12 Mar


When I showed this recent post to my friend and colleague Laura Taalman, aka mathgrrl, she suggested that another approach to creating a model of the underlying structure would be to construct the icosahedra themselves (rather than the negative space), except use wireframes of the icosahedra rather than solid ones to avoid obscuring all of the internal structure. Her encouragement motivated me to create a new OpenSCAD file for this. Taking pity on the small size of the build plate of the 3D printer that I have access to, Laura even printed the resulting STL file for me. Her machine produced the following lovely results:

Whichever one you like better, this or the Anticos, I think it’s safe to say that the two models bring out different aspects of the same edge-to-edge arrangement of icosahedra. For example, this “wirecosahedra” structure shows better how the edges of neighboring icosahedra coincide. On the other hand, I think if we had only produced this model, it would be hard to discover the octahedra lurking in the corners between the icosahedra — they get a bit lost among all of the edges.

Thanks to Laura for providing another perspective on this arrangement of Platonic solids!

09 Mar


Judging from at least one of the previous projects, Studio Infinity is intrigued with connecting polyhedra edge-to-edge. (Of course, connecting them face-to-face is interesting, too, but that’s pretty familiar from Legos and such; and vertex-to-vertex is the same as connecting dual polyhedra face-to-face.)

As you can see in the “blueprint” for the Boxtahedral Tower at right, connecting solids in this way often highlights symmetries that might otherwise be overlooked. In that case, edge-to-edge connections illuminated the threefold symmetries of cubes.

Even more surprising is that the most spherical of the Platonic solids, the icosahedron, can nestle into a cube with six of its edges just brushing the faces of the cube:

That means that if we connect icosahedra edge-to-edge, we should be able to make rectilinear structures, like with ordinary building blocks:

I wanted to find a way to show what was going on in this structure; but it seems as though the individual icosahedra get in the way of seeing the overall picture. You can’t see many of the actual edge connections at once.

Sometimes, however, what’s not there is even more illuminating than what is. Thus, the “Anticos” emerged: the negative space of the above configuration.

(More precisely, I enlarge each icosahedron by about 35% before carving it out of its respective cube. That step creates the “windows” in the sides of the cubes that lets you see the internal structure — without the enlargement, there would just be an infinitesimally narrow slit in each surface, so you would see nothing but a stack of cubes.)

Although usually S∞ focuses on structures that you can build by hand with ordinary materials, for something this intricate and detailed 3D printing seemed to be the only reasonable route to get a physical example in a reasonable amount of time. So here is the OpenSCAD source file and the resulting STL file. Since all of Studio Infinity’s content is Creative Commons-licensed, feel free to modify however you may like, and print if you have access to a 3D printer.

Now the icosahedra are revealed to embed in a lacy network of octahedra connected by rhombic prisms:

Although after the fact it may not be a mathematical surprise that octahedra show up between the icosahedra, it’s still visually striking, and I’d say that creating the physical model heightens our appreciation of how they all fit together. Moreover, physical models can reveal other surprising aspects of the underlying geometry. For example, this half-height prototype of the Anticos reveals a lovely network of hexagonal cross-sections of the packed icosahedra:

Hopefully, S∞ will have much more to say about connecting polyhedra both face-to-face and edge-to-edge in the weeks to come.

08 Mar


This MathStream post about why an icosahedron inscribes in a cube also shows that a dodecahedron fits into a cube in an analogous way. That raised the prospect that it might also be worth building an “Antidodec” analogous to the Anticos. So I quickly mocked one up in OpenSCAD (here’s the two files you need), producing the following image of what it would look like:

Surprise! it’s just a rectilinear network of (infinite golden) rhombic prisms. While it’s striking and lovely in its own way that this structure turns out to be so simple, it didn’t seem that the result would be visually interesting enough to be worth producing physically. So in the end, this post is really just a footnote to the prior project.

07 Dec

Math’s Bubbling (not) Over

Here’s a large-scale model I designed of the Weaire-Phelan space packing, built by the participants of the Fall 2019 semester on Illustrating Mathematics at ICERM in Providence. The title above is a reference to the fact that it is still not established whether this is the most surface-area parsimonious way to divide space into cells of equal volume, like an ideal foam in which each bubble encloses the same volume.

