14 March 2023

3.14 = Pi Day, and It Is Also Einstein’s Birthday

First, Einstein's birthday: March 14, 1879. Do you ever use a GPS? Then you are indebtted to Einstein for the Relativistic Time Dilation equation. (Shamelessly stolen from myself, from 2014.)

Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

I won't include the equation, since it won't add anything. You can find it in many places on the web. There are also effects because of the speed of the satellites relative to the observer on the ground. But if you know the math, you can calculate your position.

And it is Pi day. π = 3.1415926535... (That is 10 digits, and enough for most applications.)

One of the simplest ways to approximate π is to use the Gregory-Leibniz series.

π = (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + (4/13) - (4/15) ...

It is simple to understand, not simple to do. It takes 500,000 iterations to get to 5 decimal places.

More exactly, the series is expressed as:

While that series converges very slowly, it is not the most inefficient way to approximate pi. The most inefficient way to approximate Pi, is to use the Mandelbrot set.

This is the video Pi and the Mandelbrot Set from Numberphile

For those of you who don't remember what the Mandelbrot set is, here is a straightforward (if long-winded) description of the set, and a video of what the set looks like when you zoom in to examine a very small portion of the set.

(The colors in the rendition of the Mandelbrot set indicate if a given point is inside or outside the set. Points rendered in black are IN the set. Other colors are assigned to the points outside the set, and the various colors indicate how quickly the function for that point diverges under iteration. All that is explained in the video describing the set.)

OK, you've learned something. Now go plan to eat some pie with your lunch.

This was shamelessly stolen from myself, from posts from the past few years.

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