the heart of Studio Infinity, MakeStream articles show structures – and frequently how to make them – that exhibit interesting design ideas, often inspired by mathematics.
Over fifty members of the Occidental College community, organized by Prof. Jim Brown of the Math Dept., came together (Friday, 2023 Nov 10) to construct a Sierpinski-style fractal based on the regular octahedron. You can read more about it in the The Occidental weekly newspaper; more details will be posted here as time permits. But here’s the completed structure:
This construction event closed with a dramatic denouement. Studio Infinity had recommended an ambitious plan to support the resulting 17-foot-tall structure on a single vertex of the outermost octahedron, facilitated by the custom wooden cradle (fabricated by Pablo and Ricardo Lopez) that you can make out at the bottom center of the photo. This scheme worked perfectly through the third iteration of construction (out of four doublings of the size of the structure). But then:
This sculpture was just on the edge of stability — the structure was self-supporting up until the very final module, and the team was able to lift and carry it to place it atop that final module. However, once those lowest connections were made, the torque on them proved too great for the tolerances of the system being used.
The upshot is that if Studio Infinity leads another build of this intriguing fractal, it can be placed on one of its faces, rather than on a vertex, and it will remain stable. That feeling of real-time experimentation is one of the things that makes public sculpture-raising like this so exciting!
I’ve been the primary architect of a number of constructions of
So when I discovered that the Lakeshore Learning Ultimate Fort Builder construction toy allowed rods to be connected at all of the angles necessary to construct arbitrary portions of an octahedral-tetrahedral lattice, I knew that I had no choice but to eventually seek yet again to scale this tower of three-dimensional fractality.
The opportunity came in the fall of 2022, when my colleague Prof. Alissa Crans put me in touch with Profs. Derek Smith and Ethan Berkove, who were interested in holding a collaborative mathematical construction outside the math department’s building on the Lafayette College campus. After reviewing a number of proposals, they (to my delight) selected a Sierpinski tetrahedron constructed from the Fort Builder toy. They felt this structure would tie in to a number of classes being held that semester, including one on combinatorics — just how many rods and joints are there in that pyramid, anyway?
This post will be fleshed out as time permits, but for now suffice it to say that the construction — quickly dubbed Fort Sierpinski — came together very satisfyingly (more below).
The final structure came in at just a few inches shy of twenty feet tall, making it the largest Sierpinski tetrahedron I’ve been involved in yet, as well as the highest-order approximation: there are five generations of tetrahedra, from the smallest unit tetrahedra to the construction as a whole.
If you look closely, you will find just a few connections where the ability of the hub-rod sockets to take tension trying to pull them apart had to be augmented with a little strategically-placed tape (as hinted by the items on the table in the foreground). Other than this kind of patching (after all, kids do have to be able to take their creations apart by pulling out the rods, and this giant building is putting a lot of tension on certain junctions), the Fort Builder toy turned out to be quite up to this gargantuan task.
For this construction commissioned by the Dickinson College mathematics department, we chose a construction technique that goes all the way back to a construction from 2016, but this time with a twist. All previous installations done with this technique used only one length of rod, producing rigid equilateral polyhedra. But I had long wanted to construct a rhombic enneacontahedron, a shape that George Hart introduced me to with one of his artworks. Although that polyhedron is equilateral, consisting of 90 rhombi of two types, it is not rigid by Cauchy’s Theorem on polyhedra: each face is a quadrilateral, and so no face is by itself rigid. Hence, it’s necessary to brace the rhombic faces. Since there are two different face shapes, that means there are two additional rod lengths for the lengths of diagonals of those two different rhombi.
Further, the sponsors decided to proceed with a design in which the rods were colored to correspond to the ice caps, land masses, and oceans of our planet Earth, in honor of the world in which we live and a contemporaneous conference on climate change and other ecological issues.
This post will be fleshed out as time permits, but here’s a picture of the completed construction on site at Dickinson college (more below):
This photo is taken from a viewpoint over the southern tip of Mexico and the Central American land bridge to South America (since there are “only” 180 edges to this polyhedron, the representation of land masses is by necessity quite approximate). For comparison, here’s an image of the pre-build rendering of the Rhombiglobe from essentially the same viewpoint:
You may notice the actual construction has a decidedly “non-spherical” aspect on its left, in the midst of the Pacific Ocean. That phenomenon was perplexing in the midst of the build, but later analysis revealed that it was a direct side effect of there being nine different types of pieces: three colors of rod in three lengths each. One of the participants in the group construction inadvertently swapped one of the longer and one of the middle-sized blue rods in the neighborhood of the bulge. The result was that by dint of perseverance, we managed to construct a geometrically “impossible” polyhedron, which naturally adopted a non-spherical shape as it attempted to resolve all the extra stress that the incompatible edge lengths created.
