Category Archives: Geometry

Step 7: Straws in Spaaace!

… three-dimensional space, to be precise.

[This is post #15 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Now that the scaffold is complete, let’s use it! Find your straws; today we’ll use just one of the five colors. Make sure they are already cut or creased as described in Step 1 few days ago.

crease prep

To insert a straw into the scaffold: the long end of the straw enters at an arrow,


and it tracks the dotted gray line, emerging only when this line ends:


This is important: each and every straw will be inserted with this same recipe, and you don’t want them in backwards, so here are those steps again:

Insert the long straw end at an arrow, and follow the dotted line to the exit.

Let’s do the same to the other two arrows on the same hexagonal face:


Now there are four faces touching these three straws: the one whose arrows we entered with, and the three we exited from. The pink straws will touch only these four faces! So all that’s left to do is repeat with these other three. Here I’m using three more arrows on a second face (before and after):


And here’s the result with all 12 straws inserted into all 4 faces:


Tomorrow we’ll close up the corners to finish our first tetrahedron!

Step 6: Pentagons Galore

[This is post #14 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first or previous, or next post.]

Three Dee, guys! We’re so close I can taste it! (tastes like paper…)

Take five of your finished scaffold units and lay them out in a star like this.


We’re going to staple the tabs together in pairs, to create a 5-sided hole in the center. Here’s the first staple:


And here are the other 4:


It won’t lie flat, and that’s exactly what we want! Lather, rinse, repeat with the other 5:


These are two halves of a sphere that we’re now going to join together. Form another pentagonal hole by stapling one scaffold unit from one half (bottom) between two other scaffold units from the other half (top):


All that’s left is some good old-fashioned pentagon hunting. Any time you see a five-sided hole that looks like it wants to close up, help it along! When you’re done, you should be staring at a soccer ball:


And that’s the finished scaffold! Great job!

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]

Red, Alert

[This is the 4th and last installment in the current series on mini-golf and ellipse geometry. See the previous ones here: #1, #2, #3.]

We must settle one more question to round out our elliptical arc: Why does light, when shot from one focus of an ellipse-shaped mirrored room, reflect back to the other focus? To answer this question, we’ll need a Fact, a Formalism, and a Fairy Tale.

The Fact

Recall that, in the previous post, we saw that ellipses can be described by distances: Any ellipse has two focus points F1 and F2 so that the total length of broken path F1XF2 is the same for every point X on the ellipse; let d be this common total distance. In fact, more is true: the length of F1XF2 is smaller than d if X is inside the ellipse, and larger than d if X is outside.

[Ellipse Distances]
Left: The distance of the broken path is greater, equal, or less than d depending on whether the central vertex is outside, on, or inside the ellipse. Right: A proof of the “outside” case.

To prove this fairly intuitive fact, we’ll use the “straight line principle”: the shortest distance between two points is a straight line. Indeed, when X is outside the ellipse (see right diagram above), straight-line path YF2 is shorter than the path YXF2 that detours through X, and so \(\)d = F_1 Y F_2 < F_1 Y X F_2[/latex]. See if you can fill in the case where X is inside the ellipse.

The Formalism

Recall that when light bounces off a straight mirror, the angle of incidence equals the angle of reflection. But here we’re discussing light bouncing off an ellipse, which is decidedly not straight. So we need to formally describe how light reflects off curved mirrors.

[Reflection off a Curve]
Reflection off of a curved mirror behaves like reflection off of the tangent line.

If we zoom into where the incoming light ray strikes a curved mirror (illustrated above), the mirror closely resembles a straight line, specifically its tangent line. This suggests that the light should behave as if it is reflecting off of this line, with equal angles as marked. This is indeed the rule governing ideal reflections on curved mirrors: the angles of incidence and reflection, as measured from the tangent line, should be equal.

The Fairy Tale

The last ingredient involves Little Red Riding Hood and her thirsty grandmother. Red is delivering cake and wine from her mother (point M) to her grandmother (point G), but she must first fill a bucket of water at the nearby stream S, which is conveniently shaped like a straight line. She was warned by her Brothers to watch out for a big bad wolf, so she must minimize her total walking distance. Where on the stream should she fill her bucket to minimize this distance?

[Little Red's Geometry Problem]
Left: Paths from M to S to G transform into paths from M’ to G, the shortest of which is straight. Right: This straight path behaves like a mirror.

To answer this, imagine reflecting the first leg of Red’s journey across line S, so her path from M to S to G gets reflected to a path from M’ to G. The reverse may be done as well: any path from M’ to G turns into a path from M to G that stops somewhere along S. So we just need to find the shortest path from M’ to G. But this is easy: it’s just the straight path M’G. So Red’s shortest path from M to S to G is the one that stops at Z.

Notice that this shortest path is the one with equal angles as marked. This means Red’s best strategy is to pretend the stream is a mirror and to follow the light ray that bounces directly to grandma’s house. This neatly exemplifies Fermat’s principle, which says that light tends to follow the fastest routes.

The Proof

With these pieces in place, we can finish today’s question in a flash. Let’s say light from focus F1 hits an ellipse at point X, as illustrated below. Why does this ray bounce off the ellipse toward F2? If we draw the tangent line L at point X, by The Formalism above, this question is equivalent to: why does the light ray bounce off of line L toward F2?

[Reflection Off an Ellipse]
Why are the two marked angles equal? Because the white path solves Red’s Fairy Tale problem.

Let’s reimagine this Grimm scenario by thinking of F1 as Red’s mother’s house, F2 as grandma’s house, and L as the stream. I claim F1XF2 is the shortest path for Red to take. Why? If Y is any other point on line L, then Y is outside the ellipse, so by The Fact above, F1YF2 has distance longer than d. So F1XF2 is indeed the shortest. But by The Fairy Tale, we know that this shortest route behaves like light bouncing off of line L, i.e., the marked angles are indeed equal. So we’re done!