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
Lets talk about Nets.
In geometry, a Net is an unfolded 3D shape.
Imagine you’ve made a cube out of paper.
It’s six squares (2D shapes) joined on the edges and folded 90 degrees to create a 3D cube.
A Tesseract is a 4D cuboid. Basically the next one up.
A Tesseract’s Net is eight cubes, joined on their faces and folded onto one another, as a Net (called a Dali Cross in this case) it looks a bit like a minecraft tree, but fully constructed into a 4D shape the eight cubes all link together. Every face of every cube connects to the face of another cube. That might be a little hard to visualise, but Wikipedia has a rather nice animation of a rotating Tesseract which illustrates it.
If you were inside a hollow tesseract and traversing it as a 3D person, you’d be able to travel in a straight line through four cubic volumes before coming back to where you started. This applies whichever direction you pick.
A 5D Hypercube would be something like 10 tesseracts joined across their individual 3D cube-volumes, folded around so they link together into a single shape.
A 6D Hypercube would be 12 5D Hypercubes joined across their Tesseract volumes.
Basically it’s an X dimensional shape composed of a set of X-1 Dimensional objects, joined across their X-2 connections.
I have no idea how to describe what a 5D or 6D Hypercube would look like, only that if it were a hollow shape and you could traverse it in 3D, you’d be hopelessly lost very very quickly.
Perceived from a 3D perspective, it’d be either a ridiculously wibbly shape that in no way resembles a cube (most likely some kind of spike-ball) or an impossibly complex maze of cubic volumes.
Lets talk about Nets.
In geometry, a Net is an unfolded 3D shape.
Imagine you’ve made a cube out of paper.
It’s six squares (2D shapes) joined on the edges and folded 90 degrees to create a 3D cube.
A Tesseract is a 4D cuboid. Basically the next one up.
A Tesseract’s Net is eight cubes, joined on their faces and folded onto one another, as a Net (called a Dali Cross in this case) it looks a bit like a minecraft tree, but fully constructed into a 4D shape the eight cubes all link together. Every face of every cube connects to the face of another cube. That might be a little hard to visualise, but Wikipedia has a rather nice animation of a rotating Tesseract which illustrates it.
If you were inside a hollow tesseract and traversing it as a 3D person, you’d be able to travel in a straight line through four cubic volumes before coming back to where you started. This applies whichever direction you pick.
A 5D Hypercube would be something like 10 tesseracts joined across their individual 3D cube-volumes, folded around so they link together into a single shape.
A 6D Hypercube would be 12 5D Hypercubes joined across their Tesseract volumes.
Basically it’s an X dimensional shape composed of a set of X-1 Dimensional objects, joined across their X-2 connections.
I have no idea how to describe what a 5D or 6D Hypercube would look like, only that if it were a hollow shape and you could traverse it in 3D, you’d be hopelessly lost very very quickly.
Perceived from a 3D perspective, it’d be either a ridiculously wibbly shape that in no way resembles a cube (most likely some kind of spike-ball) or an impossibly complex maze of cubic volumes.
The best way for me personally to conceptualize more than 3 dimensions is to use lists of discrete numbers or objects. So, the first dimension is just a list of numbers. Maybe 1, 2, 3, 4, 5. This looks like a few dots on a page in a line. However, if we turn each of these numbers into a list, or a folder, where there is another list of numbers inside or behind it, then we would have two dimensions.
So let’s say we started out with the numbers 1, 2, 3, 4, 5 in the first dimension.
To add a second dimension, now we have folders labeled, “1,” “2,” “3,” “4,” and, “5,” that each have pieces of paper inside labeled, “1,” “2,” “3,” “4,” and, “5.” So we could refer to something like, “1,5” and know that means folder 1, page 5. We just need to decide on a standard for what the first number refers to, and what the second number refers to. Traditionally, we think of these as X and Y coordinates on a coordinate plane. The first number tells us how far left or right we are. The second number tells us how far up and down we are.
