I get 1, 2, and 3 dimensions and their visual representations. For instance, 3-dimensional space has width, length, and depth but how about a 6-dimensional space, how can we grasp it in n intuitive manner free of mathematical analysis?

In: 8

Visually you can maybe do 5 dims at the same time (with a color coded 4d object) but more than that you slice it I guess – when exploring a dataset for example.

I don’t think you should try and force yourself to intuit about n dimensional objects without mathematics. We invented mathematics for this. Same way as you want a shovel for digging a ditch. Don’t use your hands.

Short anwser. We cant. Since we haven’t found any thing in real life that is 6-dimensional there is noway to intutively represent it. It is (until we find some thing diffrent) a purely mathematical concept.

Think of a line on a piece of paper. The line represents one dimension while the piece of paper represents 2 dimensions.

For every spot on the line, there is an infinite number of spaces above and below it in the 2 dimensional space.

Now think about the 2 dimensional paper in 3 dimensions. For every spot own that 2 dimensional paper, there are infinite number of spaces above and below the piece of paper.

So the leap you have to make in order to understand extra spatial dimensions is that you have to realize there are different types of above and below than just the kind represented by the 3rd dimension. So traveling through a 4th spatial dimension would be sort of like moving either up-and-down through different three dimensional spaces, each one of which is just a cross-section of the larger 4 dimensional space you cannot fully see at any time.

For instance, a small rock might be a large 4D boulder just barely protruding into our 3 dimensional space. So going upwards in 4 dimensions might make it seem to grow bigger in front of you, while going downwards might make it seem to get smaller. This is because you are moving through different 3 dimensional cross-sections of 4 dimensional space.

First off a dimension just represents an axis you can move along without having to also move along another different dimension’s axis.

Simplest example is the 2D xy grid. You likely understand easily that you can move along the X axis without having to move along the Y, thus we can say there is an X dimension and a Y dimension to the grid.

In a mathematical sense you can easily extend this infinitely in concept if not in visualization. You just conceive of another dimension you’re moving along in. If a 2D xy grid has a coordinate written X, Y and a 3D xyz space has a coordinate written X, Y, Z we can easily extrapolate that a 5D xyzab superspace has coordinates written X, Y, Z, A, B or w/e.

So purely mathematically when it comes to higher dimensions, that should be easy to grasp.

What you’re likely though having trouble with is that we also use the concept of multiple dimensions to describe our experience of the phenomenon of spacetime, but we can grasp physical dimensions too by going back to the first sentence I wrote. A dimension just represents an axis you can move along without necessarily moving along another. In reference to actual physical space this is easy to understand as 3D as you understand you can move forward/back without having to move up/down or left/right. We can thus say spacetime has ‘width’, ‘length’, and ‘height’ dimensions to it.

But what if you want to move in some direction which isn’t those six? In physical space, that isn’t possible that dimension just isn’t there. That being said we can see our experience with the phenomenon of time and its intimate relationship with space as an additional dimension. We know we can “move” through time without having to move within the six physical directions. But we don’t HAVE to call time the 4th dimension or something, that’s just a way to think of it.

We could just as conceivably make an axis of emotions ranging from hate to love or something and describe experiencing different emotions as moving along this axis in which case we could say ’emotion’ is the 4th dimension that we can move along without moving in 3D space, etc. We could then come up with all kinds of other dimensions and claim humans actually exist in N-dimensional space but how useful those dimensions / this idea would actually be in explaining reality would be questionable.

TL;DR A dimension is just a concept. Some real life phenomenon can be described using dimensions, but a ‘higher dimension’ is just an idea like a ‘bigger number’ it doesn’t reflect anything about reality.

It’s not really something that can be explained in an intuitive manner. As 3D beings, grasping dimension higher than that is actually really hard. The concept itself isn’t that hard.

So in 1 dimension we have a single line. In 2D we add a new line that is orthogonal (90 degrees) to the first line. In 3D we add another line to which is orthogonal to the previous ones.

So far it’s quite easy. After that comes the part the is hard to imagine and visualize. We can go up further in dimensions, by adding another line that is orthogonal to all the previous ones.

Calculating with it isn’t that much harder, since instead of x,y,z we now have lets say q and w as well. We can expand the logic from going from 2D to 3D to higher dimensions.