I will be leading a group construction of a 3.4m diameter rhombic enneacontahedron next Tuesday, 2022 Oct 11, on the campus of Dickinson College. This is the first time that I will be leading a construction using this system — dubbed ZipStix at the NYC math museum — with rhombic faces rather than regular polygons. So to prepare, last weekend I built the prototype you see in the diagram at right. It has twelve faces (hence the post title) and uses both of the rhombus shapes that appear in the enneacontahedron, as well as a third rhombus that is almost, but not quite, a square. Here are the complete build instructions.
Materials
Tools
24 wood dowels, 2′ long by 3/8″ diameter
small handsaw for cutting dowels
Enough additional dowels (e.g., 11 dowels 3′ by 3/8″) to cut the following lengths:
light hammer for tapping caps onto dowels
4 dowels 32 3/4″ long, 6 dowels 27 3/4″ long, and 2 dowels 17 1/8″ long
(optionally) angle brackets for backing up caps when tapping
72 plastic end caps for 3/8″ diameter dowels, similar to these
(optionally) instant-setting glue for caps that slide off during construction
14 “zip-style” cable ties
easily-removable tape (like painter’s tape) and markers for labeling parts during construction
First, measure the lengths of all the dowels you need to cut. Here’s one of the 17 1/8″ dowels being measured (you can get two of these from opposite ends of one 3′ stock dowel), and below that, all of the 27 3/4″ dowels ready for cutting.
Next, cut the dowels to the marked lengths, and insert each end of each dowel into an end cap. Depending on the tolerances of the end caps and the dowels, you may be able to do this by pushing the caps on firmly by hand, or you may need to tap them on with a light hammer. In the latter case, you probably need to brace the caps against something to avoid breaking the tabs on the plastic caps. So place a right-angle bracket that is at least as wide as the tab is long on a flat work surface (so that the right angle is against the table), place a cap with a partially-inserted dowel so that the flat part of the cap at the end of the tab is against one arm of the bracket, and tap the other end of the dowel. To get the second cap on, first make sure that the two caps are roughly “lined up” so that their tabs are more or less in the same plane. Then, you will likely need to hold another bracket against the end cap at the other end of the dowel to have a surface you can tap on (watch your fingers!).
At left you can see all of the dowels cut, capped, and ready to go, together with the bag of cable ties we’re going to use. The assembly will proceed from vertex to vertex in the diagram above — you probably want to print out a copy to be able to refer back to it as you go through the construction process. The edges (dowels) in the diagram are color coded by length: The longest edges are orange, the six slightly-long ones are green, the 24 two-foot edges are blue, and the short edges are purple. So in the directions below, we will refer to the dowel lengths by the colors in the diagram (even though the dowels themselves are not colored). Also pay close attention to the letters labeling the vertices in the diagram. Note there are A through G and B’ through H’; the ones with “primes” on them are different from the ones without. During construction, the outside of the polyhedron will be lying on the floor as you build these vertices, so you’ll be standing and looking at them as if you were inside the polyhedron. Therefore the order that the edges are listed for a given vertex is the order that you would see them surrounding that vertex if you were standing inside the polyhedron looking outward at the vertex.
So start (step 1) by lacing a cable tie through the holes in the end caps at one end of dowels of the following lengths in order: green, blue, green, blue, green, blue. That creates vertex “A.” Feed the cable tie through its own end to secure it, and pull it some of the way through but do not pull it all the way tight yet — we will tighten up all of the vertices at the end of the construction. You should end up with something like the picture at the right. And now, it’s very important that you label the vertex you just made as vertex “A” (mark an “A” on a piece of tape and wrap it around one of the dowels at the vertex) because you will need to refer back to that vertex later in the construction.
Put vertex A aside for now, and construct the following vertices (each one separately) in the same way. Remember to label each vertex as you make it! (Step 2) Vertex B’ with orange, blue, green, blue, orange, blue. (3) Vertex D’ with orange, blue, purple, blue, green, blue. And (4) vertex E’ with orange, blue, green, blue, purple, blue. At left are pictures of these three vertices (not labeled yet, so we had to go back and figure out which was which and label them after the fact — spare yourself that chore).
Now for (step 5) find vertex A and vertex B’ and arrange them so that the free ends of one of the blue-length dowels of A and the blue-length dowel of B’ that is between the two orange-length ones are near each other. Thread a cable tie through these free ends, including one fresh blue-length dowel between those two and one more fresh blue-length dowel after the second one. Close up the cable tie and pull it partway closed like you did with the first four vertices. Immediately label the new vertex you just created “B”. Your outcome should look similar to the picture at right (see, we finally got our labels straight!).
Now by a similar process, you should (6) join the next blue from A clockwise to the blue from vertex D’ between its orange and purple, with two new blues as before, to create vertex D. (Label it!) That gives you the following: (Vertex D is the leftmost labeled one.)
