I just dont know what Is 4D space and i can’t imagine it…
In: 5
Nobody can we are built for 3D thinking.
So what do we mean by dimension?
The number o basis vectors required to reach every point in your space is the dimension number. (You can define dimensions in other ways but this’ll do.)
So what are basis vectors?
Imagine a line. Each point on the line can be reached with a vector pointing to it from the origin. So lets say we define what 1 unit is. Lets say the basis vector is 1 unit in lenght. It points from 0 to 1. Lets call this v1. Every point now can be defined as some number c times that vector v1. So two would be 2v1, -6 would be -6v1.
When we have more dimensions it means that we need v1 and v2. Now v2 has to be linearly independent from v1. This means that v2 isn’t a multiple of v1. Make them have an angle between them and this will hold true. Like make them perpenticular. Now each point on the plain can be reached with the linear combination of the two. This is basically: any point = c1v1+c2v2. This is almost like coordinates but we can add and multiply in comfortable ways. So a point (6,2) is 6×v1+2×v2.
We can add a third independent vector v3 which cannot be reached with a linear combination of v1 and v2. So it points out of the plain. Together they stretch out the 3D space.
Mathematically there is nothing weird about adding a 4th basis vector each point will have 4 coordinates, and we can reach any point by doing c1v1+c2v2+c3v3+c4v4. You can continue this as long as you want.
So that is a 4D space its streached by 4 basis vectors.
In physics you can treat time as a 4th dimensions. You have 3 spacial directions and 1 temporal. Since a lot of motion happens in a plain like a planet orbiting a star we can drop 1 spacial dimension and look at the orbit in the x,y plain and add a 3rd dimensions as time. So in this 3D diagram the orbit looks like a spiral. So that 3D diagram is the path of your object through space and time. This is called a worldline.
Now why you might want to do this is because it allows you to use different math. For moving objects you got dynamics we deal with that if we need to figure out the path of an object. It involves solving usually difficult equations. But lets say we know the path and we want to calculate different things, like time dilation in special reactivity. In general the whole path through space and time gives you a 4D diagram, but since time is one directional it behaves differently than spacial dimensions we often call a diagram like that a 3+1 spacetime diagram. But once you do that your framework is static geometry, things dont change, nothing moves its just lines, curves and coordinate transformations.
So using a 4D or higher space can be helpful in physics. Phase diagrams for example plot position and velocity. Its a good way to visualise what a system is doing. Like a pendulum will oscillate around its stabil point, and that appears as ellipsis or circles on the phase diagram. Think it through the pendulum swings left and stops. It has 0 velocity and maximum negative x direction so its in -x then it moves back to 0 gaining velocity in 0 it has +v velocity then it slows down at +x with 0 velocity and moves back with -v velocity. It draws a circle around 0. Of course if we can let that pendulum swing in any direction our phase diagram got x,y,z and v_x,v_y,v_z. Giving us a 6D phase diagram. And again looking at the curves we can geometrically understand how the system behaves. Identify stabil and instabil equilibrium states.
So higher dimensional vector spaces are like most things mathematical tools we can use.
Imagine that one day you picked up a piece of paper, and you found these 2-dimensional, living, intelligent (and English-speaking!) people on that paper. How would you describe our 3-dimensional world to them?
Well, they still have these basic 2-D shapes such as circles, squares, etc. They lack understanding of the 3-D version of these shapes (spheres, cubes). But, there is a relationship between the 3-D shapes and the 2-D shapes – a circle is just the cross-section of a sphere.
A more visual example: if you are somehow able to physically enter into these 2-D people’s space, then from their point of view, what they are seeing is a cross section of you. Let’s say that you happen to be “waist-deep”, then they’d see a roughly oval shape, with a layer of skin on the outside, cross-section of the spine off to one side, jumbled cross-section of your intestines in the middle, wrapped around half digested dinner from last night, like a CT scan.
Now, back to your question. The above thought experiment applies if you just change the 2-D/3-D to 3-D/4-D. The 3-D world that we live in is just a cross section of the 4-D space. That is another dimension out there that we literally can’t fathom, but if we were to ever meet a 4-dimensional being, we may very well be talking to a blob that happens to be a cross section of their pinky toe.
