What is Fermat's Last Theorem?

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Math? Really?

Don't worry, I'll talk about Star Trek too.

I'll start with some history. Pierre de Fermat was a French lawyer and a mathematician. I would like to say "born in ___," but as it turns out, there is some dispute on the topic. Fermat had a brother, whom their parents also named "Pierre," who unfortunately died quite young. This happened in either 1601 or 1607. The other year is the year our Pierre was born.

Pierre de Fermat was exceedingly clever, and quite a good mathematician. It was his work on geometric series that Newton poured over (in part) when he was trying to suss out that whole "calculus" thing. Fermat attracted a fair bit of (extremely nerdy) controversy in his career as a mathematician, particularly for his method of calculating minimums and maximums of functions. Descartes, of "I think therefore I am" fame had developed his own way of accomplishing this task, and particularly disliked Fermat.

Pierre de Fermat
Maybe Descartes had a point,
doesn't he look smug?

Fermat was a fan of a greek guy called Diophantus, and his book Arithmetica. Fermat owned a copy of the book, and in the margins he would write out theorems (a specific mathematical 'idea'), but he never included proofs. In fact, he would often be quite annoying on this front, describing his proofs as "miraculous," but not writing them them down due to "lack of space in the margins" (paraphrased). Then he died. These scribbled notes became known as "Fermat's Theorems," and throughout history, mathematicians have proven each and every one of them to be correct. Except one. This became Fermat's Last Theorem, a sort of Holy Grail for modern mathematicians.

Alright, there is one equation, but we'll get through it, I promise. Here it is:

What this equation says is that if you add two numbers, both raised to the same power, they can add up to another number also raised to that same power. An example would be using squared numbers: 3²+4²=5² because 9 + 16 = 25. In fact there are loads of solutions when you plug in squared numbers (n=2). Throughout history, no one has found a solution with anything higher than a squared number (no solutions for n>2). This was curious, because it asserted that this equation had an infinite number of solutions for n=2, but absolutely none for anything above n=2.

TL;DR: For this equation, nothing above n=2 is possible. No whole numbers will ever make the equation true. This is what Fermat claimed.

So mathematicians went to work trying to prove Fermat right once again. The business of actually writing down a mathematical proof is usually devilishly tricky (I'm told). You have to cover all your bases and wrap up all loose ends, so there are no flaws. Worst of all, you have to get everyone else to believe you.

At this point in the story, in steps Andrew Wiles. Andrew Wiles first heard about the theorem when he was about 10, and set out to prove the theorem. He soon learned that his knowledge was too limited, and he gave up proving the theorem, but still pursued a career in mathematics. A while later, he was presented with an opportunity to professionally try to solve the theorem by woking on another problem. He worked at it for years, and in 1994, published a proof for the Last Theorem. It didn't quite work the first time and he then published a final version a few years later that has stood up to all the scrutiny the mathematical community can muster.


So we've done it, we've solved the Last Theorem.


Star Trek. I did promise to talk about Star Trek in the beginning of this post. In an episode in second season of the Next Generation (Royale), the show opens with Picard thinking about the theorem when Riker steps in. He describes the theorem briefly and says:

"In our arrogance, we feel we are so advanced, and yet we cannot unravel a simple knot tied by a part-time French mathematician, working alone, without a computer."

This episode aired on March 27th, 1989. I enjoy thinking that a problem the writers of Star Trek thought would remain unsolved into the 24th century was solved within a decade of the writing of that episode. If one looks at the quote, we can perhaps feel a little bit arrogant about out mathematical prowess as a species, even if we've a long way to go on other fronts to catch up with Picard and his crew, it seems we are, at least, on our way.


Cheers, 

      - Scott



LINKSTORM:

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Click a video, any video. This comedian has quite an unusual style. He is very clever, and quite funny

Why Does the Moon Look Bigger Near the Horizon?

Before I start, I'm experimenting with email notification. If your'e interested, you can go to 

 https://tinyletter.com/scottsieke 
and enter an email and hit "subscribe." I'll send out a quick notification when I post something, and that's it. They don't share you email with anyone other than me.

On the the good stuff:

To start, I just read a great book by astronomer and science evangelist Phil Plait called Bad Astronomy. In it, he dispels many misconceptions that nearly everybody has about astronomical concepts, myself included. I'm going to go over my favorite here, but if you want the much more thorough version, along with much more, I highly recommend picking up a copy of his book.

If you were wondering, this is what the first page looks like.
(Results may vary)

Moving on. Nearly everyone is aware that the moon appears larger on the horizon, and nearly everyone, including myself, thought they knew why. There are several theories floating around, but the most common, and the one I believed, is that the moon looks larger near the horizon because there are trees and buildings and all that to compare it against. When the moon is high in the sky, it is all alone, and so looks smaller. Even if this is not precisely what you thought, it sounds true enough, and is therefore a quite pernicious falsehood.

This excellent photo was taken by Shay Stephens,
and was featured here on APOD.