And (7) join the remaining blue from A to the blue-length dowel from vertex E’ between its orange and purple, with two new blues, to create vertex E, like so:
Now if you look back near vertex B that you created in step 5, you should find that the following edges all have free ends near each other in this order: a purple-length one from D’, a blue from D, a green from A, a blue from B, and an orange from B’. So (step 8) lace a cable tie through all of these in order, adding one new blue so that it will end up between the orange and the purple. Close up the tie, pull it partway through, and label that vertex C. Here’s what you get:
Warning: the next step is the first one in which our partial construction will no longer be able to lie flat on the floor. (Step 9) lace up the purple-length dowel from E’, blue from E, green from A, blue from B, and the other orange from B’ with one new blue-length dowel to create vertex F. You should get something like the following, where the vertex on the floor in the middle is B created in step 5 and the new vertex is at the bottom of the picture:
Now we have four more steps, each with just the same format as the last two. So we will list the four steps, and then follow them all by a series of four pictures showing the results after each one. (10) Connect the orange from E’, the blue from E, the last green from A, the last blue from D, and the orange from D’ with one new blue to create vertex G. (11) Connect a blue from B, the last blue from C, and a blue from D’ with one new blue to create vertex C’. (12) Connect a blue from E’, the last blue from F, and the last blue from B’ with one new blue to create vertex F’. (13) Connect the last blues from E’, G, and D’ together with the last unused blue to create vertex G’.
The finale:
And now, you should have exactly 6 free ends — the ones that connect at vertex H. Bring them together and run a cable tie through them. Voilà, you finally have a complete, fully three-dimensional shape. However, it won’t look as crisp as the rendering just yet:
To get there, first check that your structure has the same general shape as shown. If not, back up and try to see where your connections don’t match the ones in the rendering at the top of the post and make any necessary corrections. Once it has taken on an overall shape roughly like a squashed ball, it’s time to tighten it up. Start by picking any vertex and pulling the cable tie there as tight as possible, working to get all of the tabs neatly arranged around the cable tie. Then go to the vertex farthest away from the one you just tightened, and pull that one as tight as possible, keeping the dowels linked to it as neatly arranged as possible as you do. Continue in this same fashion, always switching to the farthest-away vertex that you haven’t yet tightened up, until you have firmly secured every vertex.
At this point, the structure should have taken on a nice, crisp polyhedral form. You should easily be able to identify the 12 planar rhombic faces, each one split into two congruent isosceles triangles by a diagonal. Here it is in essentially the same orientation as in the initial rendering, but in real life:
And as a bonus, to get a better sense of the shape, here it is resting on one of its narrow rhombus faces (and with the vertex labels removed):
I was recently invited at the last minute to lead a mathematical construction for a seminar for math majors at Loyola Marymount University. The hope was to create something physical connected with one of the topics in the course, which linked the history of mathematics with various unsolved problems, among other things. Since there had been a fair amount of discussion about the Pythagorean Theorem, we settled on the following construction that demonstrates an interesting and less-familiar related phenomenon in three dimensions.
Given any three lengths, you can build a tetrahedron with a vertex where three right angles meet, and the lengths of the three edges meeting there are as given. (Basically, just cut off the positive coordinate axes to the three given lengths and join the resulting endpoints with a triangle.) Such a tetrahedron is called a “right tetrahedron” and those three initial lengths are called the “legs.” The following construction (erecting a prism on each face whose height is the same as the area of the face) can be performed with any lengths for the legs, but all of the calculations below are done for legs √6, √19, and √30 (which have the pleasant property that the sides of the fourth tetrahedron face are then 5, 6, and 7, as shown in the diagram to the right). To create a human-sized result, I used a decimeter as the length unit; if you wanted to make this into a tabletop-sized construction, you could scale it down by a factor of four or five — but note that when scaling down the edge rod lengths (as opposed to the altitude lengths) you need to add one centimeter, divide by your scale factor, and subtract off the one centimeter again, to allow for the extra length created by the connectors.
Materials
Tools
About 30 1/8″ diameter rods, at least 147 cm long (5′ suffices), for example wooden dowels or fiberglass rods
Measuring tape
About 90 custom connector clips (STL file, or OpenSCAD file if you need to tweak them)
Meter sticks
Three sheets of foam core, at least 70 cm by 50 cm (30″ by 20″ suffices)
Cutting pliers or small saw (for cutting rods)
Plastic wrap (ideally four colors of industrial-size rolls)
Box cutter (for foam core)
At least 225 liters of loose fill material (e.g. packing peanuts); 9 cu. ft. suffices
Scissors
If rods expand after cuts:
Optional: thick paper or cardstock for temporary lids
Gripping pliers
Drill slightly larger than 1/8″ (e.g. 9/64″)
Assemble all of the needed materials and tools, and fabricate the connectors per the supplied STL file. [vrm360 model_url=”http://studioinfinity.org/wp-content/uploads/2022/04/DowelSnapV6.stl”]
A row of freshly printed connectors before removing the support material
One finished connector ready for use
Length (cm)
Quantity
For
147
3
altitude D
119.4
3
altitude C
69
9
edges of C and D
67.1
3
altitude B
59
8
edges of B and D
53.8
7
edges of B and C
53.4
3
altitude A
49
8
edges of A and D
42.6
6
edges of A and C
23.5
6
edges of A and B
Then begin by cutting lengths of your edge rods as in the table to the right:
Because the vendor-supplied cut ends of the edge material will likely be more uniform than your hand-made cuts, especially if you choose to use cutting pliers, try to preserve as many of the manufactured ends as possible when cutting.
