So when I was doing the math for the styrofoam-dowel tetrahedron, I noticed that locating the holes

Materials	Tools
12 3/16″ dowels	thumbtacks
6 4″ closed-cell styrofoam balls	thread, wire, or thin twist-tie
optional markers	ruler
	drill with 9/64″ bit

necessary for making a regular octahedron should be easy too. Naturally, having done the theory, it’s hard to resist actually building the thing. The materials are exactly the same as for the tetrahedron, just the quantities are different. In fact, you can re-use two holes from each of the balls you made for the tetrahedron.

Then you have to measure out two different lengths of thread or twist tie, tied from one shaft of a thumbtack to another. The first is 2.1 inches, just like before; that’s the length of the sides of the square of holes we’re going to drill. The second is 3.15 inches, for the diagonal of the square. (If that value is ringing bells in your brain, you are absolutely right: if you chase down all of the math that went into this, we’re going for exactly π inches from pin to pin here, but the accuracy of this process is way below 1/100 of an inch, so even 3 1/8″ is good enough, for example.)

For each ball, you start with two arbitrary holes separated by the shorter distance (which can be previously used holes). Then you scratch (appropriate arcs of) circles of each of the two radii from each of those two holes. They should intersect in pairs (the closer arc to one hole with the farther arc from the other) to create four points for drilling arranged in a square on the surface of the sphere. Here’s a picture of the two thumbtack setups of the two different lengths, and one ball with the scratches in place.

Once you’ve drilled all four holes in each of the balls (less any holes you might be re-using), it’s time to assemble. I recommend making two triangles first (one pictured on the right), ensuring that all pairs of unused holes point upward. Then use the remaining six dowels to insert in the remaining holes of one of the triangles, leaving six free ends up in the air.

Finally, flip over the other triangle and connect it as the top face, producing the final octahedron:

Theoretically, exactly the same procedures could be employed to make a section of the oct-tet lattice, but that would require a lot of scratching and drilling…

Recently a friend of mine was giving a (math) talk and wanted as a prop “a large tetrahedron with the vertices emphasized.” This seemed like a natural for Studio Infinity, so the assignment was accepted. And my immediate first thought was of the classic marshmallow shapes that you may have made in school or Girl Scouts or wherever.

Unfortunately, for use at the front of a large room, this would have to be scaled up considerably, and I didn’t relish the thought of trying to deal with humongous marshmallows.

Materials	Tools
6 3/16″ dowels, all the same length, e.g. 2 feet	thumbtacks thread, wire, or thin twist-tie
4 4″ closed-cell styrofoam balls	ruler
permanent markers, if you want to color the balls	drill with 9/64″ bit

So the natural substitutions were two-foot-long 3/16″ dowels for the toothpicks, and 4″ styrofoam balls for the marshmallows. (Use closed-cell styrofoam, it’s much more stable and easy to work with than open-cell.) However, unlike marshmallows, styrofoam balls aren’t squishy and forgiving, they’re rigid and brittle, and you can’t really just insert a blunt dowel directly into the fairly hard surface of a closed-cell styrofoam ball. If you want your corners to be colored, do that before you begin the construction (it will also help the scratches you make in the next step show up well).

So the first thing you need to do is locate the three points where a dowel will be inserted into each styrofoam ball. Fortunately, analyzing the geometry shows us how to find those points. So get out two thumbtacks and a piece of thread. Tie the thread to the post of each thumbtack so that the length of the thread between posts is 2.1 inches. Stick one thumbtack anywhere you like on the ball, stretch the thread taut along the surface of the sphere, and stick the other thumbtack in (again, anywhere you like, so long as the thread is taut). Now gently lift one thumbtack out of the ball, and keeping the thread taut against the ball, scratch a circular arc roughly where you expect the third hole to go. Then re-insert that thumbtack in its original hole, and do the same with the other thumbtack. The two thumbtack holes and the the point where the two scratched arcs intersect are the three locations to insert the dowels.

To finish the assembly, drill holes from those points straight toward the centers of the spheres, most of the way to the center, between 1.5 and 2 inches deep. Try to make your holes uniform in depth. Repeat the process of locating and drilling holes for the other three balls. Now it’s just a matter of connecting the balls together with the dowels. The holes are deliberately smaller than the dowels so that there’s a tight fit when you insert them — push firmly. I recommend building a triangle first, then inserting the other three dowels pointing up.

