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Saturday, February 21, 2015
Science ...

I’ve been writing too much lately about all the problems of the world and about life. It’s time to take a break for a little while and do some pondering about math and science. Allow me to forget the real for a spell and wander a bit into the land of abstraction.

Today I’d like to ruminate on the notion of SYMMETRY a bit. To most of us in our daily life, symmetry isn’t all that important. We notice it now and then when two things seem basically the same in their overall design, but some important features about them have been changed or swapped. For instance, there are many different versions of Oreo cookies (of which I was a big fan as a youngster; they’re way too sweet and over-processed for me now). The basic Oreo has two dark chocolate cookies surrounding a white-ish vanilla icing layer. But there is an “inverse Oreo” on the market that has light, vanilla-flavored cookies surrounding a dark chocolate filling. The two are different, but in a symmetrical fashion; some part of the basic “Oreo-ness” is still there. Also if you like photography and remember black and white film, you may have noticed the symmetry between a negative and its print (the flip between dark and light areas on the image).

In the world of science (and engineering, i.e. the art of applying science to everyday things), symmetry is really important. In understanding how things in our bodies work on the molecular level, symmetry between molecular structures and compositions can tell us a lot. It helps to understand how things work, and also why things sometimes don’t work (i.e. disease); and further, how to fix them when they don’t. Also, our overall bodies have certain symmetries and certain non-symmetries, e.g. between the left and right sides of our bodies overall and with our individual organs. Biologists also use symmetry concepts to classify life-forms and understand how they evolve. Symmetries show up at many different levels.

The field of particle physics depends a great deal on recognizing symmetries in the characteristics and interactions of all the different fields and particles. These types of symmetries can get really imaginative and abstract, pushing the meaning of the word way beyond its most common application. Particle physicists are interested in mirror symmetry (how a mirror image is similar yet different from the original image) and the usual geometric symmetries (flipping something about an axis in 3-D space), but they also study time symmetry, gauge symmetry, charge symmetry, and now they concentrating on “super-symmetry”.

Super-symmetry is a hypothetical (for now) association between bosons and fermions. I.e., every boson, such as a photon, theoretically as a super-symmetrical fermion cousin that is alike in many ways, but still qualifies as a fermion. Likewise, every fermion (e.g. an electron or quark) has a super-sym boson cousin. These super-cousins have not yet been discovered, although the CERN Large Hadron Collider team hopes to find them once they get the big machine running again later this year.

I’m not a scientist or mathematician, so I can’t offer a solid, water-tight definition of symmetry. You can find lots of material about this with a quick search. (The basic introduction to symmetry appears to be “a physical or mathematical feature of a system that is preserved or remains unchanged under some transformation”.) Nonetheless, just for fun, I’m going to ponder symmetry in a very common, easily-recognized fashion — e.g. your basic rotational symmetry.

But even on this elementary level, symmetry (rotational symmetry here) can be manifested in many different ways. On the chart below, I’ve shown 8 different versions of a word comprised of the letters “A”, “B” and “C”. Version # 1 is your basic ABC. Version two in effect twirls the letters in a circle, but lets them stand upright and face the same direction. You just swap the inner and outer letters. If there were more than 3 letters, you would repeat this for the other inner layers. E.g., ABCDE would become EDCBA. It also works for an even number of letters; e.g. ABCDEF becomes FEDCBA.

Version #3 likewise twirls the letters and lets them stand upright, but now requires them to reverse their individual horizontal direction. You don’t see this with the A, but you do notice that B and C now face the other way. This is what happens in the mirror; version 3 is version 1’s mirror image. Version 4 likewise it version 2’s mirror image. In version 4, we have every letter retain its original position, but do an about-face horizontally.

Version # 5 takes version 1 and rotates it on its horizontal axis, so that it flips vertically. So every letter is in it’s original spot, but they are upside-down. You don’t notice this for B and C, but you do certainly notice that A is different. Version # 6 does the same thing to version 2.