Next, slide connectors onto both ends of each of the rods designated for “edges” in the table above. With the fiberglass rods I was using, the ends I hand-cut with pliers deformed and expanded, so I had to ream the holes of the connectors for these ends out to 9/64 inch by hand-twisting a drill (held by gripping pliers) into the holes.
Cut a foam-core bottom for each of the prisms, corresponding to the four faces of the central tetrahedron: the largest face D has edges 50, 60, and 70 centimeters (I just measured off 70 cm along one edge of a foam-core sheet, and then laid meter sticks down so that their corners met and they read off 50 and 60 cm respectively at the corner and marked point on the edge). The other three faces A, B, and C (in order of increasing area) are all right triangles, so I just measured the leg lengths along two adjacent sides of a sheet and then cut the resulting corner piece off. The lengths are: A – 24.5 and 43.6 cm, B – 24.5 and 54.8 cm, and C – 43.6 cm and 54.8 cm. (Note the edge lengths of the foam core faces are all 1 cm greater than the corresponding edge rods, because the connectors at each end of a rod add exactly 1 cm to their effective length.)
Assemble each of the prisms by clipping edges onto altitudes to form the desired cross section.
Every prism should have a triangle of edges at each end of the altitude, and at least one additional group of edges around its middle; the quantities above have been set so that the tallest prism D can have three internal sets of cross braces and B can have two. One set of cross braces at mid-height for prisms A and B seemed to be plenty. Note that the clips can interleave at the ends to produce triangles with coplanar rods, except for the two most acute angles (the sharpest angles of triangles A and B), where you will have to place them side by side. You can actually get a bit more stability on the taller prisms by staggering the interior cross braces slightly, rather than making each set coplanar. But on all of the prisms, the cross braces at the ends should be as close to coplanar as possible.
When you have made all the prism frameworks, attach the foam core bottoms to each prism with tape. For the finale, it matters a bit which end of each prism you attach the bottom to, and this is an aspect that was unfortunately not done correctly in the pictured build. The easiest way to get it right is to stand the D prism in the center, and then line up the A, B, and C prisms with their hypotenuses matching with the sides of prism D. Orient A, B, and C so that the legs of adjacent prisms match in length (see the diagram at left). Then attach the bottoms.
The last step in preparing for the finale is to add vertical sides to your prisms. This could be done with any sheet material (you could cut rectangles to size) but the quickest and easiest way, that also allows you to easily see what’s going on in the finale, is to wrap them with cling wrap. Industrial packaging wrap is readily available in a variety of colors, or you can use ordinary consumer food wrap (although you will likely need an entire roll). Begin the wrapping at the top by hooking the wrap onto one of the altitudes (see photo at right), and then leaving some extra wrap above the top crossbars, make one circuit of the triangular perimeter. Fold the extra down around the top crossbars, and then continue to wrap around the prism angling downward somewhat so that at least half the sheet overlaps with what’s already there at all times. Continue past the bottom panel of the prism, and then fold the excess underneath and secure with tape or by stretching and sticking the wrap to itself. The process and results are depicted below:
Once all of the prisms are wrapped, it’s time for the finale. Fill the three smaller prisms A, B, and C with loose, light filler material — we used water-soluble “packing peanuts.”
Ideally, if the bottoms are placed on the proper sides, you can now place loose temporary lids on top of prisms A, B, C, and invert them above prism D to create a space congruent to the right tetrahedron above prism D, showing the prisms of height equal to face area erected on all four faces of the tetrahedron. (See diagram at left for how that might look.) Then pull the temporary lids out and allow all of the filler material to tumble down into prism D.
If that’s too complicated or the bottom panels were not on the correct ends, simply dump all of the contents of A, B, and C in turn into prism D (as depicted to right). Here’s what you get when you’re done:
And voilà — the material exactly fills the largest prism D! Is this a coincidence? Seeing as how this is Studio Infinity, of course not. What are the volumes of these prisms? Well, letting A, B, C, and D also stand for the areas of the four faces, we have the height of prism A is also A, and so on through the height of prism D is D. And since the volume of a prism is the area of its base times its height, the volumes of the prisms are A2, B2, C2, and D2. And it turns out that for any right tetrahedron, A2 + B2 + C2 = D2 — this is the Three-D Pythagorean Theorem. So the big prism was guaranteed to fill up exactly!
