I was chatting with foldawayorigami about the hex box base, and he brought up some interesting ideas as far as using this as a base for different height/radius ratios for the same size of paper, using different shapes of paper to begin with (such as hexagon), and using this as a base for other shapes (such as a dodecagon). So, I’ll work through the math as I’m trying these out and take you on my journey! Now, mind you, I’m no mathologist, so brilliantandblonde and railehatesfun please check me (and, really anyone who’s better at math, feel free to tell me I’m wrong and how to do it right). Some of the math I’ll have to research or remember from what I was taught oh lord I’m old a decade ago, but some of it is going to be the makeshift intuitive kind. Hey, I’m the kind of guy who refuses to measure out divisions of paper with a ruler. If I can’t figure out the divisions by folding them, they’re obviously unimportant.
Before I begin with the math, although normally you would need to fold a completed model to see the emergent properties, we’ve all folded enough boxes that I should be safe with only folding two or three units. Buuuut we both know that I’ll probably end up completing the model.
So, the first thing I’m going talk about is the use of pleats to create interior angles. I don’t know the actual formula to figure out the interior angle of regular polygons, so I use the formula I came up with drunk: 180 - 360/n, where n is the number of sides, of course. The reason I like this is that, essentially, you’re attempting to take flat paper, which would be 180 degrees (how do I make the little circle thingy?) and subtracting the correct angle from half of it, which makes the paper stand up and have the correct interior angle. So for a hexagon, the interior angle is 120 and we’re subtracting 60, for an octagon, 135 is the interior and we’re subtracting 45. For a dodecagon, 150 and 30. If you’ve made Tomoko Fuse’s octagon box, I’m sure you’ve noticed the pleats. (note: I forgot that there was a specific word for the angle we’re subtracting - supplement! But, I told you it was gonna be rough, I’m just gonna leave it with the fumbling terminology.)
Now, of course, since we’re making pleats, we need to halve the angle we’re subtracting because the pleat takes 2x whatever the angle that is taken out. For Fuse, we see 22.5 degree angles and pleats, for my box, we have 30 degrees. I’m gonna try to use the same general format to make a dodecagon box, so I’ll need to make 15 degree pleats.
Now! Let’s talk radius. I was VERY LUCKY that I chose a hexagon while drunk. Hexagons are composed of equilateral triangles, which means that the radius is LITERALLY the side length. No special math needed. With the dodecahedron, using the sine rule, I got that the side:radius ratio is 1:~1.93 which means that I need to have the paper divided into fourths, use the top two horizontal sections as the rim and use the horizontal fourths each as a section (making 1:2 rectangles (not exact, but lets try it)). I should only need four pieces of paper. Hold for a moment while I fold it, please.
And it worked perfectly. So now, I’m going to try to make a hexbox with a radius:height ratio of 1:1.5. For this, I’m going to to divide the paper in vertical fourths, and fold the horizontal ½ line. I’ll take the bottom half of the paper and fold it to the middle (that’ll create the four squares which I’ll use to make the bottom of the box) and then fold the top edge to the newly created ¼ mark. That’ll make the edge. I should only need two pieces of paper.
And that worked perfectly as well. Now the only thing left is to try it with hexagonal paper - which I don’t have right now so I’ll have to fold and cut. And it’s almost time for work - I’ll do that tonight.
So far, though, what we’ve discovered is that the box will work for hypothetically any polygon as long as: 1.) the rectangles that you’re going to use for the base have one side that is equal to the edge of the polygon and one side that is equal to the radius of the polygon and 2.) you can fold a pleat for the supplement of the inside angle of the polygon.
I’ll work on the hex paper tonight.
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