The multi-dimension world (2)

    It's time to do some maths! But don't be afraid, this has nothing to do with Taylor's Theorem or Integration. It's all about imagination. 

    Suppose you are a "2-D Human", that means, you have only length and height, but no width (a good news for ones who want to keep fit lol), you happened to have an opportunity to take a trip to the 3-D world. How would you explain that fantastic world to your poor fellows who have never been there? 

    Parallel projection may be a good solution. As we all know, the projection of a sphere on the 2-D plane is a circle. Therefore, if you live in 2-D world, you can say "a circle is the projection of a 3-D object, sphere. *

    But parallel projection does not always work. What would the projection of a cube like? You may say it's a square. Well, it is in most cases. But what about if you turn it over a bit and take one of its vertices as the top instead of a whole surface? The projection would be a hexagon (try it by folding a paper cube by yourself). Thus, we need another method to describe a polyhedron in 2-D plane. And the Ancient Greek geometrist Hipparchus gave us the answer, stereographic projection.

    You will know how it works in the video shown below. Here's a brief summary in order that you won't get lost. Sphere is a very interesting 3-D object: it is 2-dimensional as it has only one surface, but occupies 3-D space, and thus we call the sphere in 3-D space a "2-D Sphere" (a similar object is the Mobius Band that we are very familiar with). By expanding the polyhedron to a sphere, the points on the surfaces of the polyhedron now lie on the same surface instead of different surfaces. Therefore, stereographic projection works in any circumstances because it can describe the relationship of any two points on the polyhedron. 

http://www.youtube.com/watch?v=2CZRR4DW6ck

    

*some important instructions: It is worth noting that even if you, as the lucky "2-D man", know what the projection is like, you will never form a explicit impression of what a sphere is exactly like because your brain is "2-D"(without the ability to construct 3-D feelings). This is a trap many people got stuck into when learning N-D geometry. The correct method to learn abstract geometry is not figuring out what they look like in N-D space, but what they look like in 3-D space(or 2-D space in the "2-D man"'s perspective).