With all of the recent activity at Studio Infinity on geometric units that can be automatically cut and scored, it was natural for the S∞ G4G14 giveaway to be the 14-sided Truncated Octahedron, which tessellates to fill space. The additional challenge here as compared to some of the earlier interlocking structures was to ensure that the resulting polyhedral units would be able to connect arbitrarily face-to-face, so that it’s possible to explore the three-dimensional structures you can build with multiple truncated octahedra.
This was accomplished by two measures. First, by leaving the square faces empty (fortunately the truncated octahedron is still rigid with its square faces deleted) and putting connection tabs similar to ITSPHUN units on the corresponding edges. That step allows any square face to connect to any other square face, in any of the four possible orientations.
Second, I cut out a triangle (not a hexagon!) from the center of each hexagonal face, and added similar connection tabs to those. That allows any hexagonal face to connect with any other hexagonal face, but only in the three orientations that allow the space-filling tessellation to continue.
These ideas result in the following SVG template:
The black lines are cuts and the magenta dashed lines are scores (all for mountain folds). If you are cutting directly from this SVG, you can also cut just the dashes of the magenta lines on very low pressure/high speed to create scores that fold well.There are two units because it takes two to create a single truncated octahedron, and to illustrate how they interleave for cutting, leaving very little unused material. (The outermost protruding tabs are for connecting these two pieces into one polyhedral unit.) In case it’s helpful, here are a PDF and the DXF design file for this unit.
See the previous post for full assembly instructions. I’ll leave you with an image of the smallest loop one can make, with six completed units.
If you build an interesting structure with these units, please post a picture in the comments!
These are the assembly instructions for the Tessellating Truncated Octahedra; you’ll find background and the cut files for them on them in the following post.
Materials
Two TTO pieces per unit you wish to build
The first order of business is to fully cut and punch out two of the pieces from the cut templates in the following post, including removing the triangular inserts in the four hexagonal faces, and separating all of the tabs and slots. That will produce a piece that looks like the following picture (with the score lines in the template highlighted by light blue lines, as otherwise they are difficult to see in the photograph):
Now make a mountain fold on each of the score lines, producing something that looks like this:
Repeat with another piece. Two of them will interlock to produce a single truncated octahedron unit. The picture on the left below shows them overlapped in roughly the relationship they will have when connected; the picture on the right shows two of the edges lined up ready to be connected.
To connect this edge, start inserting the long flap of one of the tabs on that edge into the slit opposite it on the other piece, then pull that tab all the way in, then repeat with the other tab along that edge (on the piece that has the slit in the first part of the connection, going into a slit on the piece that had the tab). This process is shown in the following sequence of three photos:
Now work your way around the unit, connecting six edges in all. The left photo below shows me working on the second edge, and the right photo shows the state with three edges connected.
When you’ve done all six edges, you’ve completed your first truncated octahedron unit, which looks like this:
But the distinctive thing about these truncated octahedra units is that they can connect arbitrarily face to face in their tessellation of space. Connecting square faces (which are empty in each individual unit) is relatively straightforward: just line up their edges however you’d like and engage the four pairs of corresponding (but opposite-pointing) edges. Connecting the hexagonal faces is just slightly trickier, because they line up in the tessellation in only three (not six) ways. (The other three ways would force two square edges to be adjacent; but as each TO unit does not have any adjacent square edges, that can’t happen in the tessellation.) So you have to line up two units so that the tabs in their triangular cutouts will engage, like so:
Then line the corresponding faces up with each other and engage the three tabs around the triangular cutouts. (You may have to bend the tabs back pretty far to get them past each other to engage, so hopefully you’ve cut your pieces out of a sufficiently flexible material, like this plastic — Roscolene lighting gel, to be exact — paper and cardstock will also work fine if you’re OK with opaque units.) Here’s me having just connected one pair of tabs around a triangular opening:
Once all three pairs of tabs are engaged, you’ll find the connection to be quite robust:
Enjoy assembling and connecting your Tessellating Truncated Octahedra! What kinds of loops, bridges, and shapes can you build?
Another aspect of the PCMI session on Illustrating Math was a series of exploratory, hands-on workshops. One of them focused, in part, on the design of modules like the one for the truncated triakis tetrahedron, but based on other existing modular origami units.
It’s more or less possible to transpose any modular origami unit to a cut-and-score single-sheet module with tabs and slits, as the following table of designs from the workshop shows:
A key difference that emerged from these workshop examples as compared to the PHiZZ units is that this plastic sheet material produced beautiful results on these geometrically exact units, whereas it did not distribute the slight geometric imprecisions inherent in PHiZZ unit constructions nearly as well as paper units do. (Many of the modular plastic PHiZZ constructions end up looking slightly lopsided, whereas the models from the workshop were all extremely crisp.) This distinction definitely plays into future material selection for geometric constructions.
