Weight Distribution of Origami Maple Seed

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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