Progress Report #5.4


Products of four or reflections

Introduction and instructions: As the title suggests, this project looks at the product of four or more reflections. Such a product can always be reduced to a product of three or fewer reflections.


Questions:

  1. Any motion of the form $H_{P}R_{l}R_{m}$ can be replaced by the product of two reflections. (i.e., two reflections followed by a half-turn can be replaced by the product of two reflections.) Prove this for the cases that the lines $m$ and $l$ intersect and that the lines $l$ and $m$ have a common perpendicular. The result is still true if the lines do not intersect and that they do not have a common perpendicular; do not worry about this case. One way to prove this result is to represent the half-turn $H_{P}$ as a product $R_{u}R_{v}$ where $v,l$, and $m$ have the property that $R_{v}R_{l}R_{m}$ can replaced by a single reflection.

  2. Prove: Any product of the form $R_{n}R_{m}R_{l}$ can be replaced by a product of the form $H_{P}R_{u}$. (i.e., three reflections can always be replaced by a reflection followed by a half turn.) One way to approach this problem in the case that $l$ and $m$ intersect is to insert the identity between $R_{n}$ and $R_{m}$ in the form $R_{u}R_{u}$ where $R_{n}R_{u}$ forms a half turn.

  3. Justify: The above two observations provide the means of reducing any product of reflections to a product of three or less.

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