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). 

Micro-photography: Not only a change of way of seeing (2)

  Take a look at this picture:

  Looks like a pelican soaring up from the water, doesn't it? The wings, the body, the neck and the long beak are clearly recognizable. Well, you will not be blamed if what you guess is a light-year away from the true answer because even professionals can't tell.
  The true object depicted in this image is the iris, ciliary body and lens of a canine (dog) eye. It is magnified 7.7X under an electron microscope. Who can ever think of a dog's eye when he/she is shown a picture like this?
  This is the fascination of micro-photography. It totally challenges our conventional thinkings. As the saying goes "seeing is believing", for a long period of time, we limit our mind to what we can see, but not things we can't see. When micro-photography is introduced, seeing is no longer believing because it is impossible for our bare eye to figure out what it a microscopic object is really like. Therefore, micro-photos can significantly improve our imagination, just like an abstract painting does.
  Another important role, and the most essential function of micro-photography, is to unveil the nature's mystery. What makes human-being different from other creatures on the plant is that we are not limited by the organic body mother nature has given us. Though we cannot see into microscopic levels, by using high multiplication microscopes, the things nature hides from our eyes can also be revealed. The meter-level world in which we are living extend to nano scale. It has not only scientific research values, but also great influence in promoting science to ordinary people.
  Compared to formulas and equations, pictures are more appealing to the public without sufficient science education. Micro-photos tend to be eye-catching due to their unique patterns, complicated structures and diversified coloration. This makes science studying microscopic world easier to accept to the public.
  OK, now it's time for some guesses!

Overwhelmed by the assignments and exams? Look at the beautiful side of maths! (1)

  Maths is, in the view of many fellows here in Georgia Tech, as one of the most dreadful and annoying courses. I can't agree more with that! The maths assignments are really hard to solve, lots of equations need to be memorized and lots of nerve-killing provings are exhausting the last bit of our brain capacity! But don't feel desperate about maths; it is, infact, the most beautiful field of science.
  Geometry is not just about triangles and circles, it is very extensive when you look beyond 2-dimensional space. In this series of posts, I would like to introduce you to the multi-dimensional world, in which you will see incredible geometric variations, the delicate microscopic structure of a mandelbrot set and intersecting Mobius Rings that construct multi-dimensional objects.
  After watching these videos, there are several possibilities. One is that you will become a fanatic of mathematics because you are so stroken by the incredible images. Or, you will end up "dead"(lol) thinking "why is it like this?", "how do mathematicians figure out what things should be like in a dimension that even themselves cannot see?" I would recommend that do not ask why, because you don't need to know why! Just enjoy the beauty of geometry!
  The video is with chinese subtitle because in China I attended a subtitle group, which translates English learning materials so that our firends can also understand them.
  Episode 1 (The video quality is very poor because this is my first time uploading a video on YouTube, I'm working on it to get an HD video format)
http://www.youtube.com/watch?v=Xif0DYi6bM4

Micro-photography: Not only a change of way of seeing (1)

  I am quite confused that those artists and scholars on art with conventional thinking are so opposed to modern photographic technique. They limit themselves to the ordinary photography, which they think contain less "connotation" and "authority" than a painting or sculpture. In their points of view, photography is a part of fast-food culture; people get information so directly that they no longer look deeply into the inner quintessense of the image. In a word, photography changes the way of seeing, making it a lot easier but in the price of fewer connotative elements.
  It is, partly true, but they are so shortsighted that they neglected the technological development in photography. These advancements in science and technology created a new type of art: microscopic images.
  Before electron microscopes were invented, it is impossible for human being to see anything on molecular level. Optic microscopes are strong enough, still what they can present is only a 2-dimensional colorless flat image. What's more, they work poorly in cooperation with cameras because the combination of the lens in the microscope and that in a camera may result in serious optic distortion.
  In the 1950s, however, electron microscopes expanded our sights 100 times further, enabling scientists to have a look of the world of molecules and cells. Soon they found out that the images they were looking at are so astonishing and delicate. For the first time, we know that the hypnotizing chemical signs written on the whiteboard or lethal bacterium can be so beautiful when we have a real look at them. Those powerful instruments created not only a new form of art, but also a brand-new world we've nevered imagined.
  After that, techniques for micro-photography developed quickly. STM(Scanning Tunnel Microscope), an enhanced version of electron microsope extended our sight to subatomic level. Advancement in coloration and computer graphics endow the image with color and a stereoscopic impression.
  This semester I would like post a series of micro-images from Nikon Small World, the world's most authoratative microphotography forum sponsored by the world's leading optic instrument manufacturer Nikon. In the first post of this series, I would like to attach one of the eye-catching images and guess what it is!