Then add the last ball to complete your tetrahedron. (And the tetrahedron shown below did in fact appear in my friend’s talk.)

So why the interest in tensegrity here at Studio Infinity? It begins with Kenneth Snelson, the inventor of tensegrity (although perhaps not of the term) as a student of Buckminster Fuller. Ken went on to become a noted sculptor, using tensegrity in many of his works. One of those sculptures, Free Ride Home, resides at Storm King Art Center. And recently, the education department at Storm King Art Center invited me to give a workshop there about tensegrity, how to make simple models of tensegrity, and how to share ideas about tensegrity with the public visiting the Center.

Materials
15 struts; recommend labeling 10 of them A through J
10 rubber bands

Thus, much of what you see here at S∞ reflects the thinking that went into that workshop. In particular for the occasion, I wanted to demonstrate a new tensegrity structure, and talk about its design process, as a means of strengthening the connections between math and art. This article will take you through building a model of that new structure.

Once you have your materials collected, you’re going to start by constructing a pentagon from five of the struts equipped with rubber bands. This pentagon will end up forming the bottom of the finished structure.

To build the pentagon, start by connecting one end of strut B to the rubber band strut A, one third of the way from one end of A to the other. Lie the two struts down so that B connects under the right side of A and extends off to the right, as shown. Then take another strut C and connect it to the right-side band of B, one third of the way along B from where B connects to A. Continue in this same fashion, connecting a fourth strut D to C in a similar spot, and a fifth strut E to D. At this point, what you have should look like this.

Now you can see that one free end of A is just in position to connect to E in the same way as all the others. So make that connection, and your pentagon is complete:

Now it’s time to make the top of the tensegrity structure, which is also a pentagon. This step gave me a great deal of trouble; I could make the pentagon just fine, but then the two pieces wouldn’t connect properly together. I struggled until the realization finally came that the top pentagon needs to be the mirror image of the bottom pentagon. So you need to reverse the earlier instructions.
           
Start by connecting one end of a strut G to the band of F one-third of the way from the end. Lay it down so that I connects to the left-hand side of F and extends off to the left. Now proceed to the left, connecting strut H to G, I to H, and J to I. Finally, strut J will connect back to F, producing the top pentagon which will look like this:

To continue the assembly, flip the top pentagon over and lay it on top of the first pentagon with the long ends of its struts lying between adjacent pairs of struts from the bottom. Now you’re going to “sew up” the two halves by connecting the ten free ends of the struts around in a circle. I will assume you’ve laid the top one down so that the strut labeled F lies between A and E on the bottom. Supposing so, you can start by connecting the free end of F to the rubber band of A. Connect it to the opposite side of A as the existing connection to A, and one-third of the way from the opposite end of A, as shown on the left.

Now just proceed around the circle, making connections in exactly the same way: the free end of G into B, the free end of H into C, I into D, and J into E.

Now you have to go back and connect the free ends of the other set of five struts. The free end of A is supposed to connect to G in exactly the same way as the previous five connections connections: on the opposite side and opposite end as the connection of H into G. It may seem ridiculous at first glance that the free end of A can get all the way over to where G now is, but just wrangle things around until it does. Then you “just” need to continue in the same fashion, connecting B into H, C into I, D into J, and finally E into F.

Unfortunately, the construction can get a bit tangled up as you go, as shown in the picture on the right, and you may find that some of the original connections are popping out. If you do, don’t despair; go back to the original pentagons and seal in the connections with additional pieces of tape on the other side of the rubber bands, as shown on the left. Then re-start the sewing process. In fact, the only reason not to simply take the precaution and secure the original connections as shown is if you think you may want to take your tensegrity structure apart again (and perhaps build something else).

In any case, if you persevere, you will eventually get to the last connection, of E back into F. To the right is a picture of what the structure should look like just before that last connection. Once the two halves are all sewn up, your structure should look like the following.

Now in theory, theory and practice are the same; but in practice, they differ, and here’s the point of divergence in this project. The theoretical plans for this construction called for a strut to extend between pairs of points as marked in red in the picture below (two of five such pairs are marked).

