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Oct17

# Straight and Curved Lines – Geometry

A long, long time ago (give or take 2,000 years) there was a culture, to which we owe a sizable part of the mathematics that we know today: Ancient Greece. The most important contribution probably came from Euclid, who gathered all the information that was known about mathematics at the time and compiled it in books called The Elements. He didn’t only gather and organize mathematics, but also “logically” organized, or shall we say, created a system that organized content material according to its implied logic (deductive reasoning). This was how axioms and theorems came into play… (an axiom implies a theorem), can you imagine how important this was? It’s hard to grasp…

Euclid reserved a large portion of his books for geometry (the Greeks loved that!), in fact, it was in this section that his work remained intact until the 19th century (but now it’s all about advanced mathematics). So really, what we study in geometry today at school started more than two thousand years ago!

We are focusing today’s post on studying straight lines and curved lines, just like Euclid studied all those years ago.

There are a lot of ways to define straight and curved lines; the most elaborate way to define them is the following:

• A straight line is a succession of points that are aligned in the same direction. Or in other words, in order to go from one point to another, we never change direction.

• On the contrary, the points of a curved line do change direction from one point to the next.

We can observe these lines in the following image:

But this isn’t the only way of defining them! The original way (and the one that they use currently in mathematics) is more akin to the method that Euclid himself used. Think of two points on a piece of paper. How many ways can you go from one point to the other?

If there aren’t any obstacles, there are plenty of ways to do this…for example:

And that’s not even all the ways! Right? So, here’s the key question: what line, of all the lines we can draw, is the shortest? In other words: which is the shortest way from A to B? That’s it! The last line, the blue one. And that’s how we define a straight line.

Between two points, the line that connects them is straight if it is the shortest possible distance between them.

If the line isn’t the shortest distance between the two points, it is a curved line.

But wait! What about the second line that we drew? This is a special case because it’s not one line, but rather, multiple lines.

• The line that connects A and C.
• The line that connects C and D.
• The line that connects D and E.
• The line that connects E and B.

So then, we still need to know what exactly is a line…how do we know if we have one or multiple lines? This is where it gets a little complicated, but I’ll try my best to explain it.

Generally speaking, a line can’t have corners (which is what we call a derivative in math). If a line has a corner, it isn’t just one line but multiple lines. In our example, we have 3 corners.

Now, we can divide what we first called a line into several parts because “the line that connects A with B” isn’t correct anymore, because it isn’t a line!

How many straight lines do you think you can draw between A and B? Just one, right? But you can draw a lot of curved lines! Euclid and all of the mathematicians that followed him thought the same for a long, long time. Until when in the 19th century came a man by the name of Gauss who thought…so what would happen if I put A and B on a sphere? For example, an airplane that leaves from Boston to Mumbai can’t follow a straight line (assuming that we can’t make a tunnel), so, what path would it take? And most importantly, is there another one that exists?

We’ll take on this mystery in a future post…

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• UtthaanSep 18 2019, 8:14 PM

That was comprehensive. Need more