17 Sep

Woven SSD

It’s high time that S∞ got back to its core: mathematical constructions you can build. Here’s an attractive star-shaped polyhedron made with a weaving technique that I am indebted to Jürgen Richter-Gebert for introducing me to. It’s called the “small stellated dodecahedron,” and is one of the MaterialsTools30 equal-length rodsScissors32 rubber bands

To build this, you first need 30 rods all the same length, made of some material that has a bit of give (hint: not glass rods). You can choose almost any material; here, I’ve used mostly aluminum tubes, with five contrasting brass tubes of the same diameter to highlight the first star you make. Possibilities include thin dowels, bamboo strips, pipe cleaners, and more.

You also need 32 rubber bands. They should be sized so that it’s easy enough to get them around five rods simultaneously, but then they should hold that thickness of material securely (of course you can double or triple up the bands if need be). You can use almost any type of elastic bands: ordinary office rubber bands, colored hair elastics, or rubber O-rings as I’ve used here.

Before you get started assembling the SSD, you should mark the locations that the rubber bands will go on each rod. To find these locations, first pick an amount of each rod that will extend past each vertex for security of the connections. For example, I used 18-inch rods, so I chose that the rods would extend 1/2 inch past each vertex. If you are making a larger version, you’d want to pick something longer; if you are making a smaller SSD, I wouldn’t go much below 1/2 inch because the vertex connections might start slipping apart too easily.

Subtract twice the amount you are leaving at the vertices from the length of the rods to get the distance between the vertices. In my case, that’s 18 – 2×(1/2) = 17 inches. Now multiply this value by 2/(3+√5); for 17 inches, this gives about 6.49 inches. (See the MathStream post that explains this calculation.) Finally add back in the amount you’re leaving at the vertices; this step gives me 6.99 inches, which we should of course round to 7 inches. That’s the distance you want to mark on the rods from each end (two marks per rod). With my 18-inch rods, I just lay them next to a ruler and mark at 7 and 11 inches.

Ok, now all is ready for construction. The basic step you’ll be doing over and over is placing two rods side-by-side and putting a rubber band around both of them and sliding it down to the first mark on the rods.

We’ll call this basic unit a “linked pair.” Take two linked pairs and put them side by side with their connections on the same side, and put a third rubber band around one rod from each pair at the ends opposite the existing connections. Slide the rubber band down so that you have a zig-zag chain of four rods.

Connect a fifth rod to one at either side, continuing the zig-zag.

Now connect the free end of this fifth rod to the free end of the first rod of the zig-zag, as shown on the right.

This is a configuration that you can now wiggle around until it forms a five-pointed star, as shown to the left. Important: looking at the two junctions on each rod, the rod must be underneath at the first junction and on top at the second junction as you proceed clockwise around the inner pentagon. Make sure that every one of the five rods follows this pattern before continuing, It’s this under-over, under-over pattern that constitutes the weaving in this construction, and that same under-over, under-over will continue throughout the entire construction.

Next, at each of the points of the star you’re forming, you can slip one rod under the other to continue the under-over junction pattern of the rods. These points of the star will eventually be vertices of the SSD.

Add a rubber band to secure each of the points, and congratulations, you’ve just made your first star out of 12.

To continue toward the SSD, take five rods and line them all up and put a rubber band around all five of them and slide it down just a short way, roughly the amount you decided to leave at the vertices above. You should then be able to splay the rods out so that the far ends are at the points of a pentagon, and there is a little clockwise swirl of the near ends of the rods (see the picture to the right).

Place each of the far ends just counterclockwise of one of the internal junctions of the star you made, in order around the star (as shown on the left).

Insert each of the far ends into the rubber band at the corresponding junction, making sure to keep it within the triangle of the star that it started in.

Now push all of the five new rods through those junctions until the second mark reaches the junction. That completes the first full vertex of your SSD.

If you look at your structure in progress from the side, you will see three out of five sides of another star. We want to complete that to a full star. So take a linked pair, and link one free end of the pair to each of the two sides of the star sticking out. Push those links in a little way and make sure the junctions all continue the under-over pattern of the first star. If all looks well, you can slide all of the connections until they reach the marks.

To complete the second star, slip the points of the star past each other as you did with the first star, and secure with rubber bands (there will already be rubber bands for this in place at two of the vertices).

