Fake projective plane
Imagine if you were drawing a picture of a plane, like the ones you see flying in the sky. Now, imagine if instead of drawing a normal plane that looks like a rectangle with wings and a tail, you drew a different kind of plane that had a curved shape to it. This is called a "fake projective plane."
To understand why it's called a projective plane and why it's fake, we need to know a little bit about geometry. In geometry, we often use something called a "projective space" to help us understand different shapes and how they relate to each other. A projective space is like a special kind of drawing board that allows us to extend lines and shapes infinitely in any direction. It's sort of like playing with a piece of elastic: if you pull it out in one direction, it stretches outwards in all directions.
Now, back to the fake projective plane. Imagine if you took a flat sheet of paper and glued two curved pieces together to make a sort of cone shape. You could then stretch this shape out over a projective space, and it would look like a flat plane. But if you tried to measure the angles in this "plane," you'd find that they don't add up like they should. In a normal plane, the angles of a triangle add up to 180 degrees. But in a fake projective plane, they add up to more than 180 degrees!
This is why it's called a "fake" projective plane - it looks like a projective plane, but it doesn't follow the same rules. It's a little bit like if you made a toy car out of a potato instead of using real car parts. It might look sort of like a car, but it wouldn't work properly.
So in summary, a fake projective plane is a shape that looks like a flat plane, but it's actually made up of two curved pieces and doesn't follow the same rules as a normal plane. It's called "fake" because it's not a true projective plane, even though it might look like one.
To understand why it's called a projective plane and why it's fake, we need to know a little bit about geometry. In geometry, we often use something called a "projective space" to help us understand different shapes and how they relate to each other. A projective space is like a special kind of drawing board that allows us to extend lines and shapes infinitely in any direction. It's sort of like playing with a piece of elastic: if you pull it out in one direction, it stretches outwards in all directions.
Now, back to the fake projective plane. Imagine if you took a flat sheet of paper and glued two curved pieces together to make a sort of cone shape. You could then stretch this shape out over a projective space, and it would look like a flat plane. But if you tried to measure the angles in this "plane," you'd find that they don't add up like they should. In a normal plane, the angles of a triangle add up to 180 degrees. But in a fake projective plane, they add up to more than 180 degrees!
This is why it's called a "fake" projective plane - it looks like a projective plane, but it doesn't follow the same rules. It's a little bit like if you made a toy car out of a potato instead of using real car parts. It might look sort of like a car, but it wouldn't work properly.
So in summary, a fake projective plane is a shape that looks like a flat plane, but it's actually made up of two curved pieces and doesn't follow the same rules as a normal plane. It's called "fake" because it's not a true projective plane, even though it might look like one.