For the actual building event mentioned in the previous post (linked above), participants could choose from a variety of target polyhedra. The origami inspiration was the PHiZZ unit, which stands for Pentgons Hexagons in Zig Zag, so the ideal targets consist of just pentagons and hexagons. With Euler’s formula for polyhedra and a little calculation you can determine that such a shape must have exactly twelve pentagons and almost any number of hexagons; the page for the event includes a table of candidates.
I chose the Truncated Triakis Tetrahedron as my target: (not only because of the alliteration, it’s a nice size and has pleasing symmetry that breaks the building down into four simple, identical sub-assemblies)
To assemble this, I worked from a diagram of the edges, graciously colorized by Elliot Kienzle:
I found it easiest to start by making each of the lighter-colored sections. They’re all the same, made of nine pieces each. You make one three-way connection all the same color (as shown in the previous post), and then turn each of the three opposite ends of each of these pieces into a three-way connection, like so:
Repeat with the other three colors.
When you have your four sub-assemblies, you can connect two neighboring loose ends of one color with corresponding loose ends of another color at the two ends of a new unit of the darker color. Here are the components laid out schematically to show how they go together:
And here they are overlapped in the actual way the first trio of connections will be made:
And here they are once both ends of the blue unit have been fully connected:
This process of connecting sub-assemblies with one additional unit happens in six places, shown by the blue edges in Elliot’s diagram above. (These six edges are actually the remnants of the six original edges of the tetrahedron that the truncated triakis tetrahedron is based on.) Once you’ve made all those links, everything hangs together like so:
(Note that this image is shown from the same perspective as Elliott’s diagram above, used as the construction guide.)
Participants took this basic concept in a variety of directions, and here’s a mini-gallery of some of the results:
Emily Z
Katherine Booth (75 edges!)
Kate Stange
Rémi Coulon
Gabriel Dorfsman-Hopkins
As a final bonus, the units are flexible enough so that you can break the pentagon/hexagon only rule: this shape is made by connecting just two of the truncated triakis tetrahedron sub-assemblies to each other in all three possible places, and it contains three quadrilaterals and six pentagons.
In many ways, modular origami is ideally suited for the type of exploratory mathematical play that S∞ is dedicated to: it’s easy to get started, very tactile, and offers nearly endless opportunities for creating interesting and beautiful objects. For example, here’s a PHiZZ unit torus that resulted from a workshop I led at The Brearley School (photo courtesy of Maggie Maluf).
But the backstory to that wonderful construction is that folding the hundreds of units required to assemble the torus took so much time that by the end of the scheduled workshop, only about a quarter of the torus was connected, and piles of PHiZZ units were strewn around the room. (Unfortunately – but unsurprisingly – I don’t seem to have a picture of this scene.) The torus was only salvaged through the valiant efforts of the school’s math club and its coach over many lunchtimes.
This anecdote highlights one weakness of modular origami – if you’re interested in building something big, folding all of the individual units necessary can be tedious and time-consuming. Hence, I’ve often wondered why there isn’t a construction toy that consists of reusable bendable pieces modeled after one of the popular modular origami units.
Two developments moved that concept from contemplation to a recent Studio Infinity project: the first is the constant improvement in hobbyist-level automated cutting machines, some of which now have dual tool slots, potentially ideal for cutting and scoring. (I use a Silhouette Cameo 4 Pro, partly because its two-foot width accommodates some materials available in bulk rolls, but there are numerous capable machines on the market – this shouldn’t be construed as an endorsement of that particular machine.)
The second was the need and opportunity for a group mathematical event for the summer 2021 Illustrating Math program at the Park City Math Institute. Since the event was virtual, it had to be easy to send any necessary supplies to the participants. Pre-cut and scored flat components seemed ideal.
I decided to stick with something modeled on the familiar PHiZZ unit for this first foray into modular origami-less construction. It also has a very simple geometry – here’s an example of a single unit:
A square sheet of paper is accordion-folded in quarters, and the resulting strip is pleated in isosceles right triangles. They usually connect in threes, as in this photo:
The idea was to create a cut/score template that would produce a piece with the same essential geometry, that could be produced in bulk on an automated cutting machine from a roll of material. In fact, choosing the material posed the first challenge. After much experimentation, I settled on the stiffest theatrical lighting gel I could find, namely Roscolene. This comes in two-foot wide rolls in a selection of striking translucent colors, and holds a crease very well, while cutting readily. Brittleness is its major drawback — I’d rather use tear-resistant mylar, but have so far had difficulty locating rolls of colored mylar sheet.
