New Sculpture Writeup: Penny Pincher

Check out the latest addition to my sculpture page: Penny Pincher!

Penny Pincher

Made with more than four thousand binder clips and two thousand pennies, this monstrosity stands at 35 pounds and just under 2.5 feet, but it is perfectly content standing on three flimsy-looking legs. The flat image above doesn’t do justice to the three-dee-ness of the sculpture, so here’s a way around(hah!) this challenge:


Head over to the full writeup for more about what it illustrates and how it was made.

MIT MoSAIC This Weekend

If you’re near Boston today (Friday) or tomorrow (Saturday), come check out MIT MoSAIC, a traveling festival, currently at the MIT Museum, celebrating Math from the perspectives of art, science, technology, and more. There are panels, presentations, workshops, and of course an extensive Math+Art exhibit from many contributing artists. A few of my recent sculptures will be shown too.

Thanks for following along for last month’s Straws Thingy series! I’ll return to give that series a proper debriefing now that I’m no longer wading through its myopic trenches of daily deadlines.

On the Origin of Scaffolds

[This is post #28 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first or previous post.]

The scaffold I’ve released this month is not the first I’ve designed. Thanks to feedback from a number of workshops and dedicated, private play-testers (more on these tomorrow!), I’ve been able to iteratively improve the scaffold’s design in a number of areas: ease of scaffold building, ease of scaffold use, and insight gleaned from scaffold (e.g., through symmetry, structure, variations). Today, let’s take a brief tour through some of these earlier designs.

Vertical Edges

My first scaffold was simply the first thing I thought of: vertical slices through each edge.


Straightforward in concept, but highly undesirable in most other respects. It takes forever to assemble the scaffold, with 30 individual pieces to cut and 60 flaps to tape or staple! It holds the straws quite firmly in place, with 3 holes per straw, which also helps with vertex weaving, since there’s just nowhere else the straw can go. But as a far stronger negative, this also means it has very little fault tolerance: punch a hole a few millimeters out of place, or tape not quite perfectly, and the straws just won’t go.


This scaffold hardly assists with assembly. It’s not clear what markings would help direct straw insertion and weaving, nor is it easy to print accurately on both sides of the paper anyway, so at this workshop I asked the students to rely on counting instead of reference marks—an extraneous and unenlightening mental burden.

On the positive side, this scaffold demonstrates a pleasing symmetry: you can make the mirror image with the exact same scaffold!

There’s a similar version for the lacier Straws Thingy variant as well.


Horizontal Edges and Snakes

My next family of scaffolds focus on horizontal bands across the edges. This first try also has 30 separate pieces (one per edge) that are nearly rectangular, which reduces cutting complexity significantly, but it still takes a while to join all the pieces.


Joining multiple edges into larger assemblies drastically and pleasingly reduces assembly time without affecting cutting time much—a clear win.


This horizontal paradigm also lets us play with helpful reference marks. Some to point straws in the right direction, others to indicate straw color or grouping. I toyed with cute little ghosties to indicate color groupings, possibly with ghost shading or ghost mouths (look closely—that’s A through E!) corresponding to color groupings.


But I still disliked this general paradigm, for a few reasons. For one, relentlessly stabbing smiling, innocent ghosties through their eye sockets felt like cute aggression taken one step too far. But mostly, this scaffold leaves vertex weaving wide open and unaided, and this is the part people struggle with the most! Unlike the previous (vertical) scaffold, this one holds the straws fairly loosely, so it won’t mind if you accidentally spiral your vertex the wrong way… until you do it 5 or 6 times. Then it minds quite a bit. And this crying infant of a scaffold isn’t expressive enough to tell you why it’s upset, or to tell you much about vertex weaving at all! For this reason it felt immature, incomplete, to the point where I even considered a separate “vertex insert” to help with the weaving step.

Vertex-centric Scaffold

This most recent vertex-first scaffold perspective nicely solves this concern.


I’d like to say I jumped here straight from the “vertex insert” idea, but the truth is more interesting: while making the set of 32 nearly identical Straws Thingys[1], a friend[2] suggested I design a reusable scaffold, instead of demolishing and restarting each time. I ultimately came up with this design, where each hexagon lifts straight over its vertex without ripping.


But even if the reusable scaffold idea doesn’t survive past these 32[3], the resulting vertex-first paradigm shift has been quite valuable!


Future improvements will be driven by your feedback! I would love to hear about your Straws Thingy experience, so don’t hesitate to let me know how it goes.


  1. No, I definitely didn’t make all of those in real time this month! []
  2. Thanks, Will! []
  3. Because really, who (else) needing Straws Thingys en masse! Correct me if I’m wrong… []

Endless Variations on a Theme

[This is post #27 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first2 previous, or next post.]

In the last few days we’ve singled out five subtle, independent tweaks to the initial Straws Thingy design. Why stop here, when there are so many other variations to consider?!

