Dual operator disambiguation

[chakravala]

@Orbots

It seems the choice of dual basis/map is, in some ways, a design decision. There are multiple “correct” choices. Especially if you have degenerate metric which is orthogonal to everything. From what I understand there are many Poincare maps you could use.

In some ways, it is a design decision. Poincare duality does not associate an orientation, so there are many variations of it. Hodge duality is very specific and there is only one variation of it. I found there are only 2 types of complement that actually matter: Hodge complement and the metric-independent complement. The other features, such as degenerate Leibniz elements do not conflict with those definition choices, as those basis elements are not anti-symmetric. The complement operation only changes Grassmann basis.

Also, it is not necessary to know differential geometry to learn my definition of complement. All you need to know is the geometric product and the reverse.

@cgunn3

For the basis 1-vectors in R3 (on p. 54) are e1e2, e1e3, and e2e3 (alphabetical order). He obtains in this way two minus signs in the co-basis corresponding the the elements e2 and e1e3. If instead he chooses the basis 2-vectors e1e2, e2e3, and e3e1, then the minus signs go away.

So in other words, you prefer to re-orient your basis to eliminate minus signs. I think this is okay to do in writing and in papers, but not in a computer program.

@ninepoints

even if the cyclic ordering is more natural when working on pen and paper, I think the lexicographic order may be more likely to come up in implementation, if only because bitfield representations of the basis blades are so ubiquitous for efficiency.

I always have an increasing index like v123 for example, which is equal to v231 by the exterior basis equivalence relation, but there is no need to create a separate instance of the class. I only need one instance of the class, so I only need v123 and don’t need a representation for v231 in my calculations, so I automatically make sure the indices are sorted. Hence, I always use increasing order, requiring a single element from equivalence class for representation.

This does not mean that I am opposed to using other orderings in writing. It only means that for computing purposes, it is simpler and more efficient to only represent a single element of each equivalence class, instead of complicating things by having multiple representations for the same object.

[ninepoints]

@chakravala

This does not mean that I am opposed to using other orderings in writing.

I think the order @cgunn3 has chosen has its origins from the Mobius edge law of edges (please correct me if I’m wrong). Given a tetrahedron, a positive sense of volume dictates that viewed from one vertex through the interior, the orientation of the opposite side is counterclockwise. Then, all cycles for each face have an ordering as described, such that tracing each cycle will run forward and backwards along each edge exactly once. This rule of course generalizes to lower and higher dimensions, but is necessary for the orientations to give consistent definitions of length, area, and volume. I actually like this ordering a good deal for those reasons (it certainly feels more natural and well motivated than the lexicographic ordering), so I just start here and introduce signs as necessary. Klein himself also adopts the cyclic ordering for the same reasons (citing Mobius).

@chakravala

I only need one instance of the class, so I only need v123 and don’t need a representation for v231 in my calculations,

I imagine this will be true for most implementations.

[chakravala]

@Orbots

To make your paper more accessible you’ll need a preliminaries section or alternate definitions that use more common mathematics. I know, you’d end up with massive volume(s) then, so maybe that can’t be done so easily.

I do plan on writing a massive non-linear book on mathematics as a whole using my VerTeX.jl graph-theoretic approach to \LaTeX editing. One of the other plans for this softwate is to enable its usage to compute all the required background information necessary to include in a customized linearization of the book for a specific user. This can be done by having the reader mark the level of understanding of topics they know or don’t know, and then a new table of contents will be computed with the relevant info.

However, this takes much more effor to complete than writing a short and concise 6 page paper, so you might have to wait a few years before it is available.