Permutations and Pappus' Theorem



	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,X₂,X₃), and let $\sigma$ be a permutation in S₃, the permutation group on threeletters. 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 $\rho(\sigma x, y)$ oreven $\rho(\sigma x, \tau x)$ for $\sigma, \tau \in S_{3}$ ?The applet below allows us to explore the results. alt="Yourbrowser understands the <APPLET> tag but isn't running the applet, forsome reason." Your browser is completely ignoring the <APPLET> tag!Get Netscape. You can move around the blue dots and the green dots.The red dots represent the points constructed by Pappus' Theorem and lieon the line $\rho(x,y)$ .You can affect the blue and the green dots by permutations by clickingthe on the menus to the right of the screen. Notice the numberings of thedots change when the menu displaying permutations is clicked. This resultsin a change in the line constructed by Pappus' theorem. The menu displayselements of S₃, here thought of as the group generatedby $\sigma$ and $\tau$ ,where $\sigma$ is an order three permutation and $\tau$ isan 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 .

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