However, at this point I could see that the assembly already had most of the structure that I intended it to have; it was just taller and narrower than the intended icosidodecahedron. So most of what I wanted to accomplish was to spread out the middle of the nascent sculpture, so it would take on more of the desired ball-like form. And I wanted to do this in a way that would preserve the five-fold rotational symmetry that the structure so far possessed. So I shifted the new struts more toward the equator so that they would spread the structure more effectively, and planned them out so they would form roughly a pentagon.

So, to finish off the piece, take a new strut with no rubber band of its own, and connect it between the points marked in blue above. (You’ll notice the middle of the structure start to widen out a bit.) Note that there is a zig-zag band of ten short rubber segments (see the image to the right) around the “equator” of the structure, and you just have to connect the midpoints of two of those that are separated by two intermediate segments. The image at the left shows the structure with one such “expander” strut in place. The endpoints of the next expander strut will each be two of the zig-zag segments of that equatorial band away from the endpoints of the first, so that the two endpoints of the second strut are also separated by two segments. Continue adding the remaining three struts, proceeding two segments counterclockwise each time. You’ll find that none of the expanders interfere with each other, and each segment of the zig-zag band will receive exactly one connection. When all five of the expanders are in place, you’re done, and the finished tensegrity, dubbed the “icloseidodecahedron,” should look like this.
If you look closely, you can see the pentagon of unmarked struts in the middle of the above image. It may not be exactly an icosidodecahedron, but it’s not far off, and the resulting structure has pleasing five-fold symmetry, with a pleasant but not excessive complexity.

OK, so now we are going to try to force a tensegrity structure to take on the shape of a truly regular icosahedron. But we don’t have to search blindly for a way; we are armed with the results of this MathStream post about how to do it. To sum those results up, if we label the ends of the struts as shown in this diagram, then we need to place twelve rubber bands along the following paths: A1-C1-A2-F1-E1  A1-B1-A2-F2-E2  D1-B2-D2-F2-E2  D1-C2-D2-F1-E1  B1-E1-B2-D2-A2  B1-F1-B2-D1-A1  C1-F2-C2-D1-A1  C1-E2-C2-D2-A2  E1-A1-E2-C2-B2  E1-D1-E2-C1-B1  F1-D2-F2-C1-B1 and F1-A2-F2-C2-B2.

Materials
6 labeled struts
12 rubber bands

These building directions are not as intuitive as the ones for the original not-quite-icosahedron, so I strongly recommend that you take your six struts and label the ends clearly before you start to build anything.

The first rubber band is easy: just take the ends of the struts named and space them out equally along the band, as shown below. Note this means that you need to leave more space on one side of strut A than the other, because there is only one connection between A1 and A2 on one side (namely C1) but two connections on the other side (F1 and E1).

The second rubber band is trickier: you have to make sure to leave the shorter segment of the band on the opposite side of strut A as last time, and stretch the longer segment around to connect to the far end of the E and F struts.

Before inserting the third or fourth rubber bands, it’s helpful to place the D strut among the others in roughly the position it will end up.

Here it is with the third rubber band (which really doesn’t yet want to stay in place).

And here it is after four rubber bands, which makes everything much more snug and gives the nascent structure much more shape.

At this point, it becomes much easier to simply wrap each rubber band around the five points it is supposed to connect, trying to keep an even amount of tension on the rubber as you go. Here’s the structure after two more rubber bands (the two that each intersect both ends of the B strut) — we’re half done!

The following two pictures show the state after eight and ten rubber bands, respectively.

    

And here we are upon first getting all twelve rubber bands in place. It doesn’t look regular yet, but maybe it just needs tweaking the rubber bands to make it symmetrical.

However, here we are after quite some time spent pulling rubber into the segments between the ends of parallel struts, to try to lengthen those segments and shorten the other segments. At this point, the rubber bands are getting pretty loose between the parallel struts, and taut to seem near the point of breaking on the other segments.

So we reach the inescapable conclusion that you can’t force a tensegrity structure on six struts like this to adopt the shape of a regular icosahedron. Is that a disappointing result? Quite the contrary; it’s always kind of exciting when things don’t work out the way you expect. It often means that something deeper is going on, waiting to be unraveled. And in retrospect, we had a clue from the original structure that this might not work out. Namely, you can observe in that structure that the ends of parallel struts are closer together than the ends of non-parallel struts, even though there are no direct rubber-band connections between these points. Intuitively, adding rubber connecting the ends of parallel struts can’t make them move any further apart, only bring them closer together. Hence, since we needed to spread those points apart to change the original tensegrity structure into a regular icosahedron, we were doomed to fail.

