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
If you are talking about 6 parameters of some algorithm then 3D = a cube, 4D = a line of cubes, 5D = a plane of cubes, 6D = a cube of cubes etc
If you are talking about 6 parameters of some algorithm then 3D = a cube, 4D = a line of cubes, 5D = a plane of cubes, 6D = a cube of cubes etc
Once I get above 4 or 5 dimensions, I like to skip directly to thinking in terms of a large number of dimensions, like “Okay if this was like 10,000 dimensions how would it work?”
Since 10,000 dimensions is so far beyond your ability to track individually, it actually helps to clarify your thinking, because the sheer size of the number 10,000 forces you to think of the dimensions “in bulk”… “Okay this is the thing that happens for each extra dimension and it happens 10,000 times”.
Think about the jump from 1 dimensions to 2 dimensions. What did you add? What new possibilities open up?
Now think again about the jump from 2 dimensions to 3 dimensions. What did you add? What new possibilities open up?
Now, instead of trying to picture it, instead think about a dimension as “adding” a new direction which “opens up” the possibility of traveling in a way that’s “perpendicular” to all the existing directions.
Now imagine “adding” a new direction 10,000 times, so you can travel in 10,000 ways that are all “perpendicular” to each other.
Of course the number 10,000 isn’t special, you could just as easily do the same sort of thinking for 6 dimensions or 10,000,000,000 dimensions.
Once I get above 4 or 5 dimensions, I like to skip directly to thinking in terms of a large number of dimensions, like “Okay if this was like 10,000 dimensions how would it work?”
Since 10,000 dimensions is so far beyond your ability to track individually, it actually helps to clarify your thinking, because the sheer size of the number 10,000 forces you to think of the dimensions “in bulk”… “Okay this is the thing that happens for each extra dimension and it happens 10,000 times”.
Think about the jump from 1 dimensions to 2 dimensions. What did you add? What new possibilities open up?
Now think again about the jump from 2 dimensions to 3 dimensions. What did you add? What new possibilities open up?
Now, instead of trying to picture it, instead think about a dimension as “adding” a new direction which “opens up” the possibility of traveling in a way that’s “perpendicular” to all the existing directions.
Now imagine “adding” a new direction 10,000 times, so you can travel in 10,000 ways that are all “perpendicular” to each other.
Of course the number 10,000 isn’t special, you could just as easily do the same sort of thinking for 6 dimensions or 10,000,000,000 dimensions.
If you are driving on a straight flat highway and somebody asks where you are, you only need one piece of information. “I’m five miles east of Bridgeport.” That’s one dimension.
If you’re on a boat, or a checkerboard, you need 2 pieces of information. “I’m at latitude x.xxx and longitude y.yyyy”. Or, “I’m in the third row, second column”. That’s two dimensions.
If you’re a fly in Rene Descartes’ study, you need 3 pieces of information – for example, how far you are from the end wall, how far from the side wall, and how high above the floor. Three dimensions
In all these examples, humans can visualize and draw or model what is going on. We could construct a miniature version of Descartes’ room and place the fly exactly where he said it was.
If we lived in a four dimensional universe, we would need 4 pieces of information to describe exactly where we were, but our brains can’t easily visualize that. One somewhat intuitive example, would be to consider time as a 4th dimension. You send your time travelling friend a message through space and time asking “Where are you?” and she replies “1625. I’m just leaving Rene’s place”. So time, in this example, is the 4th piece of information you need, to determine her location. You say “So, how was it?” “Not great,” she replies, “he kept getting distracted by this annoying fly!”
If you are driving on a straight flat highway and somebody asks where you are, you only need one piece of information. “I’m five miles east of Bridgeport.” That’s one dimension.
If you’re on a boat, or a checkerboard, you need 2 pieces of information. “I’m at latitude x.xxx and longitude y.yyyy”. Or, “I’m in the third row, second column”. That’s two dimensions.
If you’re a fly in Rene Descartes’ study, you need 3 pieces of information – for example, how far you are from the end wall, how far from the side wall, and how high above the floor. Three dimensions
In all these examples, humans can visualize and draw or model what is going on. We could construct a miniature version of Descartes’ room and place the fly exactly where he said it was.
If we lived in a four dimensional universe, we would need 4 pieces of information to describe exactly where we were, but our brains can’t easily visualize that. One somewhat intuitive example, would be to consider time as a 4th dimension. You send your time travelling friend a message through space and time asking “Where are you?” and she replies “1625. I’m just leaving Rene’s place”. So time, in this example, is the 4th piece of information you need, to determine her location. You say “So, how was it?” “Not great,” she replies, “he kept getting distracted by this annoying fly!”
Dimensions are first and foremost a math concept. It just so happens the real universe we know can be described by a specific dimensional space called R3 (3 real numbers, commonly called X,Y,Z or length, width, depth.)
However, if we happened to live on a flat plane or on the surface of a perfectly round sphere, our universe could be described with R2. (Indeed, longitude and latitude describe a position on the surface of the Earth using two real numbers!)
But notice that R2 can describe *two* different shapes of universes: surface of a sphere *or* a flat plane. That means that even though R2 can describe or limit the shape of those universes, R2 does not *define* the shape of those universes. Simply knowing the universe can be described with R2 is not enough to know the actual shape of the universe.
So, when you want to imagine a universe that can be described by R6 it could mean many different things, just like a universe described by R2 can mean many different things. There is no single R6 universe: there are infinitely many R6 universes.
Think of R6 as a category.
And your question wasn’t even about R6: it was about 6D. R6 is only one of the possible 6D spaces!
Dimensions are first and foremost a math concept. It just so happens the real universe we know can be described by a specific dimensional space called R3 (3 real numbers, commonly called X,Y,Z or length, width, depth.)
However, if we happened to live on a flat plane or on the surface of a perfectly round sphere, our universe could be described with R2. (Indeed, longitude and latitude describe a position on the surface of the Earth using two real numbers!)
But notice that R2 can describe *two* different shapes of universes: surface of a sphere *or* a flat plane. That means that even though R2 can describe or limit the shape of those universes, R2 does not *define* the shape of those universes. Simply knowing the universe can be described with R2 is not enough to know the actual shape of the universe.
So, when you want to imagine a universe that can be described by R6 it could mean many different things, just like a universe described by R2 can mean many different things. There is no single R6 universe: there are infinitely many R6 universes.
Think of R6 as a category.
And your question wasn’t even about R6: it was about 6D. R6 is only one of the possible 6D spaces!
I always found this guys video to be a good explainer for theoretical higher dimensions. https://youtu.be/0ca4miMMaCE
I always found this guys video to be a good explainer for theoretical higher dimensions. https://youtu.be/0ca4miMMaCE
Latest Answers