11 September 2014

Cool Thing of the Day: Business Card Math...[]

Okay, hear me out on this one, because I know the title isn't all that exciting. But I promise that, so long as you're the right kind of geek, it's at least something that will pique you're interest and maybe occupy you for a few minutes, or hours possibly.

Warning! Incoming Math content from a former English major!

If you're offended by mathematics or by a Creative Writing major doing mathematics, feel free to skip this post.

So I work in an office. I'm not ashamed to admit it; it pays the bills and keeps me productively busy. But I get bored sometimes (especially after an eyes-bleedingly long day staring at Excel spreadsheets), so occasionally I will reach for something to recharge my brain. Often, I grab a spare sticky-note pad and sketch something (like Batman, or ninjas, or whatever), or maybe I'll grab a piece of paper and pretend I'm good at origami. Among the things handy to snatch for compulsive paper folding are business cards:

They're small, somewhat sturdy (and so stand up to repeated re-folding), and common enough that I don't feel like I'm wasting anything (plus I have a stash of my own blank ones that I made into a glue-bound notepad a while ago). So naturally, they're a common target.

I noticed something a while ago, though, which really caught the side of me that likes to mess around with math (though another side always reminds me that I never want to do math for a living): it's really easy to fold these things into equilateral triangles.

Check this out: the standard business card in the US is 2" tall by 3.5" wide. It just so turns out that this ratio is nearly perfect* for folding into 60° angles, which create equilateral triangles, and thus also for dividing into thirds.

* The ratio isn't exact. It would need to be 3.5" × ~2.02726" or ~3.464102" × 2" to be perfect, but it's certainly close enough. 

Folding Instructions

Step 1: Obtain an ordinary 2" × 3.5" business card. [/obvious] 

Step 2: Fold one corner over to the corner diagonally opposite (e.g. bottom left to top right) so that the corners touch. Try to get this as accurate as you can.

At this point, you've just made the crucial fold. If you unfold it, you'll see that the diagonal crease intersects the long edges so that the crease is ⅓ away from either end. So now you get to pick your own adventure:

If you want to fold it into thirds, go to Step 3.
If you want to make equilateral triangles, go to Step 5.

Step 3: Stick your thumbnail on one of the spots where the crease intersects a long edge and fold over at that point so that the edge meets itself (i.e. so it's not folded over at an angle).

Step 4: Now flip it all over and fold the other edge so the short edge meets the crease you made in the previous step. You may have to finagle it a bit so that it lines up (since the ratio's not completely exact).

Step 4 (alternate): Repeat Step 2 from the other side, so that there is now an X-shaped crease in the center. Then repeat Step 3 from that side. Again, also depending on how accurately you did the folds, you may have to finagle it a bit.

You're done!

Step 5: After you've done Step 3 and folded one corner over to the other, fold over the "wings" on either side, using the "top" point, which is where the two corners meet each other and one of the "bottom" points, which is where that side hits the crease, as your end points. When it folds over, the short edge of the card should go vertically down the middle and the other edge should be flush with the crease from Step 2. Repeat this step on the other side.

You're done!

Math Content

So here's what's behind this: equilateral triangles (i.e. those with all three sides the same length) have all their internal angles at 60°. When you make that first fold, you make a rough approximation of a 60° angle with the long side of the card.

More detail: You'll notice that when you finish up Step 5, the two end flaps are right triangles (meaning one angle is a 90° corner) half the size of the big ones. In fact, they're a rough approximation of a special type of right triangle, the 30-60-90 triangle. Click that link, read the Wikipedia page, then come back, if you're not familiar.

If you're like me, you've already drawn a diagram and mapped out how the different lengths relate to each other. If you're further like me, you set up some worksheet formulas (Excel or LibreOffice) to calculate side lengths for future triangles. But that's a bit beside the point.

The main math thing that has to be true for this to work is for the ratio of the long edge to the short edge of the card being 3:√3̅, or ~1.732051. So if you do the math, you'll see that the business card dimensions aren't exact, but they're pretty close.

Anyway, hope this was at least somewhat entertaining for you. I tried to add photos of me doing the folds, but my cell phone's camera refused to focus [/badexcuse].


No comments:

Post a Comment