12.5 Polyrulers

Our theory of polyrulers perfectly mirrors our theory of polyvectors.

Definition.

The product of rulers is anti-commutative

\[ \begin{align} s \mult r &= - r \mult s \end{align} \]

and the square of a ruler is zero

\[ \begin{align} r \mult r &= 0 \end{align} \]

where \( r : \Ruler_{} \) and \( s : \Ruler_{} \) are any rulers.

Because rulers and vector have the same multiplication, we get the same standard forms for polyrulers.

degree	polyruler standard form
0	\( a \)
1	\( a \mult \bar{\standard}_{x} + b \mult \bar{\standard}_{y} \)
2	\( a \mult \bar{\standard}_{x} \mult \bar{\standard}_{y} \)

We can measure a polyvector using a polyruler of the same degree. Using the polyruler \( \bar{\standard}_{x} \mult \bar{\standard}_{y} \) to measure the polyvector \( \vec{\standard}_{x} \mult \vec{\standard}_{y} \) gives a result of one.

\[ \begin{align} \bigl\langle{\bar{\standard}_{x} \mult \bar{\standard}_{y}}\mathbin{\big|}{\vec{\standard}_{x} \mult \vec{\standard}_{y}}\bigr\rangle &= 1. \end{align} \]

Let's make a measurement in degree two.

Example.

Let's measure the polyvector

\[ \begin{align} q &: \PolyVector_{(x, y)}^{2} \\ q &= 3 \mult \vec{\standard}_{x} \mult \vec{\standard}_{y} \end{align} \]

with the polyruler

\[ \begin{align} p &: \PolyRuler_{(x, y)}^{2} \\ p &= -2 \mult \bar{\standard}_{x} \mult \bar{\standard}_{y}. \end{align} \]

We compute.

\[ \begin{align} \langle{p}\mathbin{|}{q}\rangle &= \langle{-2 \mult \bar{\standard}_{x} \mult \bar{\standard}_{y}}\mathbin{|}{3 \mult \vec{\standard}_{x} \mult \vec{\standard}_{y}}\rangle \\ &= {-2} \mult 3 \mult \langle{\bar{\standard}_{x} \mult \bar{\standard}_{y}}\mathbin{|}{\vec{\standard}_{x} \mult \vec{\standard}_{y}}\rangle \\ &= -6 \end{align} \]

That was easy!

We've now seen how to make measurements in degrees one and two. In degree zero, measuring reduces to multiplying numbers. Let's quickly summarize how measurement works with a formula for each degree.

Formulas.

We may measure a polyvector with a polyruler as follows.

\[ \begin{align} &\langle{a_r}\mathbin{|}{a_v}\rangle = a_r a_v \\ &\bigl\langle{a_r \mult \bar{\standard}_{x} + b_r \mult \bar{\standard}_{y}}\mathbin{\big|}{a_v \mult \vec{\standard}_{x} + b_v \mult \vec{\standard}_{y}}\bigr\rangle = a_r a_v + b_r b_v \\ &\bigl\langle{a_r \mult \bar{\standard}_{x} \mult \bar{\standard}_{y}}\mathbin{\big|}{a_v \mult \vec{\standard}_{x} \mult \vec{\standard}_{y}}\bigr\rangle = a_r a_v \end{align} \]

In the above,

\[ \begin{align} a_r &: \Number, & b_r &: \Number, & a_v &: \Number, & b_v &: \Number \end{align} \]

are all numbers.

Our theory of vectors and rulers extends to three-dimensional space and beyond. For those interested, we've sketched how this works in Appendix G. Rulers in Space.