Quoth Kaje:
Different set of definitions. Derleth is defining a square, for instance, to just be the four lines on the border, while you’re also including all of the region defined by those four lines. If you use the “just lines” definition, then Derleth is correct; you only need two cubes and connecting line segments to make a hypercube. If you use the “region enclosed” definition, then you’re right, you’d need a continuous set of cubes.
One way to visualize the passing of time is as follows. You can plot a point described by its relative X, Y and Z coordinates on a 3 axis graph. There’s lots of software that will let you do this and you can do it on paper, where the axes are often rotated a bit so you percieve a perspective view and no axis is obliterated from view.
Once you’ve plotted a point this way, you have a visual representation of an object’s position in 3 dimensions, at a particular point in time. You can then plot position relative to the same spatial axes at a later point in time. Connect the dots, if you so desire. Plot the positions of many objects at many points in time. It may help to standardize the time samples, so each object you track has a point plotted for the same point in time that’s plotted for all other objects tracked.
This is probably not exactly what the OP pursues, but it is a way to visualize 4-D.
And it’s not a great leap to visualize, say, my planet (I’m sure I’m a bit fuzzy on Antarctic geography, but we can summon a functional brain image) and to then think about my path through a week or a month as it would be described by plotting as described above. I can think about points, or I can summon little images of me captured in time slices along the way. A blurry path is most easy to envision, but we can insert snapshots with greater clarity at various points along the way.
From there it’s not a huge jump to add in another layer, say, what I know about my neighbor’s life and travels. And I can add in successive layers for my friends, associates and others.
This is a bit odd, but really not as weird as it seemed when I first thought about it. So, yeah, you can visualize four dimensions. A lot of the total picture will be missing from my own visualization, because I don’t sample all of the data.
I’m not psychic.
Personally, I think we can perceive the “4th dimension” and have just internalized it. This whole idea of “dimensions” is just as arbitrary a method of dissecting our perception of the universe as any other constructed method of describing reality. If we somehow managed to blot out the 4th and higher dimensions, I have a feeling everything would look rather strange, although I have no clue in what way.
I think that something that people are confusing is the difference between perceiving the fourth dimension (no matter what it may be) and representing it. It is very easy to represent four dimension on a sheet of paper (draw the representation of 3D axes and then use colors for the fourth) or by using vectors on a map, or a chart of earths movements through space. I have seen a 2D illustration of a 3D representation of a 25 (I think) dimensional shape. The key is that we aren’t perceiving these dimensions directly, but instead are making lower dimensional representations of them that we can perceive and comprehend. The two must not be confused.
Well GulDan, you’re right in that perception does not necessarily involve a visual representation. A lot of the first part of my previous post was more about mapping an event in time. Ergo, visual representation.
The latter part of the post made a weak attempt at integrating temporal and spatial perceptions in some kind of chewable model, which I suppose did not work out that well.
Oh well, you lucky folk; you get more of me tomorrow.
While it is helpful to use time to visualize a 4th dimension, time is distinct from the spacial dimensions. If you have four spacial dimensions, you can have an object rotating around two perpendicular planes of rotation. If you are using time as one of your dimensions, It becomes difficult to visualize movement when time is already taken as one of your axes.
GulDan, The problem I have with using color as one of your dimensions is what you get is a still 3D, not 4D. If you had a 4D space, you should be able to have two object with the same x, y, and z coordinates, but a different color and they would not be in contact.
I love Edwin Abbot’s Flatland. There have been many attempts at a sequel. I like Sphereland by Dionys Burger where the inhabitants of Flatland discover that their space is curved and expanding. I have both books bound together. You can get a copy here:
Flatland and Sphereland