First, I wanted each of our 32 configurations to maintain maximal symmetry: within each variant, each straw is used in precisely the same way as each other straw, except possibly for color. For example, if we decided that some vertices should be woven one way and some the other way, then these vertices, and the straws meeting there, could be distinguished. Or if some connections are short-into-long while others are long-into-short, these too would be functionally different. I simply can’t abide such untidiness!

Furthermore, the colors themselves do maintain some symmetry: when we rotate the piece to look at a different straw (or face or edge or vertex…), the worst that happens is we permute the colors: everything that was yellow is now green, everything that was green is now blue, etc.

Second, I wanted the differences to be structural. We touched on another one in the instructions: cutting vs folding the straws for insertion.


While these are indeed different preparations, the resulting geometric arrangements of straws are identical, so learning to distinguish them didn’t feel as mathematically exciting.

Finally, these 32 pass the first glance test: even when looking at all 32 Straws Thingys from yesterday, it’s not immediately apparent that there are any differences at all! You have to ponder, discover—dare I say learn!—in order to appreciate their uniqueness.


I don’t mean to say that only these 32 variants are interesting; just that they fit well together as a set. Outside of these, there are endless variations, remixes, or redesigns that are worth exploring, and I can’t wait to see what you come up with! For example, how about other shapes? Maybe consider some of the other polypolyhedra, some of which even work with the same size straws![1]


How about those too-narrow Market Basket straws from earlier—are they secretly optimized for a different arrangement? What other possibilities can you think of?!


  1. …and a different scaffold. More on this soon. []

32 Straws Thingys! Aaah!

[This is post #26 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

There’s one more modification we can make that changes the arrangement of straws without significantly affecting the overall composition: swap the roles of long and short ends! We’ve been sliding the short ends into the long ones to make the seam rest nicely along the flexy bits, but we should at least consider the opposite. In fact, my very first Straws Thingy made this choice instead, because I thought it nicely complemented the (less symmetric) 4-color color scheme.


But you know what this means: Each of yesterday’s 16 Straws Thingys can be reversed in this way, resulting in a whopping total of 32 alternative, distinguishable-but-just-barely Straws Thingys! And with 5 independent binary choices to make, we’ve built ourselves a lovely, colorful 5-dimensional hypercube of Straws Thingys.


Or maybe just an amorphous pile. Whatever you prefer.


Anyway, I’m sorry to say that this is the last day of exponential growth / straw Tribble invasion. But never fear—there are a few posts yet to come before we wrap up the series!

4D Hypercube of Straws Thingys!

[This is post #25 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

We’ve been accumulating lots of variant Straws Thingys, all subtly different, but all just as symmetrical as the others. We saw two ways of messing with the 5 macroscopic tetrahedra themselves, and then we zoomed into the molecular level and rearranged the 20 triangles that build these tetrahedra. Today, we finally venture to atomic scales and manipulate the 60 straws directly.

Let’s interrogate another scaffold feature. The arrows on the scaffold told us which way each straw should point: long end goes here, short end points this way. But what’s special about the arrows’ placement?


As with yesterday, there are two answers. Firstly, if you disregard the arrows and insert straws randomly (still following the dashed lines, of course), then the straws end up inconsistently oriented: you’ll get two long or two short ends trying to come together, when you actually need one of each.

But secondly, why not reverse all the arrows at once? Each straw gets turned around in place, so the triangles remain intact. This is a fine thing to do!

The difference is rather subtle (especially to photograph!), so here’s another way to see it: the location of the seam between successive straws. I’ve changed one straw from yellow to blue to highlight the effect:


In the original (left), the seam is neatly tucked away underneath another straw. After reversing straw orientation (right), the seam jumps outside the weave, a tad more visible. Only a tad. But for this reason (and only this reason), this arrow decision was deliberate, unlike the three prior differences where I broke ties arbitrarily.

Finally, as before, we can make this change on all 8 of our prior versions. This gives us 16 different Straws Thingys! Will it never end?! Will Target never run out of straws?!

16 different Straws Thingys along 4 binary axes: Left Cube / Right Cube: mirror.
Sixteen distinct Straws Thingys along 4 binary axes. Left Cube / Right Cube: mirror. Left/Right within cube: color swap. Up/Down: triangle weave. Front/Back: straw direction.

By the way, these 16 things are determined by 4 independent binary axes, so this is a four-dimensional hypercube of Straws Thingys!

Triangle Jitterbug: Up to Eight Versions

[This is post #24 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

We began describing Straws Thingy’s shape as 5 intersecting tetrahedra, so it is fitting that the two asymmetries we’ve identified so far (color scheme and mirror image) depend only on these tetrahedra.


Yesterday’s interloper, by contrast, requires us to look more closely at the 20 triangles, not just the 5 tetrahedra. Indeed, in yesterday’s Spot the Difference image,


the tetrahedra are in the same locations with the same color scheme, but each tetrahedron is woven differently from its four triangles.


Said differently, the spiral at each vertex rotates in the opposite direction. So this interloper occupies a middle ground between original and mirror: each tetrahedron is mirrored in place, but the tetrahedron arrangement isn’t affected.

