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
When I was in the uni doing the math (I am a sw developper) the professor told us: **WE don’t know how these multi dimensional spaces look like, but we can calculate in them.**
And he was a really good in explanation and really into the math. One time he showed us a picture of something and was really excited about it – was some visual representation of some function and big math problem. Nobody told him we don’t care, we were so happy for him as he was.
Edit> he was talking about 16-dimension space, but it still applies to 6d IMO.
In an intuitive manner? We can’t. Our cognitive system evolved to perceive and operate with a 3D environment; so there’s little in the way of analogy that can be used to intuitively grasp higher dimensional concepts. But you can use math to describe them, which is not necessarily intuitive but is a way to get around that limitation.
In an intuitive manner? We can’t. Our cognitive system evolved to perceive and operate with a 3D environment; so there’s little in the way of analogy that can be used to intuitively grasp higher dimensional concepts. But you can use math to describe them, which is not necessarily intuitive but is a way to get around that limitation.
When I was in the uni doing the math (I am a sw developper) the professor told us: **WE don’t know how these multi dimensional spaces look like, but we can calculate in them.**
And he was a really good in explanation and really into the math. One time he showed us a picture of something and was really excited about it – was some visual representation of some function and big math problem. Nobody told him we don’t care, we were so happy for him as he was.
Edit> he was talking about 16-dimension space, but it still applies to 6d IMO.
How do you imagine an n dimensional space?
That’s the neat part, you don’t. You imagine a three dimesional space and work with that knowing what properties are the same for the n dimensional space.
If you know how to drive a red card, you know how to drive an octarine car, even though no one can imagine the octarine color.
How do you imagine an n dimensional space?
That’s the neat part, you don’t. You imagine a three dimesional space and work with that knowing what properties are the same for the n dimensional space.
If you know how to drive a red card, you know how to drive an octarine car, even though no one can imagine the octarine color.
You remember what a 2 dimensional graph is? Like coordinates X,Y. That’s how you describe where a point is on a 2d plane. A point 7,2 is “7 to the right and two up” (all in inches).
You can do the same thing with 3 dimensions. Imagine a cardboard box and there’s a fly in it. To describe the point where it is, you could say it’s at 7,2,5 “7 to the right, 2 up, then 5 deep (from center origin)”.
Now imagine a whole bunch of cardboard boxes, neatly stacked up, filling an entire room. Again, we want to describe where the fly is, so we can say it’s position is 7,2,5 but it’s **in box 14**. Or we can say its position is 7,2,5,14.
But here’s the thing — the boxes aren’t *actually* stacked up as you’re probably imagining. All of the boxes are taking up the same space. The fly is in box 14. The fly isn’t in box 13, even though 13 and 14 take up the same space.
Now imagine a whole bunch of rooms, neatly stacked up, where each room is full of those cardboard boxes….
Now, you might be wondering how a bunch of cardboard boxes can take up the same space while there being a fly in one box but not another. They aren’t *actually* taking up the same space, but we can’t really imagine it otherwise. Here’s a way to kind of extrapolate it:
Go back to looking at the 2d graph that’s just X,Y. Pretend like you don’t know what 3d is, everything just looks flat. You may see the fly at position (3,8) and a caterpillar at the same position. How? Because the caterpillar is crawling around on the ground while the fly is up in the air. Yes, they’re both at (3,8), but that’s because you’re only able to see in 2d, not 3d. The fly and caterpillar are both in the same spot from a 2d viewer but not a 3d viewer.
Think of dimension as a mathematical term for “ingredient”.
The colours on our screen can be considered 3D colours. They have THREE ingredients red, green and blue.
Physics is often interested in 3 dimensions, because the space that we live in has 3 dimensions. There are three ingredients that go into WHERE you are. It takes three numbers to describe how much of each direction equates to your position.
But some problems care about more than position. A common additional ingredient is time. To describe WHEN you are requires another number (for a total of 4 dimensions). Sometimes physicists are interested in where you are and where you are moving toward (velocity), so you’d need a total of 6 dimensions (3 for your current position and 3 for the position you would be at in 1 second, if you continued at your current speed).
The more complicated the problem, the more dimensions you are likely to be interested in. Temperature, momentum, magnetic field, number of cookies, happiness, probability of snow. Every aspect is going to add a new “dimension” to the problem.
Mathematicians often imagine shapes that are 4-Dimensional. Shapes that require 4 numbers to describe where you are on the shape. You may argue that you never such a thing in real life, but mathematicians don’t care about what shapes can appear in real life and which can’t, they only care about the mathematical properties of these shapes.
You remember what a 2 dimensional graph is? Like coordinates X,Y. That’s how you describe where a point is on a 2d plane. A point 7,2 is “7 to the right and two up” (all in inches).
You can do the same thing with 3 dimensions. Imagine a cardboard box and there’s a fly in it. To describe the point where it is, you could say it’s at 7,2,5 “7 to the right, 2 up, then 5 deep (from center origin)”.
Now imagine a whole bunch of cardboard boxes, neatly stacked up, filling an entire room. Again, we want to describe where the fly is, so we can say it’s position is 7,2,5 but it’s **in box 14**. Or we can say its position is 7,2,5,14.
But here’s the thing — the boxes aren’t *actually* stacked up as you’re probably imagining. All of the boxes are taking up the same space. The fly is in box 14. The fly isn’t in box 13, even though 13 and 14 take up the same space.
Now imagine a whole bunch of rooms, neatly stacked up, where each room is full of those cardboard boxes….
Now, you might be wondering how a bunch of cardboard boxes can take up the same space while there being a fly in one box but not another. They aren’t *actually* taking up the same space, but we can’t really imagine it otherwise. Here’s a way to kind of extrapolate it:
Go back to looking at the 2d graph that’s just X,Y. Pretend like you don’t know what 3d is, everything just looks flat. You may see the fly at position (3,8) and a caterpillar at the same position. How? Because the caterpillar is crawling around on the ground while the fly is up in the air. Yes, they’re both at (3,8), but that’s because you’re only able to see in 2d, not 3d. The fly and caterpillar are both in the same spot from a 2d viewer but not a 3d viewer.
Think of dimension as a mathematical term for “ingredient”.
The colours on our screen can be considered 3D colours. They have THREE ingredients red, green and blue.
Physics is often interested in 3 dimensions, because the space that we live in has 3 dimensions. There are three ingredients that go into WHERE you are. It takes three numbers to describe how much of each direction equates to your position.
But some problems care about more than position. A common additional ingredient is time. To describe WHEN you are requires another number (for a total of 4 dimensions). Sometimes physicists are interested in where you are and where you are moving toward (velocity), so you’d need a total of 6 dimensions (3 for your current position and 3 for the position you would be at in 1 second, if you continued at your current speed).
The more complicated the problem, the more dimensions you are likely to be interested in. Temperature, momentum, magnetic field, number of cookies, happiness, probability of snow. Every aspect is going to add a new “dimension” to the problem.
Mathematicians often imagine shapes that are 4-Dimensional. Shapes that require 4 numbers to describe where you are on the shape. You may argue that you never such a thing in real life, but mathematicians don’t care about what shapes can appear in real life and which can’t, they only care about the mathematical properties of these shapes.
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