Shady Trees

Some time ago, I posted an image to Facebook of a few fractal-like trees.

Mildly interesting in many ways.

After a decently sizable comment thread, I challenged someone to calculate the area occupied by the trees by hand. Of course, someone challenged me to do the same. Thus began the journey.

First Steps

To start, I decided that I needed a reliable estimate for the area of the trees so I could judge later how accurate my final answer was. Enter GIMP. (Please don’t hate me!)

Same image with contrast turned super-high.

Increasing the contrast allows me to better count the amount of white pixels.

Histogram of the image showing values for white.

~53.7% of the pixels in the image are white; therefore, ~46.3% of the pixels in the image are black. I’ll just leave the area in percent, although I could conceivably use pixels or some other unit.

Calculation

Shapes.

Alrighty, now for the hand calculations. I printed out the picture and drawn shapes in the whites of the image. This way, I can easily approximate the area.

Then I measured each individual shape in centimeters. After adding up the areas of all of these shapes and subtracting from the area (I won’t show the extensive work necessary here), I ended up with ~147.4cm² as the area of the black. Converted to a percentage (given that the picture’s dimensions were 15.6cm x 21cm), I ended up with ~45% as the area of the black.

45% compared with our earlier figure of 46.3%. Dang.

Why do you even do this to yourself?

I needed to preserve my honor or commit sudoku.

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