Kudos to this. It also helps to think of a three-dimensional space bounded by the sides of a box, and to think of another 11 boxes stacked on top of each other.
Then, you can visualize orthogonal on dimensions higher than 3 by throwing wires between equivalent points in two boxes.
Yes, but by boxing each sub-space, you can think of the pile of boxes as a dimension separate from the Z axis.
It's like painting a house in isometric style. You draw each floor above the one below it, and put a "perpendicular" slanted axis to represent the third dimension in 2D.
You could put a 'slanted' fourth axis in a 3D origin of coordinates, but I find it easier to think of the lower corner of each box as the origin of coordinates displaced along that fourth axis.
"You could put a 'slanted' fourth axis in a 3D origin of coordinates"
Even with that the same problem occurs: "slanted" is a 3D relationship, so at best it's an analogy.
It's like "translating" the color red to some shade of gray to a person who can only see in black and white.
Or trying to describe through written words what music is like to someone who can't hear.
These are all analogies which might allow us to reason and get interesting/useful results when we use such analogies... but we should not under the illusion that we really know what those things for which we have no senses are like.
Also, there are probably all sorts of interesting/useful connections, relationships or conclusions that a being who really could sense 4D objects would find obvious or easy to make that we may never arrive at because our way of thinking of them is so limited in comparison.
That's not to mention trying to think about even higher dimensions or various other mathematical constructions that have no obvious translation to things in our ordinary experience we can relate to.
> Even with that the same problem occurs: "slanted" is a 3D relationship, so at best it's an analogy.
Yeah, but it's the same analogy you use to represent 3D figures on a 2D plane, where slanted is a 2D relationship; and we are very well versed on it and know how it works as a projection of a 3D reality.
> These are all analogies which might allow us to reason and get interesting/useful results when we use such analogies... but we should not under the illusion that we really know what those things for which we have no senses are like.
Is this some reflections about Plato's cave? You don't need much insight to understand the basis of moving over an extra dimension - if you can understand what it's like to move over parallel planes from a 2D view, you can understand the same by moving over superposed volumes from a 3D view. A hyper-sphere can be visualized as a linear collection of increasingly small spheres glued together, in the same way that you can view a sphere as a linear collection of increasingly small circles over an axis orthogonal to its center.
> Also, there are probably all sorts of interesting/useful connections, relationships or conclusions that a being who really could sense 4D objects would find obvious or easy to make that we may never arrive at because our way of thinking of them is so limited in comparison.
Yeah, and some people are really bad at 3D visualization so they have no hope to arrive at conclusions that are trivial to an architect or engineer. That doesn't prevent higher-dimension visualization techniques from being useful for the insights that it can provide, even if they can't give omniscience. For all the other insights, we have formal methods and logic reasoning, which is how 4D space was originated to begin with.
I don’t know. I feel that, for most people, it is the most visual way to see in many dimensions. That is, it’s an insight—rather than having to imagine a 5-d cube or whatever, it’s just a spreadsheet, no big deal.
Yes, but they see it as a data dump, without perceiving any spacial connections between the vectors represented by each row. The point of spatial reasoning is using our minds inner eye to visualize those connections intuitively, and the tabular data dump has nothing of it.
But I’d argue a 5d cube has even less use, as a mental model. Most people don’t know that each additional column is another dimension. People can imagine the sense of distance between varying points/entries in a table easier than a 5d cube, for instance. (Actually, that’s an empirical question)
Then, you can visualize orthogonal on dimensions higher than 3 by throwing wires between equivalent points in two boxes.