To get more understanding of 4-D objects, I'd like to talk about the fundamental building brick of the simplest objects in each dimension. In 1-D space, an object consists of one single edge and two vertices (a line is a 1-D object). In 2-D space, we can get, for example, a triangle with 1 face, 3 edges and 3 vertices. In 3-D space, we have a tetrahedron with 4 faces, 6 edges and 4 vertices. We can deduce the construction of the simplest 4-D objects using mathematical induction: as the number of vertices increases by one as dimension goes up, the object should have 5 vertices. Similarly, we can get that it has 10 edges and 10 faces (just like something people usually do in intelligence quiz). Then we get a 4-simplex, a name to call 'the simplest object in n-dimensional world', just as the image below.
It's worth noting that what we see is only the projection of 4-simplex in the 2-D world. It's not what acually is like in 4-D world. In a tetrahedron (3-simplex), we get one surface by connecting three vertices together (a triangle). In 4-simplex, we get one surface by connecting four vertices. Thus, we have a 'tetrahedral surface' which sounds ridiculous. Actually it makes sense because the term 'surface' does not only apply to 2-D surfaces we see in daily life, but also 3-D surfaces, 4-D surfaces or whatever-we-what-D surfaces.
Well I think things are really getting confusing. But don't forget the main objective of this topic is to show the stunning pictures of multi-dimensional objects. I you can't understand anything just forget about it and enjoy watching them in the video shown below! There are even more incredible 4-D objects that really look so eve-catching.
http://www.youtube.com/watch?v=YeHi1ol61eA
Wow! This is a very interesting video. Trying to draw a 4D object on a black board, and then trying to visualize a 4D object on a 3D world.
回复删除However I don't understand. As far as I know, 4D is time, so is the video trying to help us visualize the concept of time or is the video saying that 4D is just adding another side the 3D?
Interesting note, did you know that a string has 36 dimensions?
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回复删除To Hyun Ki Lee,
回复删除The term '4D' in abstract geometry is different from that in physics. In physics we combine the concept 'time' with 'space', and thus result in a '4D-world' composed of the complex interaction between a 3-D space and a 1-D time. In abstract geometry, however, ther's no such thing. There's only space, and the '4-D world' is a simple addtion of an extra dimension to 3-D space. And that's what the video talks about. Still learning abstract geometry improve our understanding on the 4-D world in physics because we can analyze it in a similar way, though not identical.
How many dimensions does a string have? It is still a question to study. I've read some articles on strings that it has been mathematically proved to have 11 or 12 dimensions. It is really interesting to know that it has 36!