If \epsilon^n=0, and e_i^2=1:

do \epsilon^n*e_i = -e_i*\epsilon^n only when n is even?

Or the anticommutativity between \epsilon and e_i breaks when \epsilon^2 \neq 0 ?

Anticommutativity is not dependent on the signature, although as stated you do use it to define the orthogonality of the basis vectors.

Reading again I see that you take the n-th power of epsilon. The minus sign is only active for \eps e_i = - \eps e_i, in all other cases this just reads 0=0. So it is more n=1 vs n>1 than odd/even.

You are confusing several types of mathematics here. Elements which square to zero are not necessarily anti-commutative. They are anti-commutative in the cheap version of geometric algebra, where they are defined with a non-invertible metric. However, in the more advanced version of geometric algebra which is first desrcibed in my work relating to Grassmann.jl software I develop, there are elements squaring to zero which do not anti-commute. The cheap version of geometric algebra used by everyone else here unfortunately is inconsistent with differential geometry, while my more sophisticated definitions allow you to bypass this issue while using an invertible metric.

However, for the average geometric algebra user, these nuances will fly over their head, and they will end up using the more common cheap geometric algebra with non-invertible metrics. In that case, yes it anti-commutes.

As far as n goes in your question, the n can only be 1, it cannot go higher in the anti-commuting version based on non-invertible metrics. In my version of more advanced geometric algebra, n is not limited to 1, but can go arbitrarily high, depending on your choices.

Note that Leo Dorst and related people all use the cheap version of geometric algebra, so Leo Dorst is only merely capable of conceiving of n=1, because they use an approach based on non-invertible metrics, which has all kinds of bad consequences. It works for Leo Dorst because he is not a mathematician, he is a computer scientist, so he does not worry about foundational mathematical issues the way I do.

Ad hominem attacks are against the forum rules. Repeat offender @chakravala has been silenced for a few weeks.

Being Dutch, I like my algebras cheap.

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When I say cheap geometric algebra, what I mean is inconsistent mathematics. The mathematics of Leo Dorst is unfortunately inconsistent in the sense of 1=0 when you try to check its consistency with other theorems in mathematics (e.g. from differential geometry).

We can give some personality points to Leo Dorst for this comment about liking his algebras being cheap, but it still doesnâ€™t change the fact that his math leads to inconsistencies such as 1=0.

I understand that Leo Dorstâ€™s ego will have a hard time accepting and acknolwedging this since this would put his entire publication record into question, which is why I donâ€™t bother with trying to teach an old dog new tricks. However, once you see the inconsistencies with Leo Dorstâ€™s and Stephen DeKenninckâ€™s work, you cannot unsee that itâ€™s inconsistent mathematics.