But this is somewhat arbitrary. The “first” dimension doesn’t have to have anything to do with left and right. The “second” dimension doesn’t have to have anything to do with up and down. We traditionally use those because we often use dimensions to talk about where something is located in space, or how it is moving through space and agreeing on what each dimension “means” helps us communicate with each other more effectively.
However, you and I could be oriented differently. If I am across from you at a table and you tell me something like, “the 3rd one from the left,” I might have to clarify whether you mean your left or my left. There is nothing inherently sacred about left and right, up and down, etc.
In fact, dimensions don’t even have to refer to space at all (although I suspect that is what you are trying to visualize and wrap your head around). Going back to my folders analogy we can create as many dimensions as we want by turning each new dimension, which is just a list of discrete numbers, into another folder.
If I tell you the piece of information I am referring to is on page 3 of folder 1 in drawer 4 of filing cabinet 2 in building 1, that’s 5 dimensions right there. Building, filing cabinet, drawer, folder, page.
If the folder analogy isn’t intuitive or is too far removed from imagining spatial dimensions, let me try one more.
A line is one dimension. A square is two dimensions. A cube is 3 dimensions. And we can split each of them up if we want. We can have a line broken up into 3 smaller lines. We can have a square broken up into 9 smaller squares. We can have a cube broken up into 27 smaller cubes.
If we label those discrete sections we just cut up, then we can use, “coordinates,” to refer to specific ones. The sections of a 1-dimensional line could just be 1, 2, or 3. But how to refer to a specific square? We’d have to specify which of the 3 horizontal positions it is in and which of the 3 vertical positions it is in. So we could have (1,1) but also (3,2) and it sounds like you intuitively understand that. The same for which specific cube we want to talk about, but now we need 3 coordinates. A 4th dimension means we would need a 4th coordinate because each of those cubes has 3 more possible options. But where can we go physically that hasn’t already been covered by the first 3 dimensions? One option that people seem to understand is time. You could make what time you are referring to the 4th dimension. Are you talking about the past, the present, or the future for that specific cube? You could refer to each as 1 (past), 2 (present), or 3 (future). But it doesn’t have to be time, that’s just one of the few things we can imagine moving through that makes sense and is somewhat similar to the spatial dimensions.
Maybe instead of making time the 4th dimension we add 3 smaller cubes inside each of our 27 cubes from the 3rd dimension. To differentiate each of the 3 new cubes inside every one of our original 27, we can refer to whichever cube is closest to the center of our larger 3D cube as, “1,” and the new cube farthest from the center of our larger 3D cube as, “3.” We do the same for every one of the now 72 new cubes (3 each inside of our original 27). We haven’t used a 4th dimension of space, but we have added a 4th dimension of information. If you want to refer to the exact position of one of the 72 smaller cubes you would need 4 pieces of information. Which horizontal section it is in, which vertical, how far back the cube is, and then how close to the center.
I’m sure that’s a little unsatisfying because we didn’t actually introduce a novel 4th spatial dimension. That’s why I prefer to picture dimensions as folders nested inside of folders (or inside drawers, inside cabinets, inside rooms, inside buildings). It’s easy to imagine having to go to the Nth level of a filing system to find a specific document, but hard to imagine finding a certain physical point of an Nth level object.
The best way for me personally to conceptualize more than 3 dimensions is to use lists of discrete numbers or objects. So, the first dimension is just a list of numbers. Maybe 1, 2, 3, 4, 5. This looks like a few dots on a page in a line. However, if we turn each of these numbers into a list, or a folder, where there is another list of numbers inside or behind it, then we would have two dimensions.
So let’s say we started out with the numbers 1, 2, 3, 4, 5 in the first dimension.
