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31 August 2001

Going Round and Round

By George E. Hrabovsky, President of MAST

News from MAST

Well, I am finally getting this column out; I only hope it gets in on time.

We have acquired some new equipment here: A Mettler Toledo AB104 analytical balance, an HP 8450A Spectrophotometer (!), an old, but working, a CAPCO model 700 gas chromatograph, an incubator, several glass petrie dishes, and both nutrient broth and nutrient agar (for microbiology experiments). We are actually doing experiments in colorimetry and should have some reports for the Bulletin in the not too distant future.

Symmetry Groups

As promised I am going to talk about symmetry groups (a little late, but what the hey). In a previous column, (print version only) "The More Things Change, The More Some Things Stay the Same," I described the importance of conservation laws. When some physical property is conserved then its behavior is symmetrical with respect to certain relations involving the physical property. The trick is, of course, to find these. We can use groups to help us.

For example, if we assume the shape of an equilateral triangle to be conserved (i.e. the distance between points never changes) then we would expect certain motions about the center of the triangle to be symmetrical. Let us look at such a triangle,

here we have labeled the vertices {1, 2, 3}, we will label this arrangement . If we rotate the triangle 120° the triangle will still look roughly the same and we then get this arrangement with the vertices now at {2, 3, 1}, we will label this .

Here we perform another such rotation we get the vertices {3, 1, 2}, we will call this .

If we imagine a line extending from 1 to the midpoint of the segment 23, then we switch 2 and 3 we get the vertices {1, 3, 2} and we will call this .

If we rotate the triangle and then perform another reflection we get the vertices {2, 1, 3} and we call this .

Finally, if we rotate the triangle twice and perform another reflection we get the set of vertices {3, 2, 1}and we call this .

We now have all of the vertices for various rotations and reflections. We will see if this forms a groups or not. Each combination of vertices forms an element of the set of orientations of our triangle. We will now construct a table of how each operation acts upon an initial orientation. We begin with as our initial orientation. If we simply apply our initial orientation (no operations) we get again. If we rotate by 120° we get , and so on. If we go through and perform all operations on all of the possible initial configurations we get the following table (called a Cayley Table).

It is fairly easy to just look at the table and see that every operation yields an element of the set (thus the set is closed under the rotations and reflections). I leave it to you to show that the operations are associative (though this too is easily seen from the table). The identity operation is . I leave it as an exercise to you to show that there is an inverse to every operation resulting in . Under these properties this forms a group.

Question: Is this group commutative?

What reveals symmetry are those operations that leave the set unchanged. We begin with and we apply a 120° rotation to get . Now we have and we apply a 240° rotation and we get . In a like fashion we can study all possible combinations that lead to our starting point, whatever configuration that is. This gives us a clue to what is conserved, since the only things not changing are the distances between the points, we would guess (even if we had not know this already) that the length of the segments remain unchanged.

A Note in Thanks

I would like to thank Allen Hubbard and Kenneth Levasseur for their work with Mathematica in gneeral and their excellent book Exploring Abstract Algebra with Mathematica, this was the inspiration for most of the illustrations and their software was used throughout. This book is published by TELOS a Springer-Verlag imprint (1999).

A Note About MAST Courses

MAST is currently offering courses in Self-Study, The Conduct of Science, and Scientific Writing. Over the next month lessons for the Basic Mathematics I and II courses will appear and I hope to have some lessons for the Basic Calculus course by the end of September. Beginning with the Basic Mathematics I course Wolfram Research (the makers of Mathematica) will be making a special offer to all SAS members who officially register for courses using Mathematica: A free six-month license will be issued to any member who takes such a course, those who take multiple courses will be treated as full-time students and will be allowed to buy the student version of Mathematica for only $140 (this is identical to the professional version, without the big reference books and with no technical support) and the ability to upgrade to the professional version for $350 later on (a total of $490, pretty good when you consider Mathematica retails for $1,200).

For Those of You on a Budget

Wolfram Research has just released the Mathematical Explorer, a subset of Mathematica that costs less than $80. I have just received my copy and will write a detailed review when I get a chance. My immediate impression is that it is very nice.

Converted by Mathematica      August 29, 2001