And of course, this can be done to all four of our prior variants, so there are really 8 different Straws Thingys! (I still can’t resist making a full set!)

8 distinct Straws Thingys along 3 axes. Left/Right: mirror. Front/Back: color swap. Up/Down: Jitterbug.
Eight distinct Straws Thingys along 3 binary axes. Left/Right: mirror. Front/Back: color swap. Up/Down: triangle weave.

This also answers a question you may have considered while building your Straws Thingy:

Why are the elbows arranged as they are on the scaffold? What goes wrong if I don’t follow the elbows and weave the corners the other way around?

The practical answer is that the scaffold gets upset. Indeed, this alternate weaving needs the scaffold holes in different locations. (In fact, this is a completely different arrangement of cylinders, and the inflation optimization even returns a different ideal length! [28.42 instead of 27.43, which, thankfully, fits comfortably in our error margins.])

But more fundamentally, nothing goes wrong! I equally could have settled on the other scaffold instead, and I don’t have a strong preference between the two. So we now have 8 different Straws Thingys, all distinguishable, but none especially distinguished from the crowd.

But Wait, There’s More!

[This is post #23 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

To finish yesterday’s train of thought, how should you proceed if you wanted to make some mirror image Straws Thingys?


You you could pull some trickery and print the scaffold in mirror image (do printers make this easy to do?), but it’s equivalent to just assemble the regular scaffold inside-out and look at the ink from behind. Follow the arrows and elbows as normal, just from the other side of the paper.


That way, you can make a nice, complete collection of all four Stra…


What?! Can it wait? I’m trying to finish a blog post here!

Well, it’s just, uh, there’s another one here to see you.

Another what?

Um, another Straws Thingy.


That’s the same as before! What does this Thingy think it is, claiming to be new and different…

Well, it is, um, very insistent. Maybe you could ask them? (Reduce to conspiratorial whisper.) The readers? (Cue strings.)

Yes, perhaps you’re right. Readers? It looks like we have another game of Spot the Difference on our hands! What makes this Straws Thingy different from the other Straws Thingys?


(Original on the left, insistent interloper on the right.)

Mirror Thingy on the Wall

[This is post #22 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

If you step in front of a mirror, don’t be alarmed if your Thingy looks a bit strange—your Straws Thingy, that is.


The Straws Thingy is not identical to its mirror image! In other words, it is chiral. Here’s a physical copy of this mirror image for closer inspection. (This is two real thingys side-by-side. I didn’t just flip the picture.)


One way to tell them apart is to look at their driving conduct. If we imagine cars driving toward and past each other along edges, starting at neighboring vertices (in this case, blue and green), the cars will keep to the right side of the road. In the mirror images, they keep left.


Seen a different way, as the straws spiral outward from the center of a face, they spiral counterclockwise for our original, but clockwise in the mirror image.


And as we saw yesterday, the original and the mirror image each have two distinct color schemes. So there are really four different Straws Thingys! I couldn’t resist building all of them.


About Faces

[This is post #21 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Knowing the color pattern around any one of the twelve faces (5-sided stars) of a Straws Thingy tells us exactly how the tetrahedra are colored, so it lets us reconstruct the entire color scheme. In other words, any one face determines all 12 faces. This is useful for quickly comparing color schemes. The “original” Straws Thingy from yesterday has these twelve faces, in no particular order:


For example, the face pictured in the top left has the colors Blue (B), Orange (O), Pink (P), Green (G), and Yellow (Y) in counterclockwise order around the face, so we’ll write it as BOPGY. It also doesn’t matter where we start—this is the same ordering as PGYBO, or YBOPG, or…—so to be consistent, let’s choose to always start at Blue. Then these twelve orderings are, respectively:


You can check that all 12 of these orderings are different, but remember that any one of them can be used to identify this original Straws Thingy.

So if we’re handed a new Straws Thingy and we want to know if it matches the original one, we only need to check a single face of the new one: if it’s in this list, the two Straws Thingys are identical, so the twelve faces of the new one will be exactly these twelve faces. If not, then there can be no face matches at all: the twelve orderings of the new faces can’t be in the above list.

So how many orderings aren’t in this list?

Well, how many orderings are there in total? We have to start with Blue, but then any of the four remaining colors can come next, then any of the three colors can come after that, then either of the two remaining colors follows, finishing with the only color remaining. So there are \(4\times 3\times 2\times 1 = 4! = 24\) face orderings in total.[1]

With 12 of those orderings belonging to the “original” Straws Thingy, there are only 12 other faces to pick from. So any other Straws Thingy, like yesterday’s “odd one out,” has to use exactly the other 12 orderings. In other words, there’s only one other Straws Thingy color arrangement! Any Straws Thingy you build will match either the “original” or the “odd one out”; which did you build?


  1. That middle term is not an excited “4”. It’s pronounced 4 factorial, and in general, \(n! = n\times (n-1)\times\cdots\times 2\times 1\) counts the number of ways to arrange n items in a line, or as we saw here, \(n+1\) items in a circle. []