Yeah, tell me about it. Every time I explore some other +5D shape, I still find out something new about 4D. It's still a strange, mysterious place that we have to explore mentally. Every voyage sharpens the view, and sometimes, there are radical breakthroughs in understanding, that completely change everything! Even the projection of the tesseract, as simple as it is, still has a lot of hidden meaning in it's logic, and is tough for normal 3D thinking to make sense of it. Take the double ring feature, for example! I believe conquering that shape, and knowing exactly why it's put together that way, is the best introduction into 4D concepts. Or any above four concept. The fifth dimension is no more strange than 4D, it's merely a matter adding another " mysterious extra direction", that extends into a new up and down. Compressing 4D flat again, means 5D is the familiar above and below that thin sheet, by perceptual analogy.
In my toratope explorations, I have learned a
fantastic concept with reducing dimensions. Especially when you apply this new understanding to how much more expansive five dimensions actually is, compared to 3D! What I learned is the cross-over patterns we see with reducing a shape by a certain number of dimensions. We have to do this, in order to actually
see it, in our 3D viewing plane. What we see, is roughly identical to a 1D cut of a 3D shape. This would be a thin needle puncturing through a 3D solid. I say roughly identical, as in the positions of the 5->3 cut shapes are in this 1D cut pattern. With any 1D cut, we always get points along a line. Since the cutting needle is so thin, we have an entire 2D plane to move in, and trace out the rest of the shape. This causes the points to dance around in complex symmetrical patterns, as a terribly reduced representation of the whole 3D shape. Same goes with a 5D shape, reduced to 3D. By a direct analogy, our 3D cutting plane is as
thin as a needle, being given a 2D array to move in, and trace out the missing 5D structure. That was a mind awakener right there

Holy crap, man, when I finally saw how that worked, it really WAS expanding my mind! Or, think of it as looking at a painting through a thin hollow tube, we see a rough estimate of 1D vision, and we have to move in 2 independent directions to find more colors.
But then, there's 6D. Which is a three dimensional loss, to get to 3D. Our natural 3D viewplane is actually
thinner than a needle, in 6D. This is the realm of where you start getting larger arrays of cut shapes. Especially in those 2x2x2, or 2x1x4, or 4x1x2 arrays. What I learned about why those arrays come about, what's making them show up as four things by 2 things, was incredible. By now, I'd have to say that it was the toratopes that really expanded my understanding into incomprehensible high dimensions. For years, I had played with all others: tapertopes, rotatopes, orthotopes, simplices, etc. Just the real basic ones, not anything CRF-like. Then came time to explore the toratopes. They ended up being highly complex, yet highly predictable with their midsections, and merge sequences. It is this predictability that makes very high-D comprehension still attainable.
I think the best example was coming to terms with the midsection cut lattice of nine dimensional triotorus tiger : (((II)I)((II)I)((II)I)) . It looks horrendous at first, but once I show what's happening here, it won't be so bad

. Ultimately, its only non-empty cut shows up as (((I))((I))((I))), which means a 4x4x4 cubic array of 64 spheres. Now, firstly, it's important to point out the notation for a torus: ((II)I) , and as we can see in triotorus tiger, there are three torii in it. This is key. Now, consider what happens when we make a 1D cut of a torus: ((I)), we get 4 points along a line. A needle puncturing through the side of an innertube will hit rubber four times. We reduced the torus by 2 dimensions. Now, consider the cubic array again: (((I))((I))((I))) , there are three torii that we have cut 2 times each, in order to get a 9D shape down to 3D. Each leg of the 4x4x4 array, corresponds to the 1D cut of a torus. When we finally reduce triotorus tiger to 3D, we get a line puncturing an innertube to the power of three! Hidden in plain sight, within the 4x4x4 array of 64 spheres, are three hollow torii, arranged orthogonally, multiplied together. The true entire 9D shape is the single surface of all three combined. When we cut it down 6 dimensions, we get four points along a line, to the power of three, making 64 spheres along a 4x4x4 cubic array. Each leg of the array contains an entire separate 2D cut array of its native torus. This gives us a full 6D cut array to move in and trace out the rest of the missing 9D structure. Our 3D line of sight is
so incredibly thin; in fact, it's
three times thinner than a needle!