Mathematically, the 2-D world has the x and y axis, and they are perpendicular to each other. Here in 3-D world, we add a z axis which is perpendicular to both x and y. In the 4-D world, we then add a 4th dimension that is perpendicular to all 3 of x, y and z!
In a 3D space, you need three values to identify any particular point in it: `(x, y, z)`, like `(1, 2, 3)`. In 2D space you only need `(x, y)` coordinates. In 4D space, you simply need one more value, like `(1, 2, 3, 4)`. You don’t necessarily need to visualize it as physical axes, that additional value could be anything. For example time. Or if you imagine any `(x, y, z)` point to be a cube itself (picture a warehouse of cubes), then the forth value could denote the side of the cube. Or you don’t imagine it like a “space” at all, and just take it as a filing system in a library: “1st floor, 2nd quadrant, 3rd shelf, 4th section”.
If you stop trying to picture *n*D coordinates as physical axes at all and just take it as “you need *n* values to identify any one location in this space”, it might make more sense.
Nobody can we are built for 3D thinking.
So what do we mean by dimension?
The number o basis vectors required to reach every point in your space is the dimension number. (You can define dimensions in other ways but this’ll do.)
So what are basis vectors?
Imagine a line. Each point on the line can be reached with a vector pointing to it from the origin. So lets say we define what 1 unit is. Lets say the basis vector is 1 unit in lenght. It points from 0 to 1. Lets call this v1. Every point now can be defined as some number c times that vector v1. So two would be 2v1, -6 would be -6v1.
When we have more dimensions it means that we need v1 and v2. Now v2 has to be linearly independent from v1. This means that v2 isn’t a multiple of v1. Make them have an angle between them and this will hold true. Like make them perpenticular. Now each point on the plain can be reached with the linear combination of the two. This is basically: any point = c1v1+c2v2. This is almost like coordinates but we can add and multiply in comfortable ways. So a point (6,2) is 6×v1+2×v2.
We can add a third independent vector v3 which cannot be reached with a linear combination of v1 and v2. So it points out of the plain. Together they stretch out the 3D space.
Mathematically there is nothing weird about adding a 4th basis vector each point will have 4 coordinates, and we can reach any point by doing c1v1+c2v2+c3v3+c4v4. You can continue this as long as you want.
So that is a 4D space its streached by 4 basis vectors.
In physics you can treat time as a 4th dimensions. You have 3 spacial directions and 1 temporal. Since a lot of motion happens in a plain like a planet orbiting a star we can drop 1 spacial dimension and look at the orbit in the x,y plain and add a 3rd dimensions as time. So in this 3D diagram the orbit looks like a spiral. So that 3D diagram is the path of your object through space and time. This is called a worldline.
Now why you might want to do this is because it allows you to use different math. For moving objects you got dynamics we deal with that if we need to figure out the path of an object. It involves solving usually difficult equations. But lets say we know the path and we want to calculate different things, like time dilation in special reactivity. In general the whole path through space and time gives you a 4D diagram, but since time is one directional it behaves differently than spacial dimensions we often call a diagram like that a 3+1 spacetime diagram. But once you do that your framework is static geometry, things dont change, nothing moves its just lines, curves and coordinate transformations.
So using a 4D or higher space can be helpful in physics. Phase diagrams for example plot position and velocity. Its a good way to visualise what a system is doing. Like a pendulum will oscillate around its stabil point, and that appears as ellipsis or circles on the phase diagram. Think it through the pendulum swings left and stops. It has 0 velocity and maximum negative x direction so its in -x then it moves back to 0 gaining velocity in 0 it has +v velocity then it slows down at +x with 0 velocity and moves back with -v velocity. It draws a circle around 0. Of course if we can let that pendulum swing in any direction our phase diagram got x,y,z and v_x,v_y,v_z. Giving us a 6D phase diagram. And again looking at the curves we can geometrically understand how the system behaves. Identify stabil and instabil equilibrium states.
So higher dimensional vector spaces are like most things mathematical tools we can use.
Nobody can we are built for 3D thinking.
So what do we mean by dimension?
The number o basis vectors required to reach every point in your space is the dimension number. (You can define dimensions in other ways but this’ll do.)