To add a second dimension, now we have folders labeled, “1,” “2,” “3,” “4,” and, “5,” that each have pieces of paper inside labeled, “1,” “2,” “3,” “4,” and, “5.” So we could refer to something like, “1,5” and know that means folder 1, page 5. We just need to decide on a standard for what the first number refers to, and what the second number refers to. Traditionally, we think of these as X and Y coordinates on a coordinate plane. The first number tells us how far left or right we are. The second number tells us how far up and down we are.
But this is somewhat arbitrary. The “first” dimension doesn’t have to have anything to do with left and right. The “second” dimension doesn’t have to have anything to do with up and down. We traditionally use those because we often use dimensions to talk about where something is located in space, or how it is moving through space and agreeing on what each dimension “means” helps us communicate with each other more effectively.
However, you and I could be oriented differently. If I am across from you at a table and you tell me something like, “the 3rd one from the left,” I might have to clarify whether you mean your left or my left. There is nothing inherently sacred about left and right, up and down, etc.
In fact, dimensions don’t even have to refer to space at all (although I suspect that is what you are trying to visualize and wrap your head around). Going back to my folders analogy we can create as many dimensions as we want by turning each new dimension, which is just a list of discrete numbers, into another folder.
If I tell you the piece of information I am referring to is on page 3 of folder 1 in drawer 4 of filing cabinet 2 in building 1, that’s 5 dimensions right there. Building, filing cabinet, drawer, folder, page.
If the folder analogy isn’t intuitive or is too far removed from imagining spatial dimensions, let me try one more.
A line is one dimension. A square is two dimensions. A cube is 3 dimensions. And we can split each of them up if we want. We can have a line broken up into 3 smaller lines. We can have a square broken up into 9 smaller squares. We can have a cube broken up into 27 smaller cubes.
If we label those discrete sections we just cut up, then we can use, “coordinates,” to refer to specific ones. The sections of a 1-dimensional line could just be 1, 2, or 3. But how to refer to a specific square? We’d have to specify which of the 3 horizontal positions it is in and which of the 3 vertical positions it is in. So we could have (1,1) but also (3,2) and it sounds like you intuitively understand that. The same for which specific cube we want to talk about, but now we need 3 coordinates. A 4th dimension means we would need a 4th coordinate because each of those cubes has 3 more possible options. But where can we go physically that hasn’t already been covered by the first 3 dimensions? One option that people seem to understand is time. You could make what time you are referring to the 4th dimension. Are you talking about the past, the present, or the future for that specific cube? You could refer to each as 1 (past), 2 (present), or 3 (future). But it doesn’t have to be time, that’s just one of the few things we can imagine moving through that makes sense and is somewhat similar to the spatial dimensions.
Maybe instead of making time the 4th dimension we add 3 smaller cubes inside each of our 27 cubes from the 3rd dimension. To differentiate each of the 3 new cubes inside every one of our original 27, we can refer to whichever cube is closest to the center of our larger 3D cube as, “1,” and the new cube farthest from the center of our larger 3D cube as, “3.” We do the same for every one of the now 72 new cubes (3 each inside of our original 27). We haven’t used a 4th dimension of space, but we have added a 4th dimension of information. If you want to refer to the exact position of one of the 72 smaller cubes you would need 4 pieces of information. Which horizontal section it is in, which vertical, how far back the cube is, and then how close to the center.
I’m sure that’s a little unsatisfying because we didn’t actually introduce a novel 4th spatial dimension. That’s why I prefer to picture dimensions as folders nested inside of folders (or inside drawers, inside cabinets, inside rooms, inside buildings). It’s easy to imagine having to go to the Nth level of a filing system to find a specific document, but hard to imagine finding a certain physical point of an Nth level object.
> how can we grasp it in n intuitive manner free of mathematical analysis?