The next challenge was to figure out the connections. The paper PHiZZ units work because the initial accordion fold creates pockets that one end of the adjacent unit slides into. Since these units were to be cut from a single layer in the final 4:1 aspect ratio, no pockets were possible. (One could cut and pre-score 2:1 rectangles that fold in half to preserve the pocket mechanism, at the expense of using twice as much plastic and losing some of the translucency in doubling the layers.) Instead, venerable tab-and-slit designs can replace the pockets as a connection mechanism. Multiple prototypes led to the following design used to produce thousands of units sent to program participants:
The pink line is a score for a mountain fold, the green lines are scores for valley folds, and the red and black lines are cuts. (Of course, you can reverse mountain and valley as long as you do all of the pieces the same way, in which case your units will come out mirror-imaged from the ones below, although matching the paper ones above.) The small “divots” in the rectangle’s perimeter help align the folds along the scores. The above image is an SVG that can be used directly for cutting with the software for many machines, but if it’s helpful, here’s a PDF of the design (you can print on paper and cut by hand — but that’s even more tedious than origami), and the DXF file used to draft it.
When you have a piece all cut out and all the tabs and slits separated and folded as above, it looks like so:
To connect two units, start as shown in the left picture below: insert the large tab from the end of one unit (the orange one here) from above into the corresponding slit about a third of the way along another unit below (the blue one in the picture).
Then reach underneath the top unit and feed the thin tab of the bottom unit up through the lower slit in the top unit, as shown in the right-hand picture above. Then bend it around and back down through the top slit as shown in the picture on the left below:
Then push this tab all the way in — it’s like a belt buckle — to finish off the connection. It should look like the picture on the right above.
To wrap up this post, here’s three units assembled like the paper triad above.
The next post will cover building full polyhedra from these units.
This is a sequel to a (pre-pandemic) post about weaving a stellated polyhedron. This time, I’d like to show how similar techniques can also be used to create a “great stellated dodecahedron” (“GSD” for short; illustration to the left). The materials are in fact the same as before, as summarized in the table above. This time, I am using 4-foot quarter-inch pine dowels for the rods, and white mini “rainbow loom” rubber bands.
Begin by marking the same points on each rod as described in the previous post; in this case, with 48″ rods and allowing 1 inch of overlap at the ends, this is 1+(46 × 2/(3+√5)) or about 18 1/2″. (See this MathStream post for an explanation of where 2/(3+√5) comes from.)
With rods this long, it’s easiest to transfer the marks from one rod to the next.
Once all of the rods are marked, make two five-pointed stars as described for the SSD, except both of these should have the reversed over-under pattern: As you follow a given stick around the inner pentagon clockwise, it must be on top at its first junction and on the bottom at its second junction. That means also that looking at any point with the center of the star away from you, the rod that comes into the point from the right should be on top. When both stars are done, put one on top of the other so that the points alternate:
(Here the star with the point directed straight up in the picture is on top.) Now take five of the rods, stand them on end, and put a rubber band around them at the higher mark. Splay them out so that the top sections make a clockwise “whorl” when viewed from above:
Stand this “penta-pod” up amidst the two stars, with each leg just to the right of a corner of the inner pentagon of the top star:
Then slip the bottom of each leg into the rubber band at the junction it’s nearest, keeping it in the same relative position to the two horizontal rods crossing at that junction:
Now slide the top star up into the air, with each leg of the penta-pod sliding through the rubber band it was just inserted into. The legs will splay out as you do this. Keep sliding until the top star has reached the lower mark on each leg and these rods have spread out to reach the points of the lower star:
Finish this step off by inserting each leg into the rubber band at the point of the star where it’s ended up. You slide it in on the left, so that the three rods that meet at the point make a counterclockwise whorl. (As they will at every outer point of the GSD in the whole construction.) At right is a picture focused on one of the points, which will hopefully make the attachment clear.
To prepare for the next step, make another “clockwise whorl” of five rods banded together at their top marks, just like the one you made before. When that’s ready, turn the construction over and rest it on what were the top ends of the five rods from the first whorl. The construction may sag a bit, but it should hold up and balance on the five ends of the rods. The two stars should still be quite close to horizontal, with the formerly lower one (that has three rods at each point) now above the other. Overall, the construction should look something like this:
Now insert each leg of the second whorl just to the right of one of the inner junctions of what’s now the top star, as shown at left. Below right is a view of all five ends inserted.
Slide all five of these legs through the rubber bands you just inserted them into, until they reach a point of the lower star. Once there, insert the leg into the rubber band at the point of the star in exactly the same way as you did with the first “pentapod” legs. See the detail picture below left:
At this point the structure should be symmetric from top to bottom. There are five unattached rod ends sticking upwards in a pentagon, and the structure is resting on five unattached rod ends at the bottom. The ends of these rods, together with the ten points of the first two stars, comprise the 20 outer vertices of the final GSD. The star points already have all of the rods attached that they will have at completion. There are ten unused rods remaining. Each one of these will extend in a straight line from one of the top five points, through two inner junctions (one from the top star and one from the lower star), and end at one of the lower five point. From each top point, there will be one rod extending down and to the right and one extending down and to the left; and arriving at each bottom point will be one rod from the right and one rod from the left. All that remains is to feed the rods through properly so that they are woven uniformly at each junction. They should create clockwise whorls at each inner junction (when viewed from outside the GSD), and as mentioned earlier, counterclockwise whorls of three rods at each outer point of the GSD.
