Interlude: What Makes Hard Mode Hard?

[This is post #10 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

What makes yesterday’s “Hard Mode” so hard? It’s more than just spacial reasoning.

In addition to puzzling out the locations of all the straws, we also need a strategy to keep them where you want them! The images so far have been perhaps a bit misleading. The single tetrahedron I photographed a few days ago?


Yeah, that’s not quite right. Here’s what actually happens when you build this tetrahedron:


It just doesn’t stay put. Indeed, looking more closely at the static image above, you may notice small pieces of tape in the back corner holding things together. Now you know why. Not until very near the end (about 4 out of 5 tetrahedra) are the straws able to hold each other in place without some form of extra support.

So all of you intrepid “Hard Mode” explorers should consider this an amendment to yesterday’s Step 2 (Hard Mode): use some tape, or string, or twist-ties, or dozens of hands[1] to hold things steady as you build.

For the rest of us, on to the actual steps tomorrow!


  1. They work better when attached to their owners. []

Step 2 (Hard Mode)

[This is post #9 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Tom Hull’s Five Intersecting Tetrahedra is beloved by origami enthusiasts (myself included!) not only for its beauty but also, in part, because assembly is a challenging and rewarding puzzle, requiring dexterous spacial reasoning and a touch of trial and error. So, for the truly adventurous among us, those who really enjoy doing things the hard way, your Step 2 is this:

Step 2 (Unnecessarily Hard Mode): Put all the straws in the right place, and you’re done!

This is an even more challenging puzzle, and therefore (hopefully) more rewarding—but at the same time, significantly less accessible. For those that choose this route, take these reference images to guide your journey, and godspeed! For those that would prefer a bit more (or perhaps a lot more) guidance, rest easy—help is on the way, in the form of Step 3, Step 4, Step 5…




Time to Build At Last! Step 1

[This is post #8 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

It’s finally time to start building! By the end of this week you’ll have your very own Straws Thingy to keep you company. If you haven’t yet acquired your Target (or other carefully measured [see the previous few posts for details]) straws, now’s the time!

Let’s begin with a simple but immensely helpful step: grab some scissors and cut your straws lengthwise. No, not all the way! Just one slit on the short straw end, and just up to (but not including!) the flexy bits.

crease prep

This will make it much easier to slide each straw into the next one in future steps. You’ll want to prep 12 straws in each of 5 colors, or 60 straws in all.

60 prepped straws

If you want to bring some origami back into the project, you can instead make a shallow crease, like this:

crease prep

Either way, remember that the flexy bits will be visible in the final piece, so avoid cutting or creasing into them.

Oh, the Places You(r Straws Thingy)’ll Go!

[This is post #7 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

OK! We’ve explored the geometry of Straws Thingy a bit, and we’ll do more of this later. But I think the rest of our mathematical discussion will be much more engaging with a Straws Thingy of your own in your hands!

michaelangelo straws

Or on your wrists!

wrist warmers

Or atop your head!


Or crowning your Christmas tree!

christmas tree topper

So let’s start talking about how you can make your own, step by careful step. What’ll you do with yours?!

Blowing Up Straws

[This is post #6 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

So how could we compute/visualize the “perfect” length to diameter ration? Let’s start with a smaller, simpler thingy made of straws (not to be confused with Straws Thingy itself):

small thingy made of straws

And here’s finding the tight, optimal packing:

inflating squares

The tubes are inflating (or equivalently, shortening) while pushing away from each other. When they can’t inflate(shrink) any more, we’ve reached the ideal, tight packing.

We can do the same for Straws Thingy proper (it’s a bit more difficult to see, hence the example above):

inflating Straws Thingy

The resulting tight packing has a length to diameter ratio of about 27.43, according to my Mathematica simulation. Why does this look so different from the 23 and change from yesterday? They are measuring slightly different things:

cutting corners with flexy bits

The 27.43 refers to the edge length of the “axial” equilateral triangle, whereas the straws “shortcut” these corners via flexy bits. Of course, the exact shortcut dimensions vary from brand to brand, which further explains why I didn’t offer more precision to yesterday’s 23-or-24 measurement.

OK! It’s time to switch focus away from discussing Straws Thingy‘s geometry and toward learning, step by step, how to make your own! So go out, find those straws, and get ready for some excitement!

No Target? No Problem

[This is post #5 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Sjoerd Visscher asked a great question on yesterday’s post: what to do if you can’t get this specific brand of straws? (Or, extrapolating, if you want a different color palette? Or just generally like doing things the hard way?) Thankfully, this is no problem. Comparing the long end of the straw (not including the flexy bits) to the diameter, you’ll want a ratio between 23 and 24. (This needn’t be too too precise. I’ve had success even at a 25-to-1 ratio—it felt only a tad loose.)