So what are basis vectors?
Imagine a line. Each point on the line can be reached with a vector pointing to it from the origin. So lets say we define what 1 unit is. Lets say the basis vector is 1 unit in lenght. It points from 0 to 1. Lets call this v1. Every point now can be defined as some number c times that vector v1. So two would be 2v1, -6 would be -6v1.
When we have more dimensions it means that we need v1 and v2. Now v2 has to be linearly independent from v1. This means that v2 isn’t a multiple of v1. Make them have an angle between them and this will hold true. Like make them perpenticular. Now each point on the plain can be reached with the linear combination of the two. This is basically: any point = c1v1+c2v2. This is almost like coordinates but we can add and multiply in comfortable ways. So a point (6,2) is 6×v1+2×v2.
We can add a third independent vector v3 which cannot be reached with a linear combination of v1 and v2. So it points out of the plain. Together they stretch out the 3D space.
Mathematically there is nothing weird about adding a 4th basis vector each point will have 4 coordinates, and we can reach any point by doing c1v1+c2v2+c3v3+c4v4. You can continue this as long as you want.
So that is a 4D space its streached by 4 basis vectors.
In physics you can treat time as a 4th dimensions. You have 3 spacial directions and 1 temporal. Since a lot of motion happens in a plain like a planet orbiting a star we can drop 1 spacial dimension and look at the orbit in the x,y plain and add a 3rd dimensions as time. So in this 3D diagram the orbit looks like a spiral. So that 3D diagram is the path of your object through space and time. This is called a worldline.
Now why you might want to do this is because it allows you to use different math. For moving objects you got dynamics we deal with that if we need to figure out the path of an object. It involves solving usually difficult equations. But lets say we know the path and we want to calculate different things, like time dilation in special reactivity. In general the whole path through space and time gives you a 4D diagram, but since time is one directional it behaves differently than spacial dimensions we often call a diagram like that a 3+1 spacetime diagram. But once you do that your framework is static geometry, things dont change, nothing moves its just lines, curves and coordinate transformations.
So using a 4D or higher space can be helpful in physics. Phase diagrams for example plot position and velocity. Its a good way to visualise what a system is doing. Like a pendulum will oscillate around its stabil point, and that appears as ellipsis or circles on the phase diagram. Think it through the pendulum swings left and stops. It has 0 velocity and maximum negative x direction so its in -x then it moves back to 0 gaining velocity in 0 it has +v velocity then it slows down at +x with 0 velocity and moves back with -v velocity. It draws a circle around 0. Of course if we can let that pendulum swing in any direction our phase diagram got x,y,z and v_x,v_y,v_z. Giving us a 6D phase diagram. And again looking at the curves we can geometrically understand how the system behaves. Identify stabil and instabil equilibrium states.
So higher dimensional vector spaces are like most things mathematical tools we can use.
Imagine that one day you picked up a piece of paper, and you found these 2-dimensional, living, intelligent (and English-speaking!) people on that paper. How would you describe our 3-dimensional world to them?
Well, they still have these basic 2-D shapes such as circles, squares, etc. They lack understanding of the 3-D version of these shapes (spheres, cubes). But, there is a relationship between the 3-D shapes and the 2-D shapes – a circle is just the cross-section of a sphere.
A more visual example: if you are somehow able to physically enter into these 2-D people’s space, then from their point of view, what they are seeing is a cross section of you. Let’s say that you happen to be “waist-deep”, then they’d see a roughly oval shape, with a layer of skin on the outside, cross-section of the spine off to one side, jumbled cross-section of your intestines in the middle, wrapped around half digested dinner from last night, like a CT scan.
Now, back to your question. The above thought experiment applies if you just change the 2-D/3-D to 3-D/4-D. The 3-D world that we live in is just a cross section of the 4-D space. That is another dimension out there that we literally can’t fathom, but if we were to ever meet a 4-dimensional being, we may very well be talking to a blob that happens to be a cross section of their pinky toe.
Mathematically, the 2-D world has the x and y axis, and they are perpendicular to each other. Here in 3-D world, we add a z axis which is perpendicular to both x and y. In the 4-D world, we then add a 4th dimension that is perpendicular to all 3 of x, y and z!