Here are some guidelines and pictures that should help to feed these rods through properly. We’ll begin with a rod (indicated by the green arrow in the following picture) that will extend down and to the right from one of the top points (at the extreme top left in the picture). It should feed through the inner junction of the top star that is immediately below and to the right of the top point; that’s the one with the white rubber band in the lower foreground of the picture. Specifically, it will feed through the leftmost sector of this junction:
It then feeds through the inner junction of the lower star just below and to the right of that, but this time it feeds through the rightmost sector of that junction. The new rod then proceeds down to the end of the lower leg that’s just below and to the right of that. The next picture shows just the one new rod added, in its final position. The new rod is again indicated by green arrows; pay particular attention to how it passes through the two inner junctions: all the way on the left at top, and all the way on the right for the lower one.
The next rod we’ll feed through will run from the upper point next counterclockwise (as viewed from the top) from the one we just attached to, and proceed to the left through the same inner junction that we just used. This will add the fifth and final rod to that particular inner junction of the upper star. As with the previous rod, we’ll insert it in the leftmost sector of that junction. Here’s the rod (we’ll continue indicating the newly-added rod with green arrows for the rest of the post) just after it’s been properly threaded through the rubber band at that inner junction of the top star:
Keep feeding it through until it reaches the inner junction of the lower star just to the left; there, as before, feed it through the rightmost sector of the junction:
Once it’s through that rubber band, continue feeding it until it reaches the end of the bottom leg below and to the left of that junction:
Make sure you secure it at the top as well. Now you’re in the home stretch — just eight more rods to insert. It’s most straightforward to insert them in pairs, just like the previous two, rotating the structure one-fifth of a turn before each pair to orient the location for their insertion toward you. Here’s the structure after just one more rod has been inserted:
For every rod that you insert, it feeds through the leftmost sector of the junction where it passes through the top horizontal star, and the rightmost sector of the junction where it passes through the bottom horizontal star. You can see this again in the following picture of the fourth rod in the midst of being inserted: (125302 goes here)
You should also see a regular icosahedron begin to emerge at the core of the GSD. Here’s a view just before the sixth rod is secured to its top point:
If you just keep going in this same fashion for all ten of the last group of rods, you’ll complete the star. Here’s a view just after the last rod is inserted:
And now you’re pretty much done! There’s really only a couple more optional steps you might want to do to get your great stellated dodecahedron into optimal shape. As depicted at right, you can examine and gently adjust each of the inner junctions of five rods (that form the vertices of the inner icosahedron) to ensure that each one makes a tidy regular pentagon at the five-way crossover. In addition, if you’ve built your GSD from rods with even a moderate amount of surface friction (dowels, paper straws, even metal rods), you can now snip the rubber bands off of all of the interior, five-way junctions and the GSD will hold together just fine. (Theoretically, you can cut the outer rubber bands at the points where three dowels come together as well. If you want to try this, I recommend just trying a couple of widely-separated ones at first, and leaving off if the outer junctions don’t seem to be holding up well. I pretty much reserve this for stars built out of really highly frictional items or ones that can be twisted a bit at the outer points, like pipe cleaners.)
It’s a bit of an involved building process, but I think the elegance and symmetry of the final construction are well worth the effort. Here’s a picture of the final result with inner rubber bands removed:
In addition, you’ll find that the finished GSD is surprisingly sturdy and robust. You can easily pick it up and handle it without fear of it falling apart. As an illustration, here’s a closing shot of the great stellated dodecahedron supported on just one point:
After seeing Laura Taalman’s inspiring 3d print, it occurred to me that one could also render the edge-to-edge cubical array of dodecahedra contemplated in this earlier post in an analogous way. Plus, I just received a new Prusa SL1 printer, and needed something to try it out on. So after just a bit moretinkering in OpenSCAD, and 400 minutes of print time using the resulting STL file, I ended up with this model:
If you recall, the negative space of the dodecahedron array was just a bit too monotonous to seem worth printing. However, I think that simplicity works very much to advantage in this wire-frame style of rendering. This model has enough complexity to afford visual richness, but is open and orderly enough not to be overwhelming.
In addition, it produces an excellent variety of shadows (or parallel projections, in more math-y lingo):
(The rhombuses in the top left photo correspond to the rhombic prism channels in the antidodec that we previously modeled.) Here’s another shot so you can see the setup for capturing the shadows. This is really best done in direct, bright sunlight — it’s tough to get such crisp, parallel rays of light otherwise.
I can’t resist one more take on this lovely model and its shadows.
If you print one of these or a variation on it, I’d love to see/hear about it.
