmetry requires more components (30 or 60) so takes more time, but is more interesting visually because you get 5-fold features at the triangle vertices. There will be 30 worms if each is 2-fold constrained and centered on a 2-fold axis, or 60 worms if each is unconstrained.  The figure at the top of this page is a 30 worm icosahedral construction (which I made about 25 years ago) with the following pattern:

D) Tetrahedral symmetry may seem simpler because it involves fewer worms (6 or 12) but it may be more confusing to build because there are two different types of places with 3-fold symmetry---one type corresponds to the tetrahedral vertices and the other type corresponds to the tetrahedron's faces' centers. And there is much more curvature, so everything closes on itself very quickly. Here is a 12-worm tetrahedral construction:

And here is the pattern for it.  Notice that the curve does not go through a 2-fold axis, the two ends are different, and twelve pieces are required instead of six.

E) "Pyrite" symmetry (which has the rotations of a tetrahedron and only three mirror planes) can be done in this technique if you make 4 "left-handed" and 4 "right-handed" triangles which assemble as an octahedron. (Each triangle abuts three triangles of the other handedness.) This is most easily accomplished in a tessellation program by choosing the "3*3" symmetry group. Below is an example of a layout illustrating this symmetry: (I drew the worm to touch the triangle edge midpoints, but that is not essential and it is not a 2-fold axis point.)

F) n orbits of worms. You can draw n disconnected lines in your tessellation and build with n shapes of worms, perhaps using a different color for each type. Everything above is a special case of n=1.