31 Jul

“Marshmallow” Snub Dodecahedron

Here’s a a student-built snub dodecahedron that resulted from a session I led in July 2018 at The College of New Jersey. It uses the classic “marshmallow and toothpick” construction technique, just with styrofoam balls in place of the marshmallows and 1/8″ diameter dowels in place of the toothpicks. For geometric accuracy, the students did a ruler-and-compass construction on the surfaces of the styrofoam balls to find the locations at which to poke holes for the dowels.

05 Apr

Boxtahedral Tower

763 6″ cardboard boxes:
232 each of 3 colors,
67 plain
2000 twist ties with beads
packing tape, roughly 1 mile

At last the day came for the installation of the Boxtahedral Tower at the Golden Gate Stem Fair. Here are all of the materials waiting to be set up.
The build started off smoothly, with double rows of interlaced boxes quickly turning into trusses.
The struts came together to form the top peak, which by the end of the first day, had turned into a tetrahedron.

On the second day of construction, the first of the so-called “unknown unknowns” hit. Here’s the construction at the end of that day:

Doesn’t look much different, does it? If you notice, though, it’s up on chairs now. When we simply tried to lift it up there at the beginning of the day, the horizontal members all but fell apart. When we tried to figure out why — after all, the struts had tested out pretty rigid in prototyping — we discovered that the packing tape barely stuck to the paint. It stuck very well to the untreated cardboard, but basically pulled right off of the paint. And it had never even occurred to test whether the tape stuck differently to painted or plain cardboard. So, to make a long day two short, it was spent solely on replacing the centers of horizontal struts with unpainted boxes, which took the tension beautifully, so that at the end of the day we were at the point we expected to be five minutes into the day.

The third day brought another unknown: the ceiling height. Turns out there was not five meters of height available in the venue. Fortunately, we could quickly scale down the bottom level by half, so that it became just the top half of an octahedron (which has a hexagonal footprint). So here’s a half-size triangular face waiting to be inserted under the tetrahedron (in place of one of the chairs). (Oh, and that means that the materials list above specifies more than was actually used, since it’s for the full tower as planned.)

And thanks to the cheerful diligent participation from all of the people pictured below, including the director of the Golden Gate STEM Fair, Marcus Wojtkowiak, (but also many others pictured and not pictured, too numerous to list), we completed the world-premiere installation of the Boxtahedral Tower.You can see it’s just about brushing the ceiling.

And here’s the obligatory shot looking up at the ceiling in the center of the structure. One of Studio Infinity’s finest constructions!

02 Jan


The title stands for “rhombic hexecontahedron of dodecahedra,” and that’s exactly what Matt Parker and I built at Studio Infinity over the 2017-2018 New Year’s break. Here’s a photo of the finished product; this post will be fleshed out further as time permits.

07 Nov

Stretching the Point

So far, we’ve created a lot of interesting small models of tensegrity structures. However, for doing public programs of the sort Storm King Art Center was planning, it’s always helpful to be able to build much larger models of things. Building giant models seems to get the ideas across more vividly, engage visitors more thoroughly, and be just plain fun.

So how to stretch the scale of these little models? I’ve experimented with using dowels instead of coffee stirrers and giant rubber bands, or sections of PVC pipe and bicycle-rack bungee cords with hooks, and several other similar systems. But Warning!: Studio Infinity does not recommend that you try any of those materials. They all share one dangerous flaw: If one connection slips, it’s all too easy for a portion of the structure to turn into a sort of crossbow, ejecting one of the rigid compression members at high velocity and potentially striking one of the people working on the structure. There is a significant possibility of injury. Do not attempt to build such structures.

Now, why wasn’t this much of a problem for Ken Snelson and his crew when they were building his sculptures? The answer is that he connected his compression members (generally large aluminum tubes) with steel cable. The steel cable has very little give, and hence there is very little opportunity to catapult one of the tubes; there will be very little motion even if tension is accidentally suddenly released on one of the cables.

However, the flip side of using a tension material like steel cable is that the tension members have to be measured very exactly for the sculpture to adopt its desired form. Too long, and the sculpture will simply hang slack. Too short, and there will be no possible chance of assembling the structure. We use stretchy materials for our models because they are very forgiving: if a section of rubber band is too short, we can just stretch it a bit extra to get the model together, and then adjust lengths/tensions afterward. That method is not possible if you’re using steel cable connections.