Thus, there remain mysteries in even the most familiar of tensegrity structures, that Studio Infinity will return to one day. Is there any way to create a truly regular icosahedron with tensegrity?

Studio Infinity is kicking off with a group of articles that feature the concept of tensegrity. What does that mean? A three-dimensional structure displays tensegrity if its compression members — rigid components that can hold their shape even when pushed inward by external forces — are connected only by tension members — materials that become taut when pulled outward, but which deform when compressed. For example, compression members might be wooden sticks (also known as struts) and tension members might consist of fishing line. The remarkable fact is that in some arrangements, even though no two of the rigid compression members touch, the structure as a whole still has a well-defined three-dimensional shape (even though any single connection between the rigid pieces, left to itself, would collapse).

We need to start with an example so you can see tensegrity in action; this example will also establish some basic techniques for cheaply and easily creating small tensegrity models.

Materials	Tools
12 wooden coffee stirrers	Scissors helpful
Tape
6 rubber bands	(See text for details.)

So, even though you can find this model many other places on the web, we’re going to start with a classic six-strut, highly symmetric, tensegrity structure.

For this construction, our struts will consist of pairs of coffee stirrers taped together near each end, and the tension members will simply be rubber bands. However, the lengths of the struts and the rubber bands need to correspond properly. With the common nominal seven-inch size of coffee stirrer, standard #16 rubber bands that you can get in bulk from a variety of office supply vendors work well. (Incidentally, you can likely get the coffee stirrers from the same vendors.) If you have different length stirrers or are using different struts (for example, those take-out chopsticks accumulating in your kitchen drawer might work well), then you’ll have to experiment with various rubber bands to find a combination that works well. If the bands are too long, then the connections will be too loose and the structure won’t hold its shape; if they’re too short, it will be difficult to assemble, and bands or struts may snap in the process.

There’s also a question of what tape to use. I like to work with gaffer’s tape, as it is strong, rips straight in both directions so you don’t always need scissors to cut pieces, and seems to stick when you want it to, yet come off when you need it to. However, gaffer’s tape is relatively expensive; electrical tape or duct tape are good alternatives. Masking tape generally doesn’t work well (in fact, for anything except for masking, really). If you get a roll of tape that is approximately one inch wide, then cutting it across the direction of the roll to produce pieces between 1/4 and 1/2 inch “long” (so that really the one-inch direction becomes the long dimension of the pieces you cut) produces results that work very well. You want to be able to wrap the tape around the stirrers one and a half to two times.

Now that you have all your supplies, you’re ready to make the struts. Here’s how. Take two coffee stirrers and place them flat against each other. Many of the stirrers will be slightly bowed or warped.
   
So if at either end the tips of the sticks angle away from each other, flip one or both over (or try different combinations of sticks) so that the tips are snug against each other when the sticks are placed flat side to flat side. (It’s OK if they bow out a bit in the middle, since we’re going to tape them together.) Since the stirrers are so cheap, just throw away any that are cracked or badly deformed.

Holding the pair of stirrers together, you want to wrap them securely with tape at least one inch in from each end, as shown. You want all of these wrappings to be the same distance from the ends of the sticks, so you should use some kind of a template. I simply place the widest part of my thumb at the end of the stick and wrap the tape just below that, quickly giving me consistent spacings. But watch out — if your thumb happens to be less than an inch wide (not uncommon, especially if you’re young) using your thumb as a template will leave the tape wrappings too close to the end of the sticks. Then the rubber bands won’t attach securely, and your tensegrity structure will tend to slip apart, making building it very frustrating. So make sure you use some template that is at least an inch long to space the tape wrapping from the ends consistently. It could be the width of two fingers (always use the same ones) or the width of the tape roll, or two marks you make on a sheet of paper lying on your work surface, or anything. Any consistent spacing between an inch and an inch and a half from the end of the sticks to the tape will work fine.

