Truncated projective plane
Okay, kiddo, imagine you have a flat piece of paper, like a square, and you want to draw some lines on it. But not just any lines - these are special lines that never end! They go on forever, like a magic road that never stops.
Now, if you draw enough of these lines on your paper, something interesting happens. They start to intersect each other in different ways, and you get little areas or sections of the paper that are enclosed by the lines. These areas are called regions.
Here's the cool part: if you keep drawing lines on your paper in just the right way, you can make sure that every region is bordered by exactly three lines. No more, no less. Any more or less, and it won't work.
This special kind of paper with magic never-ending lines, where every region is bordered by exactly three lines, is called a projective plane. It's like a new kind of land, with its own rules.
Now, you might be wondering: can you make a projective plane with just a certain number of lines? Like, what's the minimum number of lines you need to draw to make a projective plane?
The answer is, there isn't just one! There are lots of different projective planes you can make, with different numbers of lines and different ways of arranging them.
But there's one in particular that people like to talk about - it's called the truncated projective plane. "Truncated" means cut off or shortened, so that gives you a clue.
To make a truncated projective plane, you start with a regular projective plane with a certain number of lines. Then, you "cut off" or remove some of the lines and regions to make a new shape.
Here's the tricky part: you have to remove the right number of lines and regions in just the right way, so that the new shape still follows the rules of a projective plane. Specifically, you want to be sure that each region is still bordered by exactly three lines.
If you do it right, you'll end up with a smaller and slightly different projective plane, called a truncated projective plane. It's like a mini version of the regular one, but with its own special features and quirks.
So that's the story of the truncated projective plane, kiddo. It's a magical land of never-ending lines and enclosed regions, where every region has exactly three borders. And you can make a smaller version of it by carefully cutting off some lines and regions from the regular one. Cool, huh?
Now, if you draw enough of these lines on your paper, something interesting happens. They start to intersect each other in different ways, and you get little areas or sections of the paper that are enclosed by the lines. These areas are called regions.
Here's the cool part: if you keep drawing lines on your paper in just the right way, you can make sure that every region is bordered by exactly three lines. No more, no less. Any more or less, and it won't work.
This special kind of paper with magic never-ending lines, where every region is bordered by exactly three lines, is called a projective plane. It's like a new kind of land, with its own rules.
Now, you might be wondering: can you make a projective plane with just a certain number of lines? Like, what's the minimum number of lines you need to draw to make a projective plane?
The answer is, there isn't just one! There are lots of different projective planes you can make, with different numbers of lines and different ways of arranging them.
But there's one in particular that people like to talk about - it's called the truncated projective plane. "Truncated" means cut off or shortened, so that gives you a clue.
To make a truncated projective plane, you start with a regular projective plane with a certain number of lines. Then, you "cut off" or remove some of the lines and regions to make a new shape.
Here's the tricky part: you have to remove the right number of lines and regions in just the right way, so that the new shape still follows the rules of a projective plane. Specifically, you want to be sure that each region is still bordered by exactly three lines.
If you do it right, you'll end up with a smaller and slightly different projective plane, called a truncated projective plane. It's like a mini version of the regular one, but with its own special features and quirks.
So that's the story of the truncated projective plane, kiddo. It's a magical land of never-ending lines and enclosed regions, where every region has exactly three borders. And you can make a smaller version of it by carefully cutting off some lines and regions from the regular one. Cool, huh?