(ii) Following the example on symmetries of the equilateral triangle from thewritten notes, list all the symmetries of the shape above and the associatedpermutations of {1,2, 3, 4}.

Answered: Following the example on symmetries of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(ii) Following the example on symmetries of the equilateral triangle from the
written notes, list all the symmetries of the shape above and the associated
permutations of {1,2, 3, 4}.

Expert Solution

Step 1

The given symmetries of equilateral triangle is given below:

The above figure has two kind of symmetry.

(1) Rotation Symmetry.

Let line joining by point $2$ and point $4$ is considered as $x -$ axis, and line joining by point $3$ and point $1$ is considered as $y -$ axis. Then the above figure has rotation symmetry by rotating through an angle of $90 °, 180 °, 270 °$ counter clock wise. That is if $r$ is rotation of $90 °$ , then $r^{2}, r^{3}$ are rotation through $180 °, 270 °$ respectively. The rotation are shown below:

The corresponding permutation are:

$\begin{array}{rcl} r & = & (\begin{matrix} 1 & 4 & 3 & 2 \\ 4 & 3 & 2 & 1 \end{matrix}) \\ r^{2} & = & (\begin{matrix} 1 & 3 \\ 3 & 1 \end{matrix}) (\begin{matrix} 4 & 2 \\ 2 & 4 \end{matrix}) \\ r^{3} & = & (\begin{matrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{matrix}) \end{array}$