# Higher resolution

So my normal workflow for creating images here is to sketch ideas, try to create them in MATLAB, and then annotate them in Skitch, if necessary (or sketch things directly into Skitch).  Both these programs are fantastic for typical image manipulation and creating diagrams, though Skitch is (necessarily) a little underpowered.  I have run into this problem once before, where Skitch will take screenshots off your computer (from, for example, a webpage, or a MATLAB figure window), but you cannot layer multiple screenshots into the same image without working a little too hard.  Also, yesterday’s post had some pretty terrible resolution images that I produced in Skitch by trying to just put a textbox in, and then save it.

The 1000th line. Click for full resolution!

The problem with both of these is that I crossed a line into more serious image editing- creating layers, or looking for very high resolution.  I finally worked out that open-source Gimp (and I would assume it’s commercial analogue, Photoshop), a program intended for this sort of image manipulation, is a better choice.  Hence I can offer today a high res version of the 1000th line (1.3MB) and 2000th line (3.1MB) of Pascal’s triangle.  The 1000th is 6pt font, and usually readable with compression.  The 2000th line is 5pt font (otherwise the canvas was too big… it is a 100MB+ file before compression), and only sometimes readable.

The 2000th line. Click for higher resolution!

Compare this to Wikipedia’s image that helped to inspire this experiment, which they display by coloring the pixels in a grayscale from 1 to 10, one pixel per digit.  This has the benefit of higher compression, since we do not need to keep track of the value of each pixel, and easier reading from a macro level.

From Wikipedia.

Finally, just to keep things a little original, here is a similar plot of the first 2000 terms of the Fibonacci sequence.  Again, the font is pretty small, but it gives a good idea of a famous relationship, which is that for large numbers, the nth Fibonacci number is close to $\phi^n/\sqrt{5}$, where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio.  This is an easy and beautiful derivation using eigenvalues in linear algebra.  But for now, just notice that the number of digits (that is, the log of the number) looks like it increases linearly.  This could lead us to try to prove that $\log{F_n} \approx k\cdot n$, where $F_n$ is the nth Fibonacci number, and k is just some number.  It turns out we would find that $\log{F_n} \approx \log{\phi}\cdot n-\log{\sqrt{5}}$ which, while sort of ugly, is a linear function in n.

The first 1000 Fibonacci numbers, written vertically. Click for full resolution.