Materials Tools
15 cardboard tubes Drill or drill press
5/32″ bungee cord 7/32″ drill bit
box cutter

So here’s the construction method I shared with staff at Storm King Art Center. Let’s make a set of large pieces for an icloseidodecahedron. The key is to use cardboard tubes as the compression members: they’re very light, have no hard or sharp corners or edges, and yet are rigid enough to take a lot of compression from the attaching cords. You can get the cardboard tubes as mailing tubes from shipping supply companies, or as poster tubes, or even as the central tube of gift-wrapping paper (but note that often the latter kind of tube has thin walls that might buckle under compression). For the tension members, you can buy a spool of thinner-profile 5/32″ marine bungee cord. That material helps with safety as well; it just doesn’t pack the same elastic punch as its thicker cousin typically used for bungee cords with hooks.

Now, you’re going to need to drill through the walls of the cardboard tubes in order to fasten the bungee cord. To prevent the tube from rolling around as you drill it, you’re going to need to fashion some sort of a jig to hold the tube in place. You can either use whatever triangular blocks you happen to have lying around (hurray for saving the leftover bits from earlier 3-D printing projects), or you can make do with rectangular blocks as well. I recommend a drill press for this task if you have one, as you will get neat pairs of holes directly opposite each other on the cylinders. However, you can certainly make do with a hand drill, as this setup shows. You just may find that your holes don’t line up quite as well.

As far as what size of tube you should use, that’s really limited only by what you can find and handle without difficulty. You want a fairly narrow tube, with strong side walls. I have tested nominal 2-foot by 1.5″ mailing tubes (which are actually almost 25 inches long) and the center tubes of 30-inch wide wrapping paper (which again are a bit longer than 30 inches).

Once you have your jig set up and your tubes ready, the drilling is simple. You just want to make two parallel pairs of holes at each end of each tube. One pair should be two inches from the end of the tubes, and the other pair should be an inch and a half from the end. After you have drilled both holes, you want to create slits (on both sides and both ends of each tube) from the hole closer to the end of the tube leading out to the end of the tube. To make the slits, insert a box cutter in the hole with the blade pointing away from you and toward the end of the tube, and push the cutter away from you out to the end of the tube.

Next, we need to string the tubes with the marine bungee cord. For the two-foot mailing tubes, you need a 34-inch piece of the cord; for the longer tubes, a 42-inch piece worked well. Thread the cord through both holes at one end of a tube, and center the tube on the piece of cord. Then thread each end of the cord back into the tube via the hole on the opposite end of the tube but the same side, so that the cord lies flat along each side of the tube. To secure the cord, reach into the tube where the two loose ends are, pull both ends out as far as you possibly can, and then tie an ordinary overhand knot in the doubled cord. When you let go, the knot will disappear into the tube. If there are bits of bungee cord sticking out from the end of the tube, just stuff them inside.

When you’ve accumulated a pile of 15 struts strung with bungee cord, you’re ready to build.

Actually note that only 10 of them need the cord, as no struts connect to the last five “expander” struts. Putting them together uses exactly the same procedure as with the small coffee-stirrer struts; here are a couple of intermediate snapshots.

And here’s the final large-scale icloseidodecahedron:

I’ll close out this series on tensegrity with some images of the staff at Storm King Art Center building their own large-scale six-strut tensegrity structures, and proudly displaying their results.


31 Jul

Polyhedral Loops

Here’s a closeup of some snub dodecahedra that students built as part of a group construction I led at The College of New Jersey in July of 2017. They are each approximately 80cm in diameter.

And here are five of them connected by pentagonal faces into a “V” configuration.

If we’d had time to build three more, they would have closed into a loop, just like these dodecahedra:

Here’s another loop, this time of icosahedra, that the students made along the way. (Platonic ideal alert: this loop is not mathematically exact, it’s just close enough that it can be constructed with these somewhat flexible units.)

07 Apr

Octahex Ring

Here are two large-scale group constructions I facilitated the construction of as part of a session at a 2017 MAA Spring Sectional meeting at Frostburg University. The first is the one I have dubbed the “Octahex Ring” — it’s a compound of a dozen octahedra:

And the second is a higher-genus compound of 22 octahedra:
23 Apr

Elevated Icosidodecahedron

Here’s an installation I designed for the National Museum of Mathematics for the 2016 USA Science and Engineering Festival in Washington DC. The only construction materials are dowels, off-the-shelf end caps, and zip ties. It’s not actually a fully elevated icosidodecahedron; only the pentagonal faces are elevated.

And here’s another one from that event with the same construction technique; this one is a rhombicuboctahedron with its square faces elevated:

22 Dec

Solstice Star

A picture from the top of the Flatiron Building in Manhattan, NY of an installation I designed for the National Museum of Mathematics for an observation of the winter solstice. It was geometrically appropriate, in a way, because the highest angle the sun reaches in the sky on the winter solstice is quite close to the vertex angle of the regular heptagram approximated by the arrangement of lights in the photo.