In a 3D space, you need three values to identify any particular point in it: `(x, y, z)`, like `(1, 2, 3)`. In 2D space you only need `(x, y)` coordinates. In 4D space, you simply need one more value, like `(1, 2, 3, 4)`. You don’t necessarily need to visualize it as physical axes, that additional value could be anything. For example time. Or if you imagine any `(x, y, z)` point to be a cube itself (picture a warehouse of cubes), then the forth value could denote the side of the cube. Or you don’t imagine it like a “space” at all, and just take it as a filing system in a library: “1st floor, 2nd quadrant, 3rd shelf, 4th section”.
If you stop trying to picture *n*D coordinates as physical axes at all and just take it as “you need *n* values to identify any one location in this space”, it might make more sense.
Imagine that one day you picked up a piece of paper, and you found these 2-dimensional, living, intelligent (and English-speaking!) people on that paper. How would you describe our 3-dimensional world to them?
Well, they still have these basic 2-D shapes such as circles, squares, etc. They lack understanding of the 3-D version of these shapes (spheres, cubes). But, there is a relationship between the 3-D shapes and the 2-D shapes – a circle is just the cross-section of a sphere.
A more visual example: if you are somehow able to physically enter into these 2-D people’s space, then from their point of view, what they are seeing is a cross section of you. Let’s say that you happen to be “waist-deep”, then they’d see a roughly oval shape, with a layer of skin on the outside, cross-section of the spine off to one side, jumbled cross-section of your intestines in the middle, wrapped around half digested dinner from last night, like a CT scan.
Now, back to your question. The above thought experiment applies if you just change the 2-D/3-D to 3-D/4-D. The 3-D world that we live in is just a cross section of the 4-D space. That is another dimension out there that we literally can’t fathom, but if we were to ever meet a 4-dimensional being, we may very well be talking to a blob that happens to be a cross section of their pinky toe.
Mathematically, the 2-D world has the x and y axis, and they are perpendicular to each other. Here in 3-D world, we add a z axis which is perpendicular to both x and y. In the 4-D world, we then add a 4th dimension that is perpendicular to all 3 of x, y and z!
Say I invite you over for lunch and I give you my address like this: “I live at the corner of 6th street and 37th avenue, on the ninth floor, four units from the elevator.” So you write down (6, 37, 9, 4). These four numbers describe my location in space. This is technically still three-dimensional space, so I could give you this information with three numbers- my latitude, my longitude, and my altitude, but street numbers are quantized so we need the extra dimension to be more specific.
Four-dimensional space is a mathematical abstraction for people to do math in. It’s not important to be able to visualize all four dimensions. In 4D space, if you want to describe the location of a point you need to make a list of four numbers, like you did up there.
But dimensions don’t have to be spatial; they can be symbolic. Imagine a society where the only things you can eat for lunch are gyros, ice cream sandwiches, and Diet Pepsi™. Any possible lunch can be described with the number of gyros, ice cream sandwiches, and Diet Pepsi™ you ate, so every point in “lunch space” describes a unique lunch. Then, if lunch scientists figure out that you can also have an apple for lunch, then you would need an extra fourth dimension to describe every unique lunch.
Say I invite you over for lunch and I give you my address like this: “I live at the corner of 6th street and 37th avenue, on the ninth floor, four units from the elevator.” So you write down (6, 37, 9, 4). These four numbers describe my location in space. This is technically still three-dimensional space, so I could give you this information with three numbers- my latitude, my longitude, and my altitude, but street numbers are quantized so we need the extra dimension to be more specific.
Four-dimensional space is a mathematical abstraction for people to do math in. It’s not important to be able to visualize all four dimensions. In 4D space, if you want to describe the location of a point you need to make a list of four numbers, like you did up there.
But dimensions don’t have to be spatial; they can be symbolic. Imagine a society where the only things you can eat for lunch are gyros, ice cream sandwiches, and Diet Pepsi™. Any possible lunch can be described with the number of gyros, ice cream sandwiches, and Diet Pepsi™ you ate, so every point in “lunch space” describes a unique lunch. Then, if lunch scientists figure out that you can also have an apple for lunch, then you would need an extra fourth dimension to describe every unique lunch.
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