March 14th is Pi Day (3.14) and it is also Einstein’s birthday. In celebration of Pi I give you this… (And yes, I know what day it is, this is posted a day early so I can forward it to select folks.)

The most inefficient way to estimate Pi, by using the Mandelbrot 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 tomorrow.

If you want the proof that this is really pi, see this companion video. (Well, sort of… there is a little bit of hand waving, but then how much do you really remember about the complex plane?)

The administration discovered accounting mistakes last fall, but it did not notify lawmakers until the administration included adjustments to make up for the errors in Brown’s budget proposal last week.

Expanding Medicaid was a cornerstone of Obamacare. Said to be working smoothly in California. 1.9 billion dollars discovered in the fall. Before or after the election, I wonder. (Not that a few billion dollars would have changed voting habits on the Left Coast – where it is all about FEELINGS.) No one will lose their job over this error.

The 1.9 billion error is compounding the problem that CA is not collecting taxes at the rate they expected. Total budget shortfall is closer to $3 billion. [Hat tip to Small Dead Animals]

Benoit Mandelbrot was a mathematician by training. He published papers on economics and finance, on turbulence in jet engines (and other environments) and on information technology. He worked at various universities in America and Europe, but he will be most remembered for the work he did while at IBM Research in California.

While at IBM he did work on fractal geometry that would change the way we do computer graphics – so that they are much more realistic when dealing with clouds, mountains, trees, and most other things in nature. The work on fractals impacted a lot of technology and science, but most people will remember only the Mandelbrot set, which bears his name – as a result of his work on fractals. (You can see an example of part of the set in the video below.)

In nature,technology and art the most common form of regularity is repetition: a single element repeated many times, as on a tile floor. But another form is possible, in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you “zoom in” to the whole. Fern leaves and Romanesco broccoli are two examples from nature.

One might have thought that such a simple and fundamental form of regularity would have been studied for hundreds, if not thousands, of years. But it was not. In fact, it rose to prominence only over the past 30 or so years—almost entirely through the efforts of one man, the mathematician Benoit Mandelbrot, who died in 2010 just before completing this autobiography. [Stephen Wolfram from The Father of Fractals November 2012]

The work done on fractal geometry by Mandelbrot and others has influenced a large part of science and technology. It is interesting to me that it has impacted several areas of medicine around how surgery is preformed on structures – like the liver – that are fractal in the nature of the veins and arteries that are a part of it. For an hour-long documentary, see The Clouds are Not Spheres.

This video below is a zoom into the Mandelbrot set. It is 5 minutes. There are much longer zooms – some going for hours. You can find those on YouTube by searching. (This zoom at just over 10 minutes is a little better, but twice as long.)

The way the diagrams and videos are coded is that points which are black are IN the set, the other colors denote points that are NOT in the set, and the colors denote how fast the series diverges. (For a description of what the set is, that is easier to understand than the Wikipedia link at the top of the post, see the explanation given by the Numberphile.)

Before her phone call, it’s something we hadn’t thought of – and, apparently, some voters hadn’t either.

They have to pay workers more. So they have to charge more. What exactly did you think was going to happen? (The unicorns stopped pooping money for worthy causes many years ago. I know, it’s shocking!)

Heidi Perry, director of Advent Child Care, explained that many parents were not happy with the idea of the cost increase. “I think there was a little bit of sticker shock. I had a lot of parents who walked into the office last night or this morning and say ‘am I reading this right?’ And I said yes you are.”

Costs for this church-run childcare facility will go up $300 per month, per child.

We reached out to several other daycares in the area, one says they haven’t even addressed this yet and another said the tuition increase is inevitable.

The referenced article is a review of polling statistics by one of the polling organizations. They gave Trump a 30 percent chance of winning on Tuesday, so they aren’t claiming to be clairvoyant, but they do explain why the polls (and the papers) got it so wrong.

First off, the polls didn’t get it that wrong.

In fact, the error in national polls wasn’t any worse than usual. Clinton was ahead by 3 to 4 percentage points in the final national polls. She already leads in the popular vote, and that lead will expand as mail ballots are counted from California and Washington, probably until she leads in the popular vote by 1 to 2 percentage points overall. That will mean only about a 2-point miss for the national polls. They may easily wind up being more accurate than in 2012, when they missed by 2.7 percentage points.

But, as the article points out, the election is not determined nationally, but at the state level. And there are a lot of problems at the state level when it comes to polling.

They have to raise property taxes to pay for police and fire and other pensions and school construction/maintenance. 318 million dollars in new taxes need to be raised this year. With a total of $588 million of new taxes overall. (I’m not sure when that other $270 million is due to hit, but in the next year or 3 certainly.)

For years – decades – Chicago has promised the unions everything, and not funded much of anything. Now the bill has come due, and the politicos are running scared because this is going to hit people in the wallet. Where it counts.

So now they are trying to come up with a way to shift the burden away from homeowners who live in the city (and voted for this madness for 80 years or more) and onto the other property owners. Of course that assumes they can figure out how to pay for it.

[Civic Federation President Laurence Msall] it is “difficult to see how the city could afford a property tax rebate plan given the current state of its finances and lack of reserves.”

You can’t wish away the net-present-value of a string of payments, just because it is easier than really negotiating with the unions.

The 100th Anniversary of Einstein presenting his General Theory of Relativity was this past Wednesday, November 25th. He presented his work to the to the Prussian Academy of Science, in Berlin.

I wasn’t going to say much about it because I thought few people would care. But the following video is too good to ignore.

General relativity gives us spacetime, warped space, warped time, and a host of other interesting effects.

The first equation is a basic form of Einstein’s Field Equations. This gives us (among other things) the curvature of spacetime due to mass, and other effects. The second equation gives the relative time-dilation effects due to differences in frame-of-reference. (The folks who built the GPS system needed to understand those equations because there are relativistic effects to take into account between our clocks on the ground and the clocks in the GPS satellites.)