Wrapping one pair of sticks tightly together with tape near both ends produces a single strut, and for this structure, you need to make six struts.When you’re done with the struts, it is time to add the tension members. Slip a rubber band between the ends of the two coffee stirrers at one end of the strut, and slide it all the way down to the tape. Then stretch it so that it reaches the other end of the strut, and slip it between the two stirrers on that end as well. It should spring back to snug against the tape on that end also, although it may need a little encouragement to slide down all the way. That gives you a strut with two segments of rubber band exposed, one along each side of the strut.
                
Ideally, you want the same amount of the rubber band to end up along each side, so there is the same amount of tension on each segment. So try to space the two connections of the rubber band to the strut evenly along the band now, although it will be possible to adjust later.

Once you’ve equipped all six struts with rubber bands, you’re ready to begin assembly of the structure. Take two struts (let’s call them A and B; if it makes it easier to follow along, you can pencil the letters onto the struts) and slip the midpoint of one of the rubber bands of A into the space between the stirrers of one of the ends of B, as shown. We’ll call that “connecting B to the midpoint of A”. Next, take a third strut C and connect it to the midpoint of (the rubber band on) the other side of A, like so.

Now you should take a fourth stick D and connect the opposite ends of B and C to the midpoints of the two sides of D. That step produces a sort of “H” of struts, with the horizontal crossbar doubled.

At this point it is useful to know that in the completed construction, exactly two rubber bands will be nestled in each end of every strut: the original band it was equpped with, and the band from one other stick that it connects to the midpoint of. Moreover, the midpoint of every band will be connected to by some other strut. (This is a very pleasantly symmetric structure!) So, for example, struts B and C have already made all of the connections they are going to, but there are still connections to the midpoints of their bands that have yet to be made.

So let’s go ahead and make those now. Take a fifth strut E and connect one end to the midpoint of B on the upper side of the crossbar of the “H” shape, and then connect the other end to the midpoint of C on the same side of the crossbar.

Something wonderful and a little amazing has just happened here. The sticks are no longer all touching each other. To balance the tension on the various rubber bands, they have pulled apart into a stable three-dimensional arrangement. Seeing that happen is one reason that building this structure is so much fun. But we’re not done yet — it will become even nicer. So next, take the final stick F and connect one end to the midpoint of the other (lower) side of B and the other end to the midpoint of the lower side of C. That won’t change the shape much, although it should even it out a bit.

We’ve used all of the struts, but we’re not quite done. As you can see, the ends of struts A and D have not connected to anything yet, and there’s nothing connected to the bands of E and F. So that tells us exactly how to finish off: connect the two ends of A to the midpoints of E and F on the same sides of those two sticks, like so:

And then connect the two ends of D to the midpoints of E and F on the other side. If all of your connections are evenly spaced around their respective rubber bands, you will have produced a very symmetrical structure that looks like this:

On the other hand, it’s possible that what you’ve created at this point looks more like this:

If so, no worries — it’s only because your rubber bands aren’t spaced out evenly. This situation is easy to fix, and it’s completely intuitive. Any place where the ends of two sticks are closer together than elsewhere in the structure (the segment of rubber band is shorter than elsewhere), simply slide some of that connecting rubber band from an adjacent space into that connection. It’s generally easiest to do that by holding onto the rubber band in the short segment with one hand, and pulling the end of the stick along that rubber band with the other.

Conversely, any place that there is too much space between the ends of two sticks, you’ll want to slide some of that connecting rubber band into an adjacent connection, to make the long segment shorter.

With a number of tweaks like these, you should be able to get your tensegrity structure into a shape that basically looks the same from any angle; here’s another view of the same completed structure as above, in which it’s been rotated significantly from the previous picture (and yet looks nearly the same).

If you’re wondering what shape that is that we’ve just made, check out Studio Infinity’s first MathStream article.

But whatever you do, you should take a moment to appreciate what you’ve just built. No two of the struts touch each other. They’re only connected by floppy rubber bands. But the structure doesn’t collapse, and it doesn’t fall apart. Each segment of rubber band is “trying” to contract to its natural length, and the balance among all of those bits of tension compels the entire structure to adopt this particular shape. In fact, try squeezing the structure however you like, and then letting go. This tensegrity will always snap back to the same form. Hopefully you can see why, when inventor Ken Snelson and his colleagues in Buckminster Fuller’s studio first encountered these structures, they were pretty enthralled. And stay tuned; there’s more, new tensegrity fun in store here at Studio Infinity.