Google recently announced Google Photos. The basic idea is that Photos will fix the problem of photos and photo sharing. I’ll admit that there’s a problem. Even using iOS and Mac OS X, it’s never as simple as you would like to store and share photos. On top of that, we are taking more and more photos every year.

Someone once said “the best camera is the one you have with you.” Yes, this is true. Just like most humans, I tend to take pictures of tons of things. Oh look, a water fountain. Now am at my kid’s soccer match—more pictures. Is that flower near the road? You get the idea, right? We all take a bunch of pictures. It doesn’t cost anything to take a picture (well, not counting the electricity used).





The real question to consider: do I take more pictures now than I did in the past? This is obviously true—but how many pictures do I take? If I use Google Photos, how many pictures will I be storing in the future?

Total Photos vs. Time

Let’s do this. I will make a plot of the total number of pictures on my computer as a function of year they were created. In Mac OS X Yosemite, I can create a smart folder in Finder that only shows jpeg images from the year 2002. I can also do a similar thing in the Photos app. These two methods can (and will) give slightly different numbers. The Photos app will probably just be pictures I took (or maybe from my wife). The Finder smart folder will give all jpeg images. These could be from any source. Some were created by me for my blog and others were created for other purposes.

Here is the plot of the total number of pictures vs year created.

This graph shows that I seem to be taking more pictures each year. But graphs can do more than just look pretty. They can be useful too. Could I use this to estimate how many pictures I will have by next year? Of course I can. The first step is to fit some type of mathematical function to the data. If the data doesn’t look linear, how do you pick a function for the data? Well, you could just guess until you find something that looks nice. You could also fit a function based on your own ideas. I could say “hey, it looks like the number of images added increases each year – this could be an exponential function.” That would work too. Or perhaps the best source of a function is to make a different plot. If I use the same data but plot the vertical data with a log axis, you might be able to see if it’s an exponential. Like this:

An exponential function would look like a straight line on a log-plot. That looks fairly straight. So, that’s how I picked an exponential function. There is one problem. I have time values in years. If I use the year as the actual time, things might get crazy. I am going to change the time such that the year 2000 is t = 0. Here is the new plot with the fitting function.

Let me write the function in a slightly different form (and with units). With this equation I can estimate the total number of pictures I will have in the future. Let’s pick the year 2020 (which would be t = 20 yrs). If I put that value in for the time, I get a value for N at 117 thousand (117,171 actually). That’s a lot of pictures. I wonder what they will be pictures of? Maybe it will be pictures of me on my fusion powered jet-pack. How many pictures will I take each year? I can get an estimate for this by taking the derivative of the total number of pictures with respect to time. (Here is a quick intro to derivatives.) If I put in 14.5 years (my numbering scheme says that right now I am in the 14.5th year) into this expression, it says that I am currently taking 5,604 pictures per year. That might seem crazy, but yes, I take a lot of pictures. Let’s just convert that to images per day and I get 15.3 pic/day. Wow. I should slow down. Ok, one last question before I give homework. Suppose I take a picture every second of the day. What year will this happen according to my model? To start this problem, let me convert the picture rate of 1 per second into images per year. Oh, I am going to call the image rate N‘. Now I am going to take that same N‘ equation above and solve for t. There are two important things to point out. First, the units work. I have one over years to the -1 power. That gives a time in years. Second, the stuff inside the natural log is unitless (you can’t take the natural log of something with units). Now I can put in my value for N‘ and I get a year value of t = 37.653 years (or the year 2037). So at that point, I will be basically taking pictures all day long and not have any time to ride in my jet pack. That’s going to suck. No really, this is just a model. Clearly it won’t work for large values of t. At some point I am going to just max out on pictures (maybe like 100 a day).

Homework

Looking at the number of images I have isn’t just interesting (don’t laugh—you know it’s cool), it’s also great for homework. For the first homework question, let me show you this other graph. This plot shows the total number of music files on my computer as a function of year (as reported by both iTunes and the Finder).

What can you do with this data? Why is it so different than the number of images on my computer? What year will the my music purchasing rate be zero (it’s pretty close to now)?

Now for some more questions about the number of images.