I think your quadpod is a magnificent concept for illustrating your points. It's very vivid and fun, too.
I am impressed by your geometry http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html which is intriguing and persuasive.
However, if you line up the corners of the squares and also of the cubes, then you get a progression which is very helpful for teaching calculus, namely, if you consider a square x and grow it by one more bit h so you have a square of sides x+h, then: (x + h)**2 = (x + h)(x + h) = x2 + 2hx + h2 which all make geometric sense, and then you can see why you can ignore the h2 and upon subtracting x2 you are left with 2hx which, when compared with h, gives you the derivative 2x.
Similarly, (x+h)**3 = (x+h)(x+h)(x+h) = x3 + 3x2h + 3xh2 + h3 and discarding the small stuff and substracting x3 you are left with 3x2h and dividing by h gives the derivative 3x2.
This for me is a very powerful way to illustrate differentiation in a very real sense. And also these binomial expansions are very worthwhile to spend time with and very meaningful for problems in probability, heads and tails: (h+t)**3 or recessive and dominate genes, blue eyes b and brown eyes B (b+B)(b+B) for example.
So I'm curious if your triangular thinking has a nice way to talk about this all, perhaps?