## Motivation.

Suppose that we want to represent GA(2,0) (Euclidean) multivectors as a pair of complex numbers, with a structure like
\label{eqn:bicomplexGA20:20}
M = (m_1, m_2),

where
\label{eqn:bicomplexGA20:40}
\begin{aligned}
\end{aligned}

Specifically
\label{eqn:bicomplexGA20:60}
\begin{aligned}
\gpgrade{M}{1} \cdot \Be_1 &= \textrm{Re}(m_2) \\
\gpgrade{M}{1} \cdot \Be_2 &= \textrm{Im}(m_2) \\
\end{aligned}

where $$i \sim \Be_1 \Be_2$$.

## Multivector product representation.

Let’s figure out how we can represent the various GA products, starting with the geometric product. Let
\label{eqn:bicomplexGA20:80}
\begin{aligned}
M &= \gpgrade{M}{0,2} + \gpgrade{M}{1} = (m_1, m_2) = (m_{11} + m_{12} i, m_{21} + m_{22} i) \\
N &= \gpgrade{N}{0,2} + \gpgrade{N}{1} = (n_1, n_2) = (n_{11} + n_{12} i, n_{21} + n_{22} i),
\end{aligned}

so
\label{eqn:bicomplexGA20:200}
\begin{aligned}
M N
\end{aligned}

The first two terms have only even grades, and the second two terms are vectors. The complete representation of the even grade components of this multivector product is
\label{eqn:bicomplexGA20:240}
\gpgrade{M N}{0,2} \sim \lr{ m_1 n_1 + \textrm{Re}(m_2 n_2^\conj) – i \textrm{Im}(m_2 n_2^\conj), 0 },

or
\label{eqn:bicomplexGA20:260}
\begin{aligned}
\gpgrade{M N}{0} &= \textrm{Re}\lr{ m_1 n_1 + m_2 n_2^\conj } \\
\gpgrade{M N}{2} i^{-1} &= \textrm{Im}\lr{ m_1 n_1 – m_2 n_2^\conj }.
\end{aligned}

For the vector components we have
\label{eqn:bicomplexGA20:280}
\begin{aligned}
&=
+
\end{aligned}

For these,
\label{eqn:bicomplexGA20:300}
\begin{aligned}
&= \lr{ m_{21} \Be_1 + m_{22} \Be_2 } \Be_{12}
&= – m_{22} \Be_1 + m_{21} \Be_2,
\end{aligned}

and
\label{eqn:bicomplexGA20:320}
\begin{aligned}
&= \Be_{12} \lr{ n_{21} \Be_1 + n_{22} \Be_2 }
&=
n_{22} \Be_1 – n_{21} \Be_2.
\end{aligned}

Comparing to
\label{eqn:bicomplexGA20:340}
i (a + i b)
= -b + i a,

we see that
\label{eqn:bicomplexGA20:360}
\sim
\lr{ 0, n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.

If we want the vector coordinates, those are
\label{eqn:bicomplexGA20:380}
\begin{aligned}
\gpgrade{M N}{1} \cdot \Be_1 &= \textrm{Re} \lr{ n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 } \\
\gpgrade{M N}{1} \cdot \Be_2 &= \textrm{Im} \lr{ n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.
\end{aligned}

## Summary.

\label{eqn:bicomplexGA20:n}
M N \sim
\lr{ m_1 n_1 + \textrm{Re}(m_2 n_2^\conj) – i \textrm{Im}(m_2 n_2^\conj), n_{11} m_2 + m_{11} n_2 + n_{12} i m_2 – m_{12} i n_2 }.

A sample Mathematica implementation is available, as well as an example notebook (that also doubles as a test case.)

## Clarification.

I skipped a step above, showing the correspondances to the dot and wedge product.

Let $$z = a + bi$$, and $$w = c + di$$. Then:
\label{eqn:bicomplexGA20:420}
\begin{aligned}
z w^\conj
&=
\lr{ a + b i } \lr{ c – d i } \\
&= a c + b d -i \lr{ a d – b c }.
\end{aligned}

Compare that to the geometric product of two vectors $$\Bx = a \Be_1 + b \Be_2$$, and $$\By = c \Be_1 + d \Be_2$$, where we have
\label{eqn:bicomplexGA20:440}
\begin{aligned}
\Bx \By &= \Bx \cdot \By + \Bx \wedge \By \\
&= \lr{ a \Be_1 + b \Be_2 } \lr{ c \Be_1 + d \Be_2 } \\
&= a c + b d + \Be_1 \Be_2 \lr{ a d – b c }.
\end{aligned}

So we have
\label{eqn:bicomplexGA20:460}
\begin{aligned}
a b + cd &= \Bx \cdot \By = \textrm{Re} \lr{ z w^\conj } \\
a d – b c &= \lr{ \Bx \wedge \By } \Be_{12}^{-1} = – \textrm{Im} \lr{ z w^\conj }.
\end{aligned}

We see that $$\lr{z w^\conj}^\conj = z^\conj w$$ can be used as a representation of the geometric product of two vectors (setting $$i = \Be_1 \Be_2$$ as usual.)

