"In formal terms, we say that an object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation does not change the object or its appearance. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa)"
Of the various types of symmetry described in the article, I find the reflection symmetry and rotation symmetry, the easiest to understand.
Rotation SymmetryThe interesting thing about symmetries is that they apply not only to nature, physics and mathematics, but to computing as well.Consider that you are writing an API or developing a programming language or a library. By design, you the code needs to be able to undo, what it's allowed to do.Consider a "C" programLet's say we carry out the following steps
1. Allocate memory
2. Copy from user
3. Open a file
4. Write to file
What happens in case there is an error in step 4?
We carry out the following steps
1. Close the file
2. Free the allocated memory
We need to do the same when our we are done writing to the file.
Now compare what we did with reflection symmetry. Did you find anything similar? I think we should as a guideline make our API/code symmetric with respect to reflection. In lay man's terms, the code should be able to undo its own effect.
Most programming languages and API are symmetric. "C" has malloc()/free(), open(), close(). "C++" has a constructor and a destructor for each class, so we are mostly good as far as this rule is concerned. "Java" on the other hand, provides a garbage collector to maintain the symmetry. Consider what would happen if you could just allocate memory and never free it up? Open files, but never close them? The system would soon become unstable, the weight would grow on one side (memory, descriptor leaks) and the system would crash.
The symmetry created by the garbage collector makes life easy for the lazy programmer, but losing control over symmetry can throw a system in a non-deterministic state. Consider the case where a system has a lot of memory, the garbage collector never kicks in (of-course a lot depends on the actual algorithm). When a critical application needs to run, it needs most of memory, but to get free memory, the garbage collector starts running, starving the critical application of CPU time, making it wait for its memory demand. All of this could have been avoided (the CPU time overhead), but being in control of symmetry and freeing the memory when not needed.
Coming to rotational symmetry, I am yet to find a good use case for it in computer science (for an average programmer). For those involved in graphical illustrations, we could exploit it to make it easier to draw illustrations.
Tuesday, November 07, 2006
The book Discover Physics by Benjamin Crowell, talks about symmetry in Physics and in Nature. Wikipedia defines symmetry as