Symmetry

When talk about symmetry, most people will think about mirror. Yes, it is a kind of symmetry for mirror image, but there are a lot more. In fact, I found that our students’ quality is so low that, they cannot be excited by unexpected things. In primary 1 or 2, we already exposed by many kind of symmetry form rotation, translation, and reflection. How come no one can remember? When I was small, I only knew the reflection symmetry. After I learned rotation and translation, this concept is printed so dark in my mind, because of unexpected.

Anyway, back to our topic. As I mentioned above, there are rotation and translation symmetry as well. For seeing why rotation can be called symmetric, we can use 2 mirrors, making an angle with each other, and then we can product “angle reflection”. For example, making the angle between the 2 mirrors be 90 degrees, we are producing a 90 degrees rotation symmetry, because the graph is “repeating” itself for every 90 degrees rotation. Moreover, because the graph repeats by 4 times, we called it as 4-fold symmetry. When we make the 2 mirror at 60 degrees apart, then we are making 6-fold symmetry. If we accept the mirror reflection is symmetry, then we must accept rotation is also symmetry.

How about Translation symmetry? Can translation symmetry be produced by mirrors? Yes, it can. If we have 3 equal mirrors, we can make a kaleidoscope. In the kaleidoscope, beside the images in the 3 corners, we can also see a repeating pattern extended to the whole space!

From above, we can see a very important concept of symmetry – repeat. More precisely, repeated after certain transform. For reflection, the pattern repeated after a reflection (reflection is a kind of transform). Patterns repeated after rotation, after translation.  

Now, we are standing on a point for a conceptual jump. If something unchanged (repeat) from time to time, then we called it has symmetry over time. Let’s take examples, you weak up every morning at 8am, then that action is translation symmetry over time. We play a movie clip and then reverse play, and then it is reflection symmetry over time. In fact, many things have translation symmetry over time. However, if you have a clock, but after a while it broken, than the clock has symmetry breaking. It is no longer symmetry over time. Therefore, in fact, we have a lot of thing has short term symmetry and only few things have truly symmetry over time.

The physical laws and constants are the only things have forever symmetry over time. 

Can any one see now? the law of conservation of momentum and energy is just from the symmetry. From a lot experience, we believe our univerise is governed by a set of symmetry law.

for a deep understanding of symmetry, reader should study the Group theory. Group means, inside a group, every memeber share the same symmetry porperties.