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Unformatted text preview: 16 1. ALGEBRAIC THEMES R, and T .x / D Ax C b is an invertible afﬁne transformation of R3 , then
T .v/ is a vertex of the convex set T .R/.
1.4.7. Let be symmetry of a convex subset of R3 . Show that maps
vertices of R to vertices. (In particular, a symmetry of a rectangle or square
maps vertices to vertices.)
1.4.8. Show that there are exactly six symmetries of an equilateral triangle. 1.5. Permutations
Suppose that I put three identical objects in front of you on the table: This conﬁguration has symmetry, regardless of the nature of the objects or
their relative position, just because the objects are identical. If you glance
away, I could switch the objects around, and when you look back you
could not tell whether I had moved them. There is a remarkable insight
here: Symmetry is not intrinsically a geometric concept.
What are all the symmetries of the conﬁguration of three objects? Any
two objects can be switched while the third is left in place; there are three
such symmetries. One object can be put in the place of a second, the second in the place of the third, and the third in the place of the ﬁrst; There are
two possibilities for such a rearrangement (corresponding to the two ways
to traverse the vertices of a triangle). And there is the nonrearrangement,
in that all the objects are left in place. So there are six symmetries in all.
The symmetries of a conﬁguration of identical objects are called permutations.
What is the multiplication table for the set of six permutations of three
objects? Before we can work this out, we have to devise some sort of
bookkeeping system. Let’s number not the objects but the three positions
they occupy. Then we can describe each symmetry by recording for each
i , 1 Ä i Ä 3, the ﬁnal position of the object that starts in position i . For
example, the permutation that switches the objects in positions 1 and 3 and
leaves the object in position 2 in place will be described by
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- Fall '08