Like others have said, the short answer is that you can’t, but I want to point out that this is what makes math so awesome. The things you find to be intuitive feel that way because you have learned them through experience with your senses. If you have seen it or felt it many times in your life, it becomes “obvious” and “intuitive” just because your brain recognizes the patterns. However, do you believe that there is more to the universe than what you can directly see and touch? If so, you are right! There is _a lot_ more. Infinitely more, even. The amazing thing about math is that we can use it to explore those things that we cannot directly experience. We can do that because math is just symbolic logic. It’s a system for making statements that _must_ be true, no matter how complicated or far removed from what we can directly experience they are. It lets us understand things that are not intuitive, or even more incredibly, things that are _counter_-intuitive, meaning that what happens is the opposite of what you would normally expect to happen. That gives us incredible powers of prediction and lets us solve problems that would be unsolvable otherwise.
> how can we grasp it in n intuitive manner free of mathematical analysis?
Like others have said, the short answer is that you can’t, but I want to point out that this is what makes math so awesome. The things you find to be intuitive feel that way because you have learned them through experience with your senses. If you have seen it or felt it many times in your life, it becomes “obvious” and “intuitive” just because your brain recognizes the patterns. However, do you believe that there is more to the universe than what you can directly see and touch? If so, you are right! There is _a lot_ more. Infinitely more, even. The amazing thing about math is that we can use it to explore those things that we cannot directly experience. We can do that because math is just symbolic logic. It’s a system for making statements that _must_ be true, no matter how complicated or far removed from what we can directly experience they are. It lets us understand things that are not intuitive, or even more incredibly, things that are _counter_-intuitive, meaning that what happens is the opposite of what you would normally expect to happen. That gives us incredible powers of prediction and lets us solve problems that would be unsolvable otherwise.
> in maths a dimension is defined as the minimum amount of values needed to precisely describe something
Or in other words, if you can accurately describe something with three values, it is 3d. If you need four values to describe it it‘s 4d.
Let me show you an example of what n-dimensions could look like:
– you want to know an object‘s position. Thats 3 dimensions, very intuitive.
– you also care about when it was at that position. So you also need a timestamp to precisely describe it; now your object is 4d
– let‘s say you also need to know it‘s colour, so you need its red green and blue value. And boom, 7 dimensions
And you can keep adding stuff. The more complex your problem is that you‘re trying to solve the more things you have to consider, and the amount of things is the amount of dimensions
> in maths a dimension is defined as the minimum amount of values needed to precisely describe something
Or in other words, if you can accurately describe something with three values, it is 3d. If you need four values to describe it it‘s 4d.
Let me show you an example of what n-dimensions could look like:
– you want to know an object‘s position. Thats 3 dimensions, very intuitive.
– you also care about when it was at that position. So you also need a timestamp to precisely describe it; now your object is 4d
– let‘s say you also need to know it‘s colour, so you need its red green and blue value. And boom, 7 dimensions
And you can keep adding stuff. The more complex your problem is that you‘re trying to solve the more things you have to consider, and the amount of things is the amount of dimensions
I’ve thought about 6 dimensional space in terms of scale. Imagine individual atoms or molecules. They have length, width, height. So a molecule is 3 dimensional. If you zoom out to human scale – a molecule is essentially 0 dimensions. It’s a.point with (very limited) length, width, height. The scale is so different – that a molecules dimensions basically don’t exist.
So molecular scale is 3 dimensional, human scale is 3 dimensional. Boom, 6 total dimensions. The molecular scale dimensions exist inside our 3D space. The smaller dimensions are packed so tightly they cannot be viewed from the larger scale.
Extend it to cosmic scales – you have 9 dimensions + time.
I’ve thought about 6 dimensional space in terms of scale. Imagine individual atoms or molecules. They have length, width, height. So a molecule is 3 dimensional. If you zoom out to human scale – a molecule is essentially 0 dimensions. It’s a.point with (very limited) length, width, height. The scale is so different – that a molecules dimensions basically don’t exist.
So molecular scale is 3 dimensional, human scale is 3 dimensional. Boom, 6 total dimensions. The molecular scale dimensions exist inside our 3D space. The smaller dimensions are packed so tightly they cannot be viewed from the larger scale.
Extend it to cosmic scales – you have 9 dimensions + time.
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