The square as the rational or material mind is making more and more sense, to me at least. The phrase "material mind" is one Waite uses frequently and is synonymous with "natural mind" or "rational mind." The square is an excellent way to symbolize it, though this is the first time I've encountered it.
Also, it seems possible to me that the triangle represents the emergence of the soul. Here are a couple of more easily-digestible quotes from Waite that point to this:
"Above all, we are not concerned with the Old Law or the Old Temple, or with anything indeed but the Old Wine of the Doctrine, as against the comparative water of the old official religions, the indiscriminate providences, the rough ashlars of the exoteric priesthoods, which for many thousands of years have held a patent to establish the Kingdom of Heaven, and have tried as honestly as they could, but have failed always, even as external Masonry has preached the love of brothers and has not understood that love must be declared in the soul before it can sanctify the body or rule in the material mind.—The Secret Tradition in Freemasonry, "Of Christian Symbolism in Connection with Temple Building."
"It comes about in this manner that we recognise the presence within us of Mind in another mode and after another aspect, which is or may become our guide into all truth. Beyond the vision of the rational mind there is this deeper state which has been known to some, according to records of the ages, and it is known at this day even to a few thinkers in the sacred heart of their being. . .There are the intimations of living experience behind official doctrine, the mind of the Soul behind the rational or so-called material mind."—Shadows of Life and Thought, Ch. 26.
The triangle represents the awaking of consciousness, the "mind of the Soul" as Waite calls it. The soul, filled with love, "rules in the material mind."