Examining paper models of maple seeds that do not rotate, even when we tried to build them so that they would, suggests that the distribution of weight is critical. If you cut a strip of stiff paper and drop it, it whirls around its long axis as it drops. If you weight one end of the strip heavily, it falls straight down without whirling at all. The maple seed must fit somewhere between these two extremes, but where? To refine our models it would be helpful to visualize the weight distribution in the seed and the model. You can find out the average distribution of weight in a maple seed if you have a lot of them, and can cut them into pieces carefully so that you can measure the average weight of the seed itself, the rib near the seed, the rest of the rib, and the wing. I didn't have enough seeds to do this. Why don't you do this and post your results on your web site? But there is another way to estimate the weight of the origami model's parts. We will borrow a trick from calculus, and divide the paper that we fold to make the model into a 30x32 grid of squares.
This technique is used to compute the centroid of irregular shapes as an introduction to integration in calculus. In our case it lets us count the number of squares that are in the layers of folded paper. In the figures below you can see how the origami instructions were diagrammed, with the numbers of layers generated by each fold totalled and drawn onto their respective areas.
Steps three and four were critical because the repeated folds made it too difficult to assign weights to the resulting areas, which became very small. Therefore I lumped the weights together and made the assumption that this estimate was accurate enough to use in the remaining steps.
The numbers of layers in each grid square of the finished model's shape give us the weight distribution. Coloring the squares according to the number of layers gives us a 2D contour plot.
I visualize this kind of data better if it is shown as a 3D surface. The weight is proportional to the height of the surface, and the heights are easily separated by color. You can make such a plot by entering the grid's values into a spreadsheet and plotting the grid as a 3D surface, coloring it to distinguish the different weights. I use an old Macintosh program called WingZ, which is still very useful although it is now more than seven years old! Here is the data entered into a 16x16 matrix...
...and here is the resulting graph, colored to match the 2D contour plot, smoothed out, and rotated to show the distribution of weight in the origami model. How does it compare to the weight distribution of real maple seeds? How would you design and build a cut paper model that would be very likely to rotate?
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