Scaling Factors

To keep myself going on this epic journey of weight loss, I've started making a photographic record. I took the first set of nine photographs on January 3rd, the first day of my diet. The photo set starts with a close-up of my face and includes full-body shots from the front, back, and both sides, two closer shots of me raising my arms, a side shot of me bending over, and a shot of me in a weightlifter's muscle pose just for kicks.

I weigh myself every week, just to prove I'm still losing weight, but I didn't want to take pictures that often. First of all, taking the pictures is a lot of work. I setup the camera on a tripod and start the timer then quickly pose for the shot. After the camera snaps the picture, I check the image, figure out what went wrong and where to stand or what to do to prevent it, and go again. What with sorting through the pictures and processing them, it takes about 2 hours.

Second, weighing myself is a test I need to perform regularly to confirm that I'm continuing to lose weight. The photos, on the other hand, are intended as an illustration of weight loss rather than a test, so they only need to be taken when they will show something.

I decided, more or less arbitrarily, to photograph myself again when I dropped through 360 pounds, which would mean I'd lost 40 pounds, or 10% of my body mass. That happened officially this last Friday, March 18, when I hit 358.5 pounds. That night, I took a second set of photos.

I added one of the photos next to the original photo in the header above so you can see the change 40 pounds brings. (Yes, that's always been me in the photo.) Not that much of a difference, is there? Side-by-side, I think I can see a very slight change. I was hoping it would be more obvious, but I guess I weigh so much that it takes a lot more than 40 pounds to make a difference. The problem is that fat is three dimensional, although only two dimensions matter.

Let me explain.

If I were to suddenly double in size, my weight would far more than double. My body is 3-dimensional, so I'd be twice as big in every direction: twice as tall, twice as wide, and twice as thick from front to back. That would give me (2 x 2 x 2 =) 8 times the volume and therefore 8 times the weight.

Working the math backwards, in order to double my weight, my size would have to increase by an amount that when multiplied by itself three times gives 2. That is, my body would have to be scaled to the cube root of 2, which is about 1.26.

However, my body doesn't gain wait by scaling equally in all directions. When I gain weight, it's not because I've gotten taller. I gain weight by adding to the layer of fat that's wrapped around my torso and limbs. My bones don't get any longer, they just accumulate fat around them. Seen end-on, my limbs and torso increase in cross section. I gain weight radially.

This radial expansion is essentially two dimensional, so my weight is proportional to the square of the change in the width of a part of my body. Doubling my weight would require my size to scale up by the square root of 2, which is about 1.41.

Now let's look at what's happend to me in the last two and a half months. I lost 10% of my weight, so my size must have scaled by roughly the square root of 0.9, or about 0.95. In other words, the visible width of my body is still about 95% of what it was at the beginning of the year. In concrete terms, I only lost 5% of my stomach.

That's just barely enough to be visible. Both images above are cropped from larger ones. I just cropped the first one back in January to show my torso. It's 141 pixels wide. A 5% reduction should shrink it by 7 pixels to 134.

I processed the second image this weekend, cropping it to look as much like the first one as possible. I didn't count the pixels, and it was much larger when I cropped it. I shrunk it to a vertical height of 200 pixels like the first image and posted it up there. Only then did I do the math in this post. I just went back to check and, amazingly, the width came to exactly 134 pixels.