Homework 2 - Solutions

1. Using the notation suggested in the following diagram, answer the following questions:


bhmwk2sol__1.png




a. Find all of the equivalent descriptions of the mapping $R_{1}R_{2}$ and the mapping $R_{1}R_{3}.$ (4 pts.)





MATH (Notice that these each rotations of $90$ degrees in the counter clockwise direction.)

MATH (Notice that these are all half turns.)


b. What is the smallest group containing the reflection $R_{1}$? Be sure your list only contains one description of each mapping. (4 pts.)




The smallest group (or least of mappings generated by $R_{1})$ is MATH Here is the multiplication table:




$I$ $R_{1}$
$I$ $I$ $R_{1}$
$R_{1}$ $R_{1}$ $I$

c.. What is the smallest group containing the mappings $R_{1}$ and $R_{2}R_{4}$? Be sure your list only contains one description of each mapping.(4 pts.)




Recall that MATH and hence MATH and that MATH It follows that smallest group (or least of mappings generated by $R_{1}$ and $R_{2}R_{4}$ is MATH Here is the multiplication table:

$I$ $R_{1}$ $R_{3}$ $R_{2}R_{4}$
$I$ $I$ $R_{1}$ $R_{3}$ $R_{2}R_{4}$
$R_{1}$ $R_{1}$ $I$ $R_{2}R_{4}$ $R_{3}$
$R_{3}$ $R_{3}$ $R_{2}R_{4}$ $I$ $R_{1}$
$R_{2}R_{4}$ $R_{2}R_{4}$ $R_{3}$ $R_{1}$ $I$




d. What is the smallest group containing $R_{1}$ and $R_{2}?$ Be sure your list only contains one description of each mapping.(8 pts.)


This turns out to be all of the symmetries!. The list is
MATH
Here is the multiplication table:

$I$ $R_{1}$ $R_{2}$ $R_{3}$ $R_{4}$ $R_{1}R_{2}$ $R_{2}R_{1}$ $R_{1}R_{3}$
$I$ $I$ $R_{1}$ $R_{2}$ $R_{3}$ $R_{4}$ $R_{1}R_{2}$ $R_{2}R_{1}$ $R_{1}R_{3}$
$R_{1}$ $R_{1}$ $I$ $R_{1}R_{2}$ $R_{1}R_{3}$ $R_{2}R_{1}$ $R_{2} $ $R_{4}$ $R_{3}$
$R_{2}$ $R_{2}$ $R_{2}R_{1}$ $I$ $R_{1}R_{2}$ $R_{1}R_{3}$ $R_{3} $ $R_{1}$ $R_{4}$
$R_{3}$ $R_{3}$ $R_{1}R_{3}$ $R_{2}R_{1}$ $I$ $R_{1}R_{2}$ $R_{4} $ $R_{2}$ $R_{1}$
$R_{4}$ $R_{4}$ $R_{1}R_{2}$ $R_{1}R_{3}$ $R_{2}R_{1}$ $I$ $R_{1} $ $R_{3}$ $R_{2}$
$R_{1}R_{2}$ $R_{1}R_{2}$ $R_{4}$ $R_{1}$ $R_{2}$ $R_{3}$ $R_{1}R_{3}$ $I$ $R_{2}R_{1}$
$R_{2}R_{1}$ $R_{2}R_{1}$ $R_{2}$ $R_{3}$ $R_{4}$ $R_{1}$ $I$ $R_{1}R_{3}$ $R_{1}R_{2}$
$R_{1}R_{3}$ $R_{1}R_{3}$ $R_{3}$ $R_{4}$ $R_{1}$ $R_{2}$ $R_{2}R_{1}$ $R_{1}R_{2}$ $I$

2. Using your MIRA, carefully plot the path a ball would follow in order to achieve a hole in one. Prove that your path is the shortest path the ball could take to the hole. (6 pts.)





bhmwk2sol__133.png

In order to show that $IK+KL+LJ$ is shortest possible (two bank path using the bottom and right hand rails) from $I$ to $J,$ one can argue by selecting two points $M$ and $N$ with either $M\neq K$ or $N\neq L$ and then showing that $IK+KL+LJ<IM+MN+NJ.$ To argue this, let $I^{\prime }$ be the reflection of $I$ over the bottom rail and $J^{\prime }$ be the reflection over right hand rail. Note that $K$ and $L$ are, respectively, the interesections of MATH with the bottom rail and the right hand rail. Now, as reflections are being assumed to preserve distance, two applications of the triangle inequality yield that:
MATH
As either $M\neq K$ or $N\neq L$, one of the above inequalities must be strict and hence $IK+KL+LJ<IM+MN+NJ.$

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