What's on the back of the envelope?

The name of this blog is "Back of the Envelope".  One of my earliest blog posts was a long explanation about why I named the blog that.  The short version, from that post, can be summed up quickly:

The expression is common enough, but if you're not familiar with it, a back of the envelope calculation is a quick, simple calculation done as an estimate. It's called "back of the envelope" because it can be written out on a small sheet of paper . . . When I first applied for this address on blogspot, the idea was to name the blog after myself . . . Nothing really felt right, though, so I started thinking of other names, a name appropriate for an engineer writing about things he was distinctly unqualified to discuss. It took surprisingly little time to come up with "Back of the Envelope."

I've always used the image of an envelope with something written on the back as the symbol of this site.  In fact, this is the one I had for a long time:

The old back of the envelope symbol.

With the new template, I wanted to change the design while keeping the concept.  The new design is the background of this page.  Unfortunately, since I used the full size of the background that Blogger recommends (1800x1600 pixels, or close to it), you probably can't see the whole thing unless you have a super-high resolution display, even without the blog contents covering it.  So here's the full image, at a reduced resolution:

The new background.

As you can see, the central equation is the same.  This is the bra-ket notation used in quantum physics, and shows the inner product between two quantum computation values, 0 and + (which is a superposition of 0 and 1), so the overlap of 0 and + (technically it's the inner product, but it's the degree to which the two are the same) is one over the square-root of two.

What about the rest of the calculations?  Are they legitimate, or just random doodlings?  They're all legitimate, and equations I've used before, though it's been years.  Hopefully there aren't any mistakes.

The next equation, in red at the top, is just a circle divided into six parts, with one part divided in half.  The equation calculates the area of that section, but it's mainly an excuse for me to estimate pi as three.  That's a common estimate to use for pi when you're just doing a back of the envelope calculation.  Another useful estimate is 5 dB, or the square-root of 10.

On the left side is a charged particle over a ground plane.  This results in an image in the ground plane.  The charge in the ground plane responds in such a way that it's equivalent to an equal and opposite charge reflecting the placement of the first charge.  This results in the equation below, which is also the equation for the potential for a charge dipole.  Charge dipoles consist of equal and opposite charges close together, so that they minimize each other's effects.  A ground plane effectively converts a charge into a charge dipole, which is why ground planes help reduce noise coming from the circuits they're placed under (they also tend to minimize noise coupling into the circuit).

Below that, at the bottom of the page, is the time-invariant form of Schrodinger's Equation, since I figured I needed that on the back of the envelope.

On the right side is a 3-bit Gray code.  This is a binary sequence where only a single bit changes for each step of the sequence.  This was originally used as a method of binary counting for mechanical switches. Since mechanical switches don't change instantaneously, switching from 011 to 100 (3 to 4 in binary), could result in spurious outputs as each switch changes at a different time.  By making it equal the change from 010 to 110 instead, there are no spurious values between them.  In modern digital computers, this particular reason is not as relevant, but it is still useful for error correction.  A Gray code can be visualized as a cube, shown above, where each step travels along the edge of the cube.  I included the cube, with convenient arrows, mainly to give people a clue that I was doing a Gray code, rather than let them think I was trying to count in binary and getting it wrong.  I'm not sure whether it worked or not.

So that's everything.  I hope you enjoyed this boring math post.  I also hope I didn't mess up any of these equations.