Products of four or more reflections

Lemma

Every product $H_{P}R_{v}R_{u}$ can be replaced by a product $R_{l}R_{m}$.

Proof

Let $H_{P}R_{v}R_{u}$ be the motion we wish to analyze. We need to consider three possibilities: the lines $u$ and $v$ intersect, the lines $u $ and $v$ have a common perpendicular or neither.

First suppose that $u$ and $v$ meet at point $Q$. Let $s$ be the line containing $P$ and $Q$ and let $l$ be a line perpendicular to $s$ passing through $P$. Note that $H_{P}=R_{l}R_{s}$ and that, as $u,v$ and $s$ all meet at $Q$, there is a line $m$ such that MATH. It follows that MATH.

Now suppose that $u$ and $v$ have a common perpendicular $t$. Let $s$ be a line perpendicular to $t$ that contains the point $P$ and let $l$ be a line perpendicular to $s$ containing the point $P$. Note that $H_{P}=R_{l}R_{s}$ and that, as $u,v$ and $s$ are all perpendicular to $t$, there is a line $m$ perpendicular to $t$ such that MATH. It follows that MATH.

Lemma

Every product of the form $R_{w}R_{v}R_{u}$ can be replaced by a product of the form $H_{P}R_{l}$.

Proof

Let $U$ be a point on $u$. Let $v^{\prime }$ be a line in the pencil formed by $w$ and $v$ that passes through the point $U$. (This is the join theorem.) Let $w^{\prime }$ be a line in the pencil such that MATH. Now drop a perpendicular $l$ from $U$ to $w^{\prime }$. Observe that $l,v^{\prime }$ and $u$ are concurrent and hence there is a line $a$ such that MATH. Set MATH and note that MATH. Hence
MATH

Theorem

Let $M$ be a motion. Then $M$ can be described as the product of three or fewer reflections.

Proof

First suppose that $M$ has been described as the product of four reflections, i.e., MATH. Note that there is a half-turn $H_{P}$ and a reflection $R_{u}$ such that MATH and hence $M=H_{P}R_{u}R_{1}$. Now, by the first lemma in this section, there are lines $l$ and $m$ such that $M=R_{m}R_{l}$. Let also recall that a reflection can also be described as the product of three reflections, i.e. MATH for any line $m$.

We proceed by induction to establish the result for products of five or more reflections. First we make the induction hypothesis: Suppose that $M$ is a motion which has been described as the product of $n$ reflections. If $n$ is odd, then $M$ can be described as the product of three reflections and if $n$ is even then $M$ can be described as the product of two reflections.

It is clear that the induction hypothesis is true when $n=3$ or $n=4$.

Now suppose that $M$ is the product of $n+1$ reflections, i.e., MATH. First suppose that $n+1$ is even. Then $n$ is odd and henceMATH can be reduced to a product of three reflections, say $R_{w}R_{v}R_{u}$. Thus MATH and hence $M$ can be further reduced to product of two reflections. Now suppose that $n+1$ is odd: then $n$ is even and thus MATH can be reduced to a product of two reflections, say $R_{v}R_{u}$. Thus MATH, which is the product of three reflections.

Note that we have shown any proper motion can be reduced to the product of two reflections and that any improper motion can be described as a product of three reflections.

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