Physics Rediscovered #3: The shapes of reality

Did you notice that we live in a tyranny?

Look around. You must see it. It's everywhere and at no point did any of us agree to this.

Time to wake up sheeple! Down with the tyranny of rectangles!

Yes, that's right. Everything is a goddamn rectangle.

From the bed you wake up in, through tables, desks, windows, doors, screens, parking spots, laptops, tablets, phones, keyboards, books, postcards, banknotes, to the coffin they bury you in.

I suspect I might be rectangle myself.

I get that arranging things in rectangles is efficient and all, but ffs is it boring.

Sometimes, I wonder if aliens will also share our fetish for all things rectangular.

I'm just gonna say what we're all thinking - circles matter!

Ok, I digress, but only a little.

You see, triangles, for example, seemed to go out of fashion in modern times, even though they are the basic building blocks, from which all other shapes are made.

However, the ancients had a special thing for triangles.

Especially, the chillest and the illest philosophers, big democracy mobsters, the OGs of formal mathematical proofs...

Give it up for the Greeeeeks!

Sokrateños 4 life!

Never skip geometry day

Triangles, squares or circles, the Greeks knew how to keep it real.

In the last episode we've covered how looking intensely at the night sky can give you mathematics.

This is what happened to the Babylonians, Egyptians, Mayans, and others. However, as far as we can tell, for them, it never went beyond the practical.

"Practical, shmactical" - ancient Greeks

For the Greeks it wasn't enough. They wanted to see how far the idea would take them, regardless of practical applications.

More so, they didn't want to just base it on really, really strong feelings. Send hard, formal proofs or GTFO, as they used to say back in the day. My Hellenic homies were just built different.

Suddenly ideas like axioms and theorems began to pop up everywhere like cookie consent forms.

For the uninitiated - an axiom is something I say and you better agree with me or there's gonna be trouble.

A theorem is like a mathematical murder case that you solve with logic and deductive reasoning, ending up with a proof … and a body.

Probably the most famous of all the theorems and certainly the foundation for a lot of things to come is the Pythagorean theorem.

E = mc^2

Where today we write it as:

a² + b² = c²

I say today, because for a long time the Greeks didn't use algebraic representations like the one above. They just couldn't be bothered as they were confident that they can express everything they needed through pure geometry.

For example, take the following, familiar, quadratic equation:

x² + ax = b

The Greeks would think about this as:

• take the area of a square with the side equal to x

• add to that the area of a rectangle with sides a and x

• all of that area needs to be equal b

They might then go about finding x by building a new square out of those components and doing some geometrical voodoo on it. I'll drop a Greek-style solution at the end of this article so as not to give you algebra shock.

I'm not irrational, you're irrational

This is just the tip of the iceberg. So much stuff is happening because of geometrical exploration.

By investigating an isosceles right triangle, Hippasus of Metapontum brought about the end of the world ... sort of.

You see, the early Greeks were convinced that whole numbers are the perfect representations of Nature itself. "All is number", as Pythagoras allegedly said. Even if something isn't a whole number, like 1.5, you can surely represent it as a ratio of whole numbers, like 3/2.

So now take the √2.

This didn't seem to be a problem, at first. It must be greater than 1, but smaller than 2. It could be difficult to pinpoint exactly, but maybe it's something like 7/4 or maybe 127/113. Either way it has to be expressible as a ratio of whole numbers. Right? Right?!?!

Well, Hippasus showed everyone that they couldn't be more wrong. He proved (perhaps not as the first one ever - sources sketchy) that nowhere in the whole wide world is there a ratio of whole numbers that can express √2. You just can't do it, ever. Today, we call these numbers irrational.

This was a devastating revelation as it went against the mainstream philosophy of mathematics and how it relates to the Universe. It also shook some of the foundation of already existing proofs. One legend has it that people got so pissed, that they've drowned Hppasus. Others say that he was so shocked at his result that he drowned himself. Either way water seemed to be involved.

Then you have Archimedes - the Eureka dude, who's plausibly annoyed that the Babylonians and Egyptians can't get π right (the ratio of the circumference of a circle to its diameter). So he proceeds to showing everyone how it's done.

He takes a circle, puts a hexagon inside and calculates the circumference of the hexagon. He compares that to the circumference of the circle - it's obviously not right, specifically less than it should be as there is some space left.

So he doubles down, getting a polygon with 12 sides, like this:

Persistence is required for burnout

This one fits much better, but still not good enough. Calculating the lengths of sides of these polygons is tedious work so he repeats the process 3 more times, at which point he has enough. Archimedes ends up with a polygon of 96 sides. He then goes at it again but with the circle encased in such polygons.