In versions #7 and #8, we have some real fun. We twirl versions 1 and 2 respectively on their horizontal axes (flip them vertically), and then spin the overall assembly on the vertical axis (flip it horizontally). This overall horizontal flip requires that each letter exchange its position with its opposite end partner, and also flip the way it faces.

Each of these versions is like the first version in some way; they all have some form of symmetry with the basic ABC. And yet, they each relate to it in a somewhat different fashion. Nonetheless, something of the old “ABC-ness” is always preserved. If you later came across some similarly scrambled variant of XYZ and knew the symmetry transformation rule, you could work backward and figure out that you were dealing with XYZ (as opposed to ZXY or ZYX).

It’s up to your imagination to see and recognize this. In applying such imagination to real-world problems, mathematicians and scientists have been able to figure out deeper underlying relationships, and thus better understand the processes that make physical things happen. And that has made all the difference between humans living in log-cabins and riding horses, and living in modern society and getting a ride from Uber via your smart phone.

Oh, let’s think about another little symmetry on my chart, one that manifests itself on the level of the individual letter (my explanations focused on the overall word). The capital letter A can be rotated horizontally (around its vertical axis) and yet look the same. But turn it upside down, and it looks Cyrillic. B and C don’t hold up too well with a horizontal flip, but rotate them vertically and they appear about the same. Some capital letters, like X and I and O, can flip both ways and survive intact. Poor old P, R and L get messed up by any kind of flip. So, we find another layer of symmetry within our overall example of symmetry.

All of this isn’t going to change the world, given how trivial my little example is. But apply it to molecules and atoms and atomic particles, and you can change the world. Even in your own routine daily life, it’s always good to keep an eye out for symmetry. You never know what you might otherwise have passed up.

◊   posted by Jim G @ 4:55 pm      
 
 


  1. Jim, I’m afraid that my comments today will be literally childish. Once the land of “math and science” is entered, I am lost and have no clue, except to notice things as a child would. Therefore, the best I will be able to do is the following few childish remarks. And by “childish” I might describe them further as noticing the little, unimportant, concrete things that have nothing to do with the thinking process going on. Be that as it may, that’s about all I’m able to do when it comes to math and, science, to say nothing of engineering. So here are my childish remarks:

    I noticed with interest the fact that you called engineering an “art”. How interesting that engineering would be called an “art” by someone who knows the field; and I do not doubt for a minute that it surely must be an art. Considering engineering an “art” is something that never occurred to me. I’d be interested in where the “art” comes in when one is doing engineering.

    I “got” mirror symmetry and geometric symmetry, but after that I was completely lost with the rest of the “symmetries”. Any comment I would have I’m sure would elicit a response I once got from an engineering prof. I mistakenly tho’t that auditing his course might be “interesting”; the second mistake I made was to ask a question (no longer remembered). He told me that if I had to ask that question, I should get out of the class”; and I left.

    Lastly, and here I confess this is yet another lack on my part, I was completely unable to understand how symmetry can “make things happen”. *Make*?? It’s not that I don’t believe you; I think my problem is that there is too big a jump in the explanation between “symmetry” and the “make” of “make things happen”. But, again, this is due to my complete lack of mathematical and scientific ability. I’ve often tho’t I’m dyslexic when it comes to numbers; and over the years, I’ve come to understand the problems people have who are dyslexic when it comes to words. I can look at a 2, for instance; and if I do not concentrate hard enuf on what that number is, *2*, I am liable to come up with a 9 or a 5 or anything else. I was the despair of my father when my report card show a “C” in math; and his joy in adding random numbers on license plates was completely lost on me. I preferred to wonder about the lives of the people who lived in the houses I passed as we drove along.

    I can see you completely enjoy the fun of noticing symmetries in the world. As much as I’d like to share that with you (or even argue a point with you) in your post; bu it’s a lost cause when it comes to me. Sorry. MCS

    Comment by Mary S. — February 22, 2015 @ 2:14 pm

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