oxenholme wrote:My brain is obviously too ingrained with 3D!
Actually having thought some more, I think you were more right than I first thought! Here's my current thinking...
Viewing an object in any number of dimensions presumes light (or similar) travelling from the object to the viewer, and light travels along 1-dimensional lines. We can't get a sense of depth along that line, so the viewer always loses one dimension. This is why 3D objects only appear to us as 2D projections.
When this N-dimensional viewer looks at a M-dimensional object, they still lose one dimension. If the viewing direction lies in the hyperplane of the MD object, then we lose the ability to perceive that dimension, and the object appears a (M-1)-dimensional. Eg, when we view a ploygon side on so it appears as a line, or when we view a line end on to appear as a point. And as you said, you can never view a polygon to appear as a point. You can only lose at most one dimension.
On the other hand, if the viewing direction is not aligned with any of the M-dimensions, or even better when it is orthogonal to all those dimensions, then we can see the full structure of the object. This however is not possible when M = N, since there are no spare dimensions for our viewing direction. So N dimensional objects always appear as (N-1)-dimensional.
You can of course project an N-D object onto any number of dimensions, but this is a bit different from how a viewer might be able to see it.
You have your X, Y and Z axes. How do you go beyond these?
Well, in our universe, there are only 3 dimensions, so it is not possible to build anything with more dimensions. Mathematically there is nothing special about 3 though. No reason to stop at 3. It's just our physical world that disallows it. The fact that it seems so intuitive that there can't be more than 3 is a human issue, not a mathematical one
You can convert a tessellation into three dimensions by removing shapes and folding the edges. In so doing the remaining shapes or polygons are not distorted in any way. They are merely re-oriented.
Is the four dimensional equivalent of a polygon or shape a cell or polyhedron? What is the "fold" like that will take it into four dimensions? To go from two to three only two polygons are necessary. How many cells are necessary to go from three to four?
One of the most difficult things to visualise in 4D is rotation. Our initial intuition says that since orientation in 3D requires 3 values (eg heading, pitch and roll), that it should require 4 values in 4D, right? Actually no, it requires 6! Think of 2D. There you only need 1 value. These are triangular numbers: 1, 3, 6 (the difference between successive numbers increasing by one each time).
The reason is that we think of rotation as being "about a line", ie we concentrate on what is not
changed by the rotation rather than what is. We imagine rotation is still about a line in 4D, but no. In 4D rotation is about a plane! Think of 2D again, rotation is not about a line there (unless you think of a line that extends out of the 2D plane, but then it's 3D). In 2D we rotate about a point. The key is that rotation always happens within
a plane. Think of what is
changing rather than what is not.
Now you're ready to think about 3D nets for 4D shapes. The nets are like 3D shapes all stuck together face-to-face. It's hard for us to imagine, but the folding happens about the planes of those shared faces, by rotating in the plane of the two remaining dimensions. Just like 2D faces in 3D, each 3D cell rotates into 4D without distorting, but still maintaining full contact at their shared face. In 3D it would be locked solid, but the extra dimension allows it to rotate. Hard to imagine!