Geometry

Hey Guys!

The other day I was idly switching channels on TV and happened to come across one of the most overplayed movies ever: The Avengers. Now I'm really not a huge fan of the movie (No offense Marvel Fanboys) but I happened to notice something really cool.

The whole plot of the movie revolves around this random blue thing from Asgard which is the energy source of the future: The Tesseract. It's quite flashy in the movie, being one of the infinity stones, but what most people probably didn't notice is that a tesseract is, in fact, a real mathematical object. A tesseract is the 4 Dimensional equivalent of a Cube. Yep, that's right, the entire movie is based around a 4D cube.

Now, there are probably tons of questions rampaging around in your mind. But what is a 4D cube? What is the fourth dimension anyway?! How can something so outrageous exist? Well, look no further, for all your questions are about to be answered. In this post, I'll be defining what a dimension is.

Dimensions

The first thing that we need to define is a dimension. Now, what is a dimension? We've heard the term multiple times (high school math classes come to mind), but what is a formal definition for it? For that, we need to define a point in space.

What's a point? What's space?

Euclid defined a point as 'that which has no part'. Now, if I take a piece of paper and make a jab with a pencil, I'll get a small area. If I take a magnifying glass and look at it, I'll be able to see smaller points within it. But that's not a point in Euclidean geometry. What Euclid says is that it has no length, no breadth, and no height. If I take a magnifying glass, I won't see it any bigger.

Now the whole concept of a point seems counter-intuitive, but it works. Points are the smallest we can get. If we are able to define a point, we are essentially able to define everything else- or rather, everything else is defined from the definition of a point.

Space

This is just a collection of points.

Space is nothing but a collection of points. In fact, it's a collection of an infinite amount of points. This definition brings a truckload of complexities upon itself, but, in essence, that's really all there is. If you want the math, space is defined as the set of points which all fulfill a common relation. But space can be vastly more complex: In fact, we now see space much more than simply a collection of points. Space is a dynamic, living, and ever-changing entity, as we'll get to when we progress in geometry.

If you want to know more about how an infinite amount of things can add up to a finite thing, just wait- I'll soon be making another post about infinities.

Defining Dimensions

Well, essentially, a dimension is a measurement. It helps us to locate a point in space.

Now, let's say that in our hypothetical universe, there's only one point. We don't have anything else to define, so we don't need any measurement to locate the point- that's all there is. A point is then, quite easily, zero-dimensional.

Lines and creating numbers

Let's say I line some more points up in just one direction; they're now aligned in a horizontal strip stretching from here till there. That's what we call a line. Lines have infinite length, so now we can begin talking about abstract quantities like length, which is another way of measurement.

So how am I going to measure points on a line? I can simply take a random point and mark it as zero. Then, I measure equal intervals and start labeling them as 1, 2, 3, -1, -2, -3, and so on. I can extend this nomenclature indefinitely.
But wait! There are many points which I've left in the middle! I can just treat them as fractions of the equal intervals and label them 1/2, 1/3, 1/4, and so on.

If you think carefully, there are still some points left to label. These are numbers which I cannot express as fractions (in the form p/q, where p and q are random numbers and q is not zero), and these are termed as irrational numbers. A fitting name, because they don't obey some laws the others do.

I can still mark the numbers on the line though. For instance, √2 is an irrational number, and I can express it using a bit of the Pythagorean theorem.

But there are still numbers left! These evasive numbers are those which I cannot express on the number line with normal methods because they don't obey any equation which I can plot. They are called the transcendental numbers, and a good example is pi. I can apply a couple of special tricks to plot them, and then I can safely say that there are no more numbers I need to plot (or are there?).

Planes

No, we're not taking off to flight.

Now, we've settled what a line is. Picture in your mind that you are standing on a surface and a line is stretching in the left-right direction on the ground. We've assigned every point on that line a label. All I need to do to summon that point up is say its label.

We're going to do some very neat work here- imagine stretching the exact same line in the back-forth direction. We now have what's called a plane- a flat ground! To define any point on it, we're going to need another number- The label of the left-right line (let's call this the x-axis, and the label the x-coordinate) and the label of the back-forth line (y-axis and y-coordinate).

And look- I can now define any point in that region with just two numbers! By convention, we name the points as A(x,y), where x and y are the coordinates of the point A. 

Space

We can keep repeating this process indefinitely. I can draw another up-down line and label it the same way, and call it the z-axis. We need another number to label the point now, the z-coordinate, so our point becomes A(x,y,z). 

I can keep doing this, but dimensions after 3 become very counter-intuitive. This is because the world that we live in has only 3 dimensions, but that doesn't mean that the dimensions can't hypothetically exist.

Geometry

That brings us to the end of the first geometry page. Geometry's going to keep us up for a long time- and it's going to be amazing every step of the way. 

The reason I'm making these pages is to give an idea to people that geometry is something that anyone can understand, and showing people the beauty hidden in the magic of the shapes. From Euclid to Riemann to Einstein, we'll be exploring the beauty of geometry in many ways. We'll be getting our hands dirty, of course, and doing the calculations: after all, math is something that has to be done. 

This was just an introductory page, introducing the layman to the wonderful world Euclid dreamed up and Descartes built upon. But this just marks the beginning- soon we'll be inventing axioms, making predictions, building remarkable proofs, and exploring what these people did hundreds of years ago. We'll see how dimensions manifest in exotic ways, how we can still map what happens inside space, and, if we're lucky, create strange new universes out of nothing.

Thanks for reading and be sure to subscribe!