Measuring directly is certainly an option, especially if you have easy access to some calipers, but for a lower-tech solution, just line ’em up! These straws below are perfect, at 23 and change.

length to width ratio: 23 and change

Keep in mind that there will be some manufacturing variation (these are not precision-critical parts in their intended use, after all!), so don’t worry too much about getting the absolute perfect ratio, or measuring each straw overly carefully.

length variation
Slight variation in length: when lined up along the right end, the flexy bits are not perfectly aligned.

So practically speaking, if you buy straws that are too long, cut to size. And if they’re too short, just leave them sticking out a bit:

pushing straws in
Left: with ideal length, push until flush with flexy bits. Right: compensating for short straws.

But where does this “perfect” 23ish ratio come from? How might we compute it, without resorting to trial and error? We’ll talk about this tomorrow.

Straw Size Matters

[This is post #4 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Two days ago, I recommended Target’s up&up brand of flexible straws. Why so specific? Is Target sponsoring these posts? In a word: no. (Hint hint, Target! =D) So why these in particular?

Up&up straws from Target

For this model, the size of the straws is critical—specifically, the length-to-diameter ratio. If the straws you use are too long/skinny (like the ones below from Market Basket), the finished model turns out wobbly and unsteady, as if it (appropriately!) drank a bit too much.

My straws are too big!

On the other hand, use straws that are too short/fat and they simply can’t squeeze past each other. If you find those perfect Goldilocks straws—not too long, not too short, but just right—they keep Straws Thingy perfectly snug and sturdy. And in this regard, Target’s up&up straws hit a bulls-eye.

Target logo

(Image source)

The Shape of Thingys To Come

[This is post #3 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

OK, so what is this shape? As a first approximation, Straws Thingy is simply the Compound of Five Tetrahedra, a favorite among origami folders thanks to Tom Hull’s popular and supremely elegant design.

Straws Thingy as a compound of five tetrahedra.

But on closer inspection, these tetrahedra break down into even smaller parts: each tetrahedron is made from four interwoven triangles, making 20 triangles in all. Here’s one such tetrahedron (left) with one of the four triangles distinguished (right):

One tetrahedron of straws

Robert Lang describes this shape as a “polypolyhedron” (specifically, it is the mirror image of Polypolyhedron 44), and he originally classified all such shapes (there are 54) for origami—his folding of this particular polypolyhedron, a model he named K2, is especially stunning and took a full day to assemble!

I therefore like to think of Straws Thingy as origami without the origami. The straws’ flexy bits (technical term, I’m sure) do the folding for us, and instead of holding everything together with creases, we stick straws inside each other—the same beloved trick for blowing bubbles in my chocolate milk from across the room…

Straw triangle

Enter Straws Thingy

[This is post #2 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first or next post.]

Without any ado whatsoever (other than yesterday’s post, and now this sentence, and now this parenthetical, and now…), here’s a pretty thingy made of 60 drinking straws:

[Straws Thingy]

After much deliberation and Simple English Wikipedia browsing, I decided to call it Straws Thingy.

So what is it? How does it work? What is it good for? Why does it induce vertigo if you stare at it too long? And how can you make one yourself?

I’ll cover all this and more in the coming weeks. (Except for that 4th question. You might want to ask your doctor about that one). And yes, I’m confident that you can indeed make your own. To start, I suggest heading to Target and picking up a box of up&up 5-color flexible straws, making sure you have at least 12 of each color.

[up&up Straws from Target]

Get Ready for NaBloPoMo 2015!

[This is post #1 in a mini-blog-post series for NaBloPoMo 2015. Jump to the next post.]

Welcome back to Three Cornered Things!

I’ve been hoping to restart this blog for a while now, and I have lots of topics and new projects that I’m excited to share with you! What I haven’t had lately includes: (1) a clear picture of exactly how I want to present this new content, and (2) the confidence and self-compassion to charge ahead without part (1).

That’s why I’m challenging myself to this month’s NaBloPoMo. With one (tiny!) blog post each day this month, let’s delve gradually but deeply into a small, recent project that I have thoroughly enjoyed designing, iterating on, and sharing locally through friends and workshops and events, but haven’t yet been able to write up for public viewing.

This challenge sets me up to accomplish this gradually, confident that by the end of the month I’ll have shared everything I hope to on this one project, and insecure in the fact that, as of this first post, I have no clue what the rest will look like beyond a very preliminary outline.

See you tomorrow!