# 32 Straws Thingys! Aaah!

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

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!)

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…

PSST!

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:

BOPGY, BPYGO, BGYOP, BPGOY, BGPYO, BYPGO, BOYPG, BYOGP, BPOYG, BOGYP, BYGPO, and BGOPY.

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?

### Notes

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. []

# Scheming Colors

[This is post #20 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first , previous, or next post.]

Yesterday I asked what differentiates one of these from the other two:

The answer: color scheme! They all have 5 tetrahedra, one in each color, but they’re not positioned in the same way every time. The two Straws Thingys on the edges are identical, though it’s not apparent until we rotate one to match the other:

To avoid motion sickness, here’s that final frame again. Check that the colors match perfectly in the two assemblies:

By contrast, when comparing one of these “original” Straws Thingys with the remaining “odd one out” (pictured above, center), the colors refuse to match up, try as you might to rotate them into alignment. We can get pretty close, though: below, only green and yellow have swapped places. (The “original” is on the right, with the “odd one out” on the left.)

Does this mean we get a third unique color scheme if we swap, say, green and blue instead? How many colorways are possible in total? Let’s investigate this tomorrow.

# Odd One Out?

[This is post #19 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Congrats again on finishing your Straws Thingy! (If you had or are having trouble, please let me know.)

During the next week we’ll prod more deeply into a few features of this sculpture’s symmetries, especially those that become apparent during construction.

To begin, a puzzle: all three of these Straws Thingys were built precisely according to the instructions outlined earlier, and yet, they’re not identical arrangements! What differs?

This is not meant to be a trick question: their dissimilarity is clearly visible in this image without zooming in or guessing. And in fact, two of these three are identical. Who’s the odd one out? Are there other ways this difference can manifest?

# Step 10: A Savage Ending

[This is post #18 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

The straws are locked and loaded, but how do we get the scaffold off?

Violently!

It’s finally complete. Congratulations on building your very own Straws Thingy!

Look forward to a return to mathematical musings tomorrow.

# Step 9: Quintuplicate!

[This is post #17 in a mini-blog-post series for NaBloPoMo 2015. Jump to the first, previous, or next post.]

Here’s where we left off yesterday:

Now just repeat four more times!