## Another simplification.

We have sums of the form
\label{eqn:bicomplexGA20:480}
\textrm{Re}(z) w \pm \textrm{Im}(z) i w

above. Let’s see if those can be simplified. For the positive case we have
\label{eqn:bicomplexGA20:500}
\begin{aligned}
\textrm{Re}(z) w + \textrm{Im}(z) i w
&=
\inv{2} \lr{ z + z^\conj } w + \inv{2} \lr{ z – z^\conj } w \\
&=
z w,
\end{aligned}

and for the negative case, we have
\label{eqn:bicomplexGA20:520}
\begin{aligned}
\textrm{Re}(z) w – \textrm{Im}(z) i w
&=
\inv{2} \lr{ z + z^\conj } w – \inv{2} \lr{ z – z^\conj } w \\
&=
z^\conj w.
\end{aligned}

This, with the vector-vector product simplification above, means that we can represent the full multivector product in this representation as just
\label{eqn:bicomplexGA20:540}
M N \sim
\lr{ m_1 n_1 + m_2^\conj n_2, m_2 n_1 + m_1^\conj n_2 }.

## Ontario elections are done. Did you choose your oppressor?

How is it that this election farce is viewed as “Civic Duty”?  How is it that “Democracy” and our bloated cancerous Government is viewed with almost religious overtones?  How is it that a once every four years pick between two or three identical oppressors is perceived as representation?

In Canada, we have what is effectively a three party system, the PC, the Liberals, and the NDP parties.  All of these want to continue the status quo, and steal your money at gunpoint.  If you object to having your money stolen, then you can go to jail, or have your wages stolen at the source, or have your property taken.  It would be somewhat amusing to itemize all the ways that government steals from me, but actually thinking through the details of that is a very depressing road to walk.

The PC party portrays themselves as pushing for limited government, and will bandy about terms like privatization to support that claim.  However, this is privatization as implemented in the USA, where it means that they will continue to steal from you, but will give your money to their friends.  They do occasionally promise less taxes, but I’d be surprised to see numbers that demonstrated this ever occurred while they were in power.  If they did ever manage to reduce taxes, was there a corresponding increase in debt spending?

The NDP doesn’t hide it’s goals.  More government, less freedom, more taxes, less personal choice.  They know better how to use your resources than you ever can.  It’s basically a weakened communist party, with socialist ideals that have been shown to fail again and again and again.  Reward the unproductive, feeding the welfare system that feeds on itself.  Impose central planning, making decisions on how to spend the money they steal from you, from their ivory tower positions of power.

The Liberal party is the middle ground between the two, positioned to deceive people into thinking that they have a choice between the two radical extremes.

My partner is very happy (perhaps overjoyed?) that the Liberal party has won the election (in our riding as well as provincially).  It is surprising to me to observe enthusiasm over results that I consider meaningless.  I don’t want the government ruling over me, no matter who the puppet figurehead is.  I don’t want my money stolen to pay the debt on past and continuing warfare.  I don’t want my money stolen from me to support a welfare system that hurts the people it claims to support, while ensuring that they can never break their dependency.  I don’t want to support an insane health care system that is in the pockets of the big pharma companies, and introduces idiotic government policies that doctors have to comply with at every step.  To get my daughter’s skin condition looked at, we have to first see a GP, so that he can get paid to refer us to a skin doctor.  It makes sense to the GP because he gets paid for it.  No doctor or public service worker will object to big government, since they have a vested interest in big government, because that is where their pay check comes from.

It necessary to consider this pay check bias, to make sense of any enthusiasm for the liberal party election win.  She works for the university, an institution that receives their funding from monies stolen from me, from you, and even from her.  Nobody is likely to vote for loosing their job, and perceived the PC party as a potential threat to job security.  The government provides the universities with an educational cartel, setting both the regulations that allows them to rubber stamp useless degrees, as well as the funding they require to operate.  Calling most university degrees useless is not a bias against that poor sucker that comes out of school with debt in exchange for the double major in politics and art history, and minor in psychology.  I also consider my own undergrad engineering degree largely useless.  Almost everything that I learned for my job I learned on the job, and suspect that is the case for the bulk of people that do work their way through university.  If the government wasn’t propping up these bumbling institutions with the money they take from us, I am sure that more targeted and effective job centric learning institutions would have an incentive to establish themselves.

People vote for the “free” stuff that governments claim to give out, but the argument is a weak one.  Vote for us and you can have one unit of “free” stuff.  Sure, but they’ll take 1.2 units of stuff from you, but do so in a way that makes it seem like they are taking 0.7 units from you and 0.5 units from somebody else.  Free things from the government are like the “free” in popular commercial marketing, only free if you also buy two of equal or greater cost.  If you think that the free stuff promised by some specific political party happens to help your specific financial situation, you will still lose when all is said and done.  Government and bureaucracy will always continue to grow, manufacturing its own demand for itself.

I don’t believe that any elected government, especially with the party system that forbids individual choice, can ever be representative.  Democracy embodies the insanity of decision by committee.  How many times have you ever observed a meeting with more than ten people come to any sort of reasonable consensus, or even make progress?  How can we fool ourselves into thinking that this will work when scaled to hundreds of people.  How can we fool ourselves into thinking that this works when so many politicians are bought and paid for, and will work for their buddies and their financiers.

No person that I would ever vote for will win a seat, and even if they did there’s no way that they could fight the institutional inertia of government once in place.  I’ve voted before for the Green party as a statement of “none of the above.”   This time I noticed that there was a Libertarian running in my current riding, and ended up voting for him.  I had very mixed feelings about voting at all, since not voting expresses dissatisfaction with the system in a way that is perhaps stronger than voting anti-strategically.   While the Libertarian party is in many ways consistent with my anarchist views, they will never get elected.  Too many people are busily sucking on the public sector tit, and a fundamental shift in attitudes would be required to get enough buy in for any Libertarian to ever win an election.  This vote was therefore yet another vote for “none of the above”, one made knowing full well that it was completely pointless.

I don’t get any more representation having done my civic duty than anybody else. </rant>