Now rotate the structure 1/5 of a turn around the original star. You’ll see another star with only three sides in place. You want to complete this in a similar fashion. The only difference is that this time, you only make one entirely new link; at the other end of the linked pair you’re adding, you feed the rod through the existing rubber band at an internal junction. Make sure to continue the same under-over pattern throughout. The new rod should pass through a triangle in the second star that does not yet have a rod passing through it (it should not pass outside the second star or through the pentagon of the second star).

When that one’s done, add two more linked pairs to complete two more stars in exactly the same fashion.

The next linked pair is very similar to these; you just have to feed both of its free ends through existing junctions. When that’s done, you’ve finished six of the twelve stars in the SSD.

To continue from here, take a single rod (not a linked pair) and feed it through any of the partial vertices and the nearest internal junction that only has two rods at it, as shown.

Adding that rod will create a new partial star with three edges in place. Complete that one just the way you did the third star.

You’re getting close now. Complete two more stars like the third, and then one like the sixth.

That leaves just a single line of four junctions that does not have a rod going through them (shown down the middle of the image above). Carefully insert the final rod through all four junctions – it’s probably easiest to pass it through the two internal ones first, then tuck the two ends in at the vertices, to produce something like this. (This last edge is again down the middle of the image below.)

Now comes the exciting final part: cut away all of the rubber bands at the internal junctions. They were just there to stabilize things during construction; they’re not actually needed to hold the SSD together once it’s built, thanks to all of that weaving that we did. And voila!

In fact, if you used a high-friction material like bamboo strips or pipe cleaners, it’s often possible to cut away the rubber bands at the vertices as well. I don’t have a picture of this, as the metal rods are too slippery for the vertices to hold. If you want to try, just proceed carefully, and if any of the vertex junctions “pop,” tie it back up immediately and don’t cut any more (unless you want to quickly revert to a pile of separate rods).

If you like this construction and happen to be in Providence on 2019 Sep 28, come to the ICERM Open House at WaterFire Providence and help build a glowing version with 10-foot pipes for the rods!

29 Jul

More Life at a glance

A few days after the event at TCNJ, students at the PROMYS program at Boston University built another “Life sculpture” in which each layer is a generation and time proceeds downwards. Here, we explored questions of how you might know things like whether the resulting “sculpture” would be connected, or whether it would be self-supporting. For these types of questions, what one really needs is to solve the (more computationally thorny) “inverse Life” question: what colonies of cells can give rise to a given configuration in the next generation?

This sculpture begins with a pattern in its top layer that will eventually result in four Life “gliders” proceeding in different directions, which would then serve as four “legs” for the sculpture to stand on. Unfortunately, we didn’t quite have time to build enough generations to see the four gliders diverge.

27 Jul

x-y-t Life

Here is a photo of the first 13 generations of the evolution of the “R” pentomino pattern in John Conway’s Game of Life. Each layer represents one generation, and time proceeds downwards. In each layer, live cells are represented by boxes. The color of the box indicates how many generations that cell has been alive: yellow for one, orange for two, and blue for three or more. These “Life sculptures” were built by middle-school students at The College of New Jersey in 2019 July.

(If these boxes look familiar, you’re right – they are the same ones from the Boxtahedral Tower reused for a completely different construction.)

17 Apr

Spectral Snub

Here’s an image from inside one of a pair of mirror-image snub dodecahedra built by passersby on the Harvard Science center plaza in 2019 April. The completed work, “Spectral Snub”, was on display inside the Science Center for the following four days. Photo courtesy of Stepan Paul.

07 Nov

Deltahedral Torus?

Here’s a torus built from equilateral-triangle Geometiles that I used as a prop for an undergraduate talk at Harvard University in the Fall semester of 2018. Actually, the structure it is based on is not mathematically exact; the triangles theoretically are isosceles triangles of sides 1, 1, and 0.998, but there is plenty of give in the real world to construct it physically. As far as I know, it remains an open wuestion whether there is a true deltahedral torus in which all vertices are hexavalent.

21 Oct


This is a placeholder post for pictures of an installation I led on 2018 Oct 21 at the Mathematical Sciences Research Institute, entitled “Tetrahelix”. It consisted of a double helix, one strand of which was composed entirely of regular tetrahedra connected face-to-face (such compounds can reach any point in space and come arbitrarily close to closing in a loop but can never make a mathematically perfect loop), and the other strand of which was the combinatorial dual of the first, realized by a geometric structure that can only be thought of as a “polyhedron” in a relaxed way. When I get a chance, I will post the construction techniques and math behind this installation.

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.