Check out the latest addition to my sculpture page: 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.
[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.
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.
If you want to bring some origami back into the project, you can instead make a shallow crease, like this:
Either way, remember that the flexy bits will be visible in the final piece, so avoid cutting or creasing into them.
[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!
Or on your wrists!
Or atop your head!
Or crowning your Christmas tree!
So let’s start talking about how you can make your own, step by careful step. What’ll you do with yours?!
[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):
And here’s finding the tight, optimal packing:
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):
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:
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!
[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.
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.
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:
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.
[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?
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.
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.
[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.
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):
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…
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.
[I have spent the last few weeks worth of expository writing preparing the content (linked to) in this post, hence the longer-than-usual wait. Enjoy!]
Here’s a rather complicated collection of 54 loops:
No two loops link through each other, and yet, these loops really don’t want to come apart. How hard is it to separate them? Well, suppose they were made of thin, bendable string, and we could move and (un)tangle them as much as we wanted as long as we kept the strings intact. Could we separate them? Absolutely not!
What if we allowed ourselves to cross strands through each other—real string can’t do this, of course. Then we could certainly get the loops apart, but how many such crossings would we need to use? As it turns out, we would need at least 108 strand crossings! This is a good first approximation of how much these loops don’t want to separate.
If the strands had open ends, then there would be no such knot-theoretic obstructions to building this shape. So we could, for example, make it with paperclips! (This is not easy…)
I call this Pair o’ Boxes, and it is one of seven new sculptures that I have recently uploaded to my Mathematical Sculpture page. More of these sculptures are shown below, and each has a detailed mathematical description (and larger pictures) at my sculpture page. Go check them out! The Pair o’ Boxes page in particular explains much more about the knot theory of the above structure, including a proof that 108 crossings are necessary to fully separate the strands.
Sorry for the shameless advertising; we now return to our regularly-scheduled posting.