Such as?

The main two were order-independent transparency, which produces nicer and more realistic transparency effects, and deferred shading, which is what a lot of modern-day engines employ. It's just that the former wasn't really that useful when you generally only had two texture layers going on at once, and the Dreamcast wasn't poweful enough to do much with the latter other than create some nice-looking spheres.

(And I should probably have said other companies didn't incorporate them in hardware until PS4/XB1, since you probably can pull off said effects on PS3/360; you'd just have to do it in software)

It's very surprising to me that there are people who think the GC had better looking games than the XB. Is it an exaggeration to say that literally every multi-plat game looked better on the Xbox? I don't think it is. Even if the differences were minor, they always favored the Xbox. Some publishers would also put half assed effort, into the GC version of certain games. EA did this with Madden. I was blown away by how shoddy Madden was on the GC.

it brought me to halo, and hours of local blood gulch play

You can use Depth Peeling on the Original Xbox and Xbox 360 in order to accomplish a form of order-independent transparency by leveraging the Depth/G-buffer anyway... And was actually a requirement on the Original Xbox for the game 'Shrek' as it used Deffered Rendering.

The Original Xbox had programmable shaders and the necessary buffers, so it was capable of some impressive techniques like that.

In short the games are the proof.
Halo 2 was leveraging some impressive lighting, bump mapping, shadowing and shader effects at the time, even had depth of field.
Morrowind was a stupidly massive game with some impressive shadered water.
Ninja Gaiden Black has some impressive texture work.

The Original Xbox also had games in High Definition... So I think it's safe to say, the Xbox had the graphics edge that generation.
Although the Gamecube certainly had bang-for-buck.

In some ways, but using something like dire ports of Madden from EA onto the GC was more an indication of what EA thought about the console at the time more than the graphical power that it was capable of, using software to try to directly say "look what the hardware is capable of" in terms of a system where devs don't have confidence (or coding ability) would mean that someone could jump to games like WWE2k18 on the Switch to suggest the machine isn't capable of playing a x360 title above 10fps

Of course if you have an entire library of games which look better on one machine than the other then you have other things to consider and could say that software overall does say that the machine was more powerful, but keep in mind as well on the ps2/xbox games had access to an absolute ton more storage space, even a 2 disc game on the GC would still be around half the space of anything on a DVD, never mind a dual layer disk.

Just saying, that "software = representation of the machines power" could lead to someone posting this

As the ability of the Switch. Seriously though... love that video so much, watch the framerate when their name bar appears at the end, you can pretty much count the frames.

When it comes to the Sixth generation though, if I'm not mistaken wasn't factor 5's Star Wars game the highest polygon shifting title of that generation (I could be wrong on that, but have heard it uttered a few times, not saying that misinformation can't spread)

Sure was beautiful though.

What I am more or less saying is you take a well made game for Xbox or Gamecube and break down the rendering pipeline, it removes all subjectivity.
The best looking games on Xbox were certainly using rendering techniques that typically didn't appear on Gamecube for one reason or another.

That game was a powerhouse.
The Original Xbox easily had superior culling to pretty much any GPU released up to that point in time as it was a big focus by nVidia, thus poly counts would be a tough one to gauge as they can be counted differently or counted in a way that doesn't account for that technology.

But from a raw numbers perspective... The Original Xbox did have a higher theoretical count in that aspect over the Cube.

Just keep in mind on one hand you are saying that "games looked nicer on the xbox meaning the hardware was stronger" but when it's a game shifting the most poly's of the generation you flip to "that game is great.... but the xbox hardware is the thing which shows its hardware was stronger, not the software but the raw numbers"

I will point out, that I agree with you that the xbox is the more capable machine, I don't believe that the GC could hold a candle to it in raw power, but I'm just saying that the GC was capable of great things with great devs and equally places like EA could make the machine look like dogshit when they lost interest in it. I just always try to stay away from the "software running great = great hardware" angle.

Nope. I said "The Games are the proof".
Then elaborated on the games being the proof by breaking down said games and looking at the rendering pipeline and the technology behind them.
You need to remove subjectivity as much as possible.

And I have recognized and mentioned the Gamecubes Strengths in aspects such as texturing.

The GC had decent FP from both CPU and GPU, most other only had decent from one. It could do closer to 11 GFLOPS.
It also had the most impressive matrix transformations, able to do 3 billion/sec. (PS2 and XBOX at 2 billion).

Overall, I would say the graphics on the GC were more impressive and the original, except for one thing... the biggest limitation with the gamecube in terms of graphics being more obvious was the media was only 1.5GB on the mini dvd and no internal hard drive, and Xbox could read 6.8GB on a dvd. That limited or extra storage played a factor for many games in the graphics quality...