All this grind gives him the lower and upper bound for π, which end up being 3.1408 and 3.1428, respectively. Pretty good, right?

Another irrational number, though. Zeus dammit!

This is not a heading

To make things more annoying, Zeno of Elea decides to throw a wrench into the machine.

He comes up with 4 paradoxical problems, which gave everyone over a millennium long migraine.

Here's one to give you a taste. This one is supposedly inspired by Homer's Iliad, specifically Achilles chasing Hector around the wall of Troy.

You have a slow and fast runner. The slow one has a head start and the fast one has to catch up.

However, as the fast one reaches the last position of the slow one, the slow one moves by some amount.

So the fast one needs to catchup again, but by that time the slow one moved again. And so on, ad infinitum (which is Latin and not Greek, but the whole text is in English so back off).

In the end the fast runner seems never to be able to catchup with the slow one. On that one reality seems to disagree. Guess reality is wrong then.

Why is this such a problem in the first place?

It's because, for all their swag, the Greeks didn't know how to deal with infinites.

Here's how we would handle this.

Let's say Achilles is twice as fast as Hector. Let's also say that Hector has a meter of a head start.

This means that as Achilles covers the initial 1 meter, Hector moves by 1/2 of a meter. As Achilles the covers that 1/2 meter, Hector makes 1/4 of a meter, and so on. So the distance covered by Achilles (or Hector) is:

d = 1 + 1/2 + 1/4 + 1/8 + ...

Now, let's pull out 1/2 from that fraction tail:

d = 1 + 1/2 * (1 + 1/2 + 1/4 + 1/8)

Whoa, check this out. The thing in the parenthesis is d again! So:

d = 1 + 1/2 * d

which makes d = 2. Boom!

Achilles is going to overtake Hector at the second meter. Take that, Greeks!

Achilles and Hector reading the Illiad and not liking it one bit

Let's do some lines

As you can see, this geometrical mindset started taking mathematics into the abstract.

Virtual numbers, unsolvable paradoxes, infinite divisions and much more. No one remembers what practical means anymore.

Then comes in my main man, Euclid. He lives in Alexandria so has access to practically the entire knowledge of the world contained in its famous Library.

He decides to take everything he knows, add his own stuff on top of it, and put the whole thing into a book (actually more like 13 books).

He calls it the Elements.

The Elements go viral immediately, arguably becoming more popular than the Twilight Saga, 50 Shades of Gray and all the Harry Potter books combined.

Euclid tells us many things there. Golden ratios, prime numbers, cones, pyramids, Platonic solids, you name it. However, planes are especially important.

I don't mean the flying cattle wagons that we cram into, trying to get to our vacation spot, arriving already exhausted. I mean infinite flat surfaces, on which everything happens.

On those planes Euclid places lines in different configurations. Paraphrasing one example, he says that if you have two straight lines, which are not parallel to each other, they will cross at some point. Nothing special here.

However, we can restate that to mean that if the lines are truly parallel then they will never cross. Like ever, even in infinity.

If you are bothered by the concept of an infinite, ever expanding Universe, then the idea of straight lines extending into infinity should be equally disturbing.

A geometry where parallel lines never cross, even in infinity is something we call a Euclidian geometry. Freeing ourselves of this Euclidian regime will be one of the most important breakthroughs in science, but that is a story for later.

Are we there yet?

We've just scratched the surface but this articles feels like it's going to be infinite so I'm leaving you with the promised Greek-style solution to the quadratic equation. Knock yourself out.

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Let's start with a general form of the equation:

ax² + bx = c

Let's divide by a to make it easier to work with:

I know it looks weird but there's a purpose.

They might go about solving this by building a giant, uber-square in the following way:

1. Take the square x².

2. Take the rectangle b/a, cut it in half. Now each one is a b/2a rectangle.

3. Attach each of the halves to the sides of the x² square.

Eventually, you end up with the following:

There's no reward under the ??? - just more algebra...

Cutting and pasting our original figures created the need to add additional area (???) to complete our uber-square. Since we are dealing with squares, we know that the area of the new addition has to be:

But we can't just add stuff for free. We know that the area of our original figures amounted to c/a. This means that the area of our uber-square is:

Because this is the area of the uber-square, then its side must be square root of that. But in the picture above we also see that the side is equal to x + b/2a. This means that:

By rearranging, we finally get:

Should look familiar if you paid attention in school.