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Permutations and Pappus' Theorem
Let x
=(X1,X2,X3) and
y=(Y1,Y2,Y3) be
ordered triples of collinear points in the plane.Then we can define
to be the line containing the three points
,
,and
,which must be collinear in the projective plane according to Pappus' theorem.
Now, let x be the the ordered triple of collinear
points, (X1,X
2,X3), and let
be a permutation in S3, the permutation group on three letters.
Then
can be thought of as acting on the space of ordered triples of points.For
example, if
then
.The question is: What effect do permutations have on the line constructed
by Pappus' Theorem? That is, Is their any relationship between
and
or even
for
?The applet below allows us to explore the results.
You can move around the blue dots and the green
dots.The red dots represent the points constructed by Pappus' Theorem and
lie on the line
.You can affect the blue and the green dots by permutations by clicking the
on the menus to the right of the screen. Notice the numberings of the dots
change when the menu displaying permutations is clicked. This results in a
change in the line constructed by Pappus' theorem. The menu displays elements
of S3, here thought of as the group generated by
and
,where
is an order three permutation and
is an order two permutation. It should be easy to check that
for all
.
Next Step: Observation One
.
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