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 \(\rho(x,y)\) to be the line containing the three points \(\overline{X_{2} Y_{3}} \cap \overline{X_{3} Y_{2}}\) , \(\overline{X_{1} Y_{3}} \cap \overline{X_{3} Y_{1}}\) ,and \(\overline{X_{1} Y_{2}} \cap \overline{X_{2} Y_{1}}\) ,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 \(\sigma\) be a permutation in S3, the permutation group on three letters. Then \(\sigma\) can be thought of as acting on the space of ordered triples of points.For example, if \(\sigma = \left( \begin{array}{ccc}1 & 2 & 3\\2 & 3 & 1\\\end{array} \right) \) then \(\sigma x=(X_{2},X_{3},X_{1})\) .The question is: What effect do permutations have on the line constructed by Pappus' Theorem? That is, Is their any relationship between \(\rho(x,y)\) and \(\rho(\sigma x, y)\) or even \(\rho(\sigma x, \tau x)\) for \(\sigma, \tau \in S_{3}\) ?The applet below allows us to explore the results.

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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 \(\rho(x,y)\) .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 \(\sigma\) and \(\tau\) ,where \(\sigma\) is an order three permutation and \(\tau\) is an order two permutation. It should be easy to check that \(\rho(x,y)=\rho(\nu x, \nu y)\) for all \(\nu \in S_{3}\) .

Next Step: Observation One .