Yes, that’s right, the Hodge complement is a well established formula, most likely you are using the incorrect interior product and therefore your Hodge complement is wrong.
If you actually watched my video, you’d know that the interior product used by Leo Dorst is incorrect and not the one which Grassmann and Hodge used in their work.
If you wanted to make a correct definition of Hodge complement you’d also need to use the correct interior product. Most likely you learned the incorrect definition because your work seems based on Gunn, Dorst, etc.
Once you switch to the correct interior product, perhaps then you will arrive at a correct definition of Hodge complement.
The formulas of the Hodge complement certainly require the original Hermann Grassmann interior product and not the incorrect stuff perpetuated by Dorst etc.
@chakravala I understand what you are saying here. I’m not ready to add any coauthors. I will try to look into what you have said more, about incorrect products or definitions in any established literature, or any prior works / preprints. No one should just hand over new results without some co/author credit. I’m just hoping to make sure I’m not wasting my efforts too. Not sure I will talk much more here, because I doubt this will really help me now. Take care, thanks.
If your goal is to make something compatible with Grassmann’s original works, then you will inevitably retrace my steps and arrive at the same result I do.
My work is named after Grassmann for a reason.
However, it may be that you can arrive at something satisfying a Hodge complement condition but incompatible with Grassmann’s original work, then your work will inevitably differ from mine. If you take that route, you will be incompatible with Grassmann algebra and you will be doing something different. That’s probably the path your current work is took. That path will lead to something inconsistent with what I consider historically important (Grassmann algebra).
People need not be embarrassed for being wrong and incorrect, it is inevitable that 99% of people will stumble and make many mistakes along the way, we are seeing some growing into old age, never getting it correct.
I’ve tried many times to point out mistakes to people but I have grown grumpy over time with this community.
I remain somewhat vague these days, because I already lived through many times of trying to help people, I have gone through these type of discussions countless times now and never receive any credit or reward, and mostly I am ignored as these issues are subtle and nuanced.
@chakravala Thanks again for your chat. I may pop in here and talk a little more sometime. But, I’d just like to say, I think many are happiest when quiet and just doing our research. I know I am. I’m not good at talking or trying to convince anyone of my work. I have been curious to get some feedback on it though. It is kind of difficult so far to get feedback. I guess I should just continue the research quietly and try to submit a paper to a conference or journal and hope to get peer review etc. I think it is going to be difficult! You have your focus on getting Grassmann’s work right, and that is good, keep at it. I may be on to something a little different, as you said. The dual Je, has great results that keep working, but maybe it is not the way Grassmann was doing it. But, didn’t Grassmann only work in fully degenerate algebras with only the wedge product? He did not have an inner product as geometric algebra has. But then, I am not an expert on Grassmann’s algebra. I think geometric algebra and grassmann algebra are going to have differences.
I tried to check some stuff on
but this wiki will not load correctly for me. The formatting looked broken. Maybe I try again later.
The people in academia reviewing papers are all clueless about this topic, they will maybe give a review but they will be ignorant of the issues I speak about specifically.
Grassmann is the originator of what we call geometric algebra. While Clifford is credited with the geometric product, Grassmann also had it and called it the central product. Grassmann certainly had interior products and he created the foundational concepts which lead to what we now call geometric algebra. Clifford made some contributions as well, but Grassmann started it, Grassmann is the root of it.
What Grassmann was lacking is a metric. Grassmann was merely using the identity metric, so he never modified the metric of the algebra, although he had interior products. That’s why I call my complement the Grassmann-Hodge complement, as it was the Hodge complement which extended the Grassmann complement to general metrics.
However, modern day authors 100 years later have diverged from Grassmann’s original roots, which I consider problematic, as this results in inconsistencies.
This clarifies a lot of mud in which I found myself trying to figure why different sources use different definitions.I thought they were all talking of the same thing.
You do not clarify if you are only using versors squaring to 1.
If you include nilpotents, that dual will be zero when \omega has nilpotents \epsilon_i^2=0.
Actually I do clarify this, but you may have missed it.
Yes, the Hodge complement is by definition zero for a degenerate element. It cannot be otherwise, since the Hodge complement is defined by the metric.
If you don’t want the complement to be zero, then you need to use the Grassmann complement defined by the identity metric, not the Hodge complement defined by the degenerate metric. In Grassmann.jl this is ! instead of the Hodge complement.
Hence, you can always use a Grassmann complement instead of a Hodge complement, if you desire. In Grassmann.jl both options are available in any algebra, the Grassmann complement always uses the identity metric regardless of algebra, while the Hodge complement always uses the modified metric of the algebra.
It’s something I do talk about, but of course such subtle nuances are easy to overlook.
I know I am. I’m not good at talking or trying to convince anyone of my work. I have been curious to get some feedback on it though. It is kind of difficult so far to get feedback. I guess I should just continue the research quietly and try to submit a paper to a conference or journal and hope to get peer review etc. I think it is going to be difficult! 
You have your focus on getting Grassmann’s work right, and that is good, keep at it. I may be on to something a little different, as you said. The dual Je, has great results that keep working, but maybe it is not the way Grassmann was doing it. But, didn’t Grassmann only work in fully degenerate algebras with only the wedge product? He did not have an inner product as geometric algebra has. But then, I am not an expert on Grassmann’s algebra. I think geometric algebra and grassmann algebra are going to have differences.