The best way to test an idea is not to try to prove the idea right, but rather to try to prove it wrong. In our case, how could we prove this moon hypothesis wrong? If we think trees and buildings are the cause of the illusion, let's find an area with no trees or buildings, such as a beach or cruise liner. As it happens, the illusion holds true even on the "scientifically idealized" horizon of the ocean, where there is nothing to compare the moon against. Now for the other counter-example: when in the center of a large city or a "non-idealized horizon," buildings (and light, but that's beside the point) block your view of much of the sky. However, when the moon is located within the same reference frame as the building, and you can see both high up in the sky, the moon still looks small, even with the building to compare it against.

It turns out the answer is indeed an interesting illusion, but not that exact illusion.

When people depict the sky, we draw it as if it is about the same distance away in all directions:

Image: TWCarlson

... or so it seems. In illustrations such as this, we draw the sky to be equidistant in all directions, but as it turns out, in actuality we perceive the horizons to be farther away than the zenith.

I'll get back to the moon in a minute, but this is the crux of the illusion, so I'm going to flesh it out a bit. Rather than the celestial sphere looking like a hemisphere as in the above drawings, we perceive it as more of a shallow or flattened bowl, with the perceived zenith much closer to the observer. A great way to test this is to gather up: 1) Some friends you don't mind pestering and 2) a protractor. Head outside, and ask your friend(s) to point up so their arm is at a 45 degree angle to the ground. Measure the angle and record the results. What you will get is a measurement that is almost certainly between 30 and 40 degrees, and certainly not 45 (unless you have clever friends). Using the protractor, point your own arm up to 45 degrees and it will almost certainly surprise you just how far up 45 degrees in the sky is. This is because, for whatever reason (feel free to speculate wildly here) the horizon just seems much farther away than the zenith.

On to the moon! The moon is always roughly the same size in the sky. It varies by about 4%, but that's pretty imperceptible to us.

Image: Marco Langbroek

Remember the "supermoon?" That's it.

If the whole sky takes up 360 degrees all the way around, the moon takes up about half a degree, or less than your pinkie nail at arms length (try it).

The moon looks bigger near the horizon, because the horizon seems farther away. An object the same size on a background that seems farther away will look larger than one on a "closer" background.

The lower line is analogous to the moon at zenith (closer) and the upper line is analogous to the moon near the horizon (further).

Here is a great graphic that pulls everything together in one image:

Image: Lloyd Kaufman and James H. Kaufman

Cheers,

     - Scott

LINKSTORM:

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This topo map may help (Warning: this image is monstrous)

Possibly the most time sink-y page on Wikipedia, but also an important one

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Why is Nepal's Flag Strange Looking?

I'm experimenting with email notification. If your'e interested, you can go to 
 https://tinyletter.com/scottsieke 
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To start off with, here is Nepal's flag:

The current flag of Nepal

The first thing you probably noticed was that it is not a quadrilateral like every other national flag on the planet. The second thing was probably either "that is wonderful," or "why would anyone choose that as their state flag?"

To answer the question of why the flag is shaped so distinctly, one must go back quite a ways. In fact to around 800 BC (or much earlier by some accounts) when the Kirat people inhabited the Nepal area. Their flag looked like this:

Image: Danesh Rai

So the answer to the question may be "it is... because it always was." Because that's not very satisfying, at least not to me, maybe learning about the meaning of the flag will help. The two most oft-quoted reasons are that it symbolizes the Himalayan Mountains, and it's representative of the two dominant religions in the area: Hinduism and Buddhism, but these both seem a bit retcon to me.

This site has a much more satisfying answer to both the question of the flags meaning and the reasons behind its shape. Two separate branches of the Rana Dynasty which ruled the area in the late 19th and early 20th centuries had two different pennons (which are the essentially the same as the triangular collegiate pennants) for each branch, and in 1962, when the constitution was written, the flags were combined into the familiar flag we see today

...But how would you go about making it, or writing down that shape? In 1962, they couldn't simply attach a .jpeg to their constitution, so they came up with another way.

<sidenote>
Remember back in high school geometry class when you learned all about geometric constructions? What real world application did that ever have? The answer, it turns out, is basically zero, but not quite zero. As it happens, if you want to create your very own Nepalese flag you'll need to call upon these ancient skills.
</sidenote>

Written into Schedule 1, of the Nepalese Constitution (Scroll to the very bottom) is a detailed geometric construction of the county's flag. This is in fact the only correct way to produce the flag according to the countries law.

Here is a great video produced by Brady Haran at Numberphile that led me to this topic.

After watching this video, I decided to make my own Nepalese Flag and I headed off to the FedEx store. I acquired a poster board and a sheet of brown paper that measured about 6'x5' and went to work. After about an hour, using nothing but a compass a straight edge and a length of string, I came up with this:

My very own Nepalese flag

It looks lovely above the television in my living room.

If, for some reason you didn't get you fill of geometric constructions after the Nepal flag video, this should certainly slake your odd thirst for geometry:

How to construct a 17-gon

Cheers,

   - Scott



LINKSTORM (I'll add possibly interesting links to the end of each post; just unrelated things I've run across):

Here is a great video of a group that sang the same song though many different genres of Western Music

This is a long video, but to me it's downright therapeutic to watch someone make something from scratch that turns out beautifully

Did you know humans emit visible light??

This is a great map of interconnected flavors
(Here is a high resolution)
(Here is an NPR story about it)

This is just lovely. Back to the Future meets surfing

This is an interactive topography map, and I never knew I wanted one of these.