When we start learning geometry in school, one of the first things we learn is the difference between a "2-dimensional shape" (that exists on a piece of paper), and a "3-dimensional shape" (that exists in the real world). But, can we go further than this? Is it possible for us to draw a "4-dimensional shape"?
At first, this sounds like a pretty mind-boggling question. After all, we're completely used to living in a 3D world! Sure, people sometimes treat time like a '4th dimension', and in some ways it does act like one - but that feels like a bit of a cop-out, because we're stuck going forwards in time: we can't speed time up, slow time down, or travel backwards in time (as much as we might sometimes like to ). We have control of our own movement through 3D space, but we have no control over our movement through time, so thinking of "time as the 4th dimension" doesn't really give us a full idea of what "4D space" would be like.
Fortunately, there is another approach to this. We can return to the 2D and 3D shapes that we're already familiar with from our school days. We could imagine a hypothetical 2D universe on a piece of paper, which is inhabited by flat, 2D people. They can move left and right, and they can move forwards and backwards, but they have no 'up' and 'down': everything is just flat to them. Sometimes, these people might ask themselves, "What would a 3D shape look like?" - and, they'd run into the same problems that we do when we try to visualise a 4D shape!
So, if these flat people are trying to visualise a 3D shape, from their own 2D perspective, how might they do it? Well, one way in which they might do it is by visualising a 3D shape as a stack of familiar 2D shapes. For example, by visualising a stack of squares, they'll get some idea of what a 3D cube might look like. Likewise, in our 3D universe, we might be able to visualise a 4D hypercube by using a similar technique. (For example, I like to think of a series of different-sized cubes, nested inside one inside another, like Russian dolls. In this model, 'outside-inside' is effectively playing the role of the '4th dimension'.)
Here are two videos by YouTube user The Lazy Engineer, which explain the concept further. The first video is about our hypothetical 2D people, living in a 2D world known as 'Flatland'. It explores how they might visualise a 3D shape (and, therefore, how we might visualise a 4D shape):
The second video introduces some specific 4D shapes, and demonstrates how we might visualise them in 3D. (Of course, being a video on a screen, it can only give us a 2D representation of a 3D visualisation of a 4D shape, so it's far from perfect, but it does its best ) . This video does contain a fair amount of mathematics, but it also has some nice visuals:
To be honest, my head is still spinning after all of that . For something like this, I think we really do need a real-world 3D model, rather than just a 2D representation on a screen!
At first, this sounds like a pretty mind-boggling question. After all, we're completely used to living in a 3D world! Sure, people sometimes treat time like a '4th dimension', and in some ways it does act like one - but that feels like a bit of a cop-out, because we're stuck going forwards in time: we can't speed time up, slow time down, or travel backwards in time (as much as we might sometimes like to ). We have control of our own movement through 3D space, but we have no control over our movement through time, so thinking of "time as the 4th dimension" doesn't really give us a full idea of what "4D space" would be like.
Fortunately, there is another approach to this. We can return to the 2D and 3D shapes that we're already familiar with from our school days. We could imagine a hypothetical 2D universe on a piece of paper, which is inhabited by flat, 2D people. They can move left and right, and they can move forwards and backwards, but they have no 'up' and 'down': everything is just flat to them. Sometimes, these people might ask themselves, "What would a 3D shape look like?" - and, they'd run into the same problems that we do when we try to visualise a 4D shape!
So, if these flat people are trying to visualise a 3D shape, from their own 2D perspective, how might they do it? Well, one way in which they might do it is by visualising a 3D shape as a stack of familiar 2D shapes. For example, by visualising a stack of squares, they'll get some idea of what a 3D cube might look like. Likewise, in our 3D universe, we might be able to visualise a 4D hypercube by using a similar technique. (For example, I like to think of a series of different-sized cubes, nested inside one inside another, like Russian dolls. In this model, 'outside-inside' is effectively playing the role of the '4th dimension'.)
Here are two videos by YouTube user The Lazy Engineer, which explain the concept further. The first video is about our hypothetical 2D people, living in a 2D world known as 'Flatland'. It explores how they might visualise a 3D shape (and, therefore, how we might visualise a 4D shape):
The second video introduces some specific 4D shapes, and demonstrates how we might visualise them in 3D. (Of course, being a video on a screen, it can only give us a 2D representation of a 3D visualisation of a 4D shape, so it's far from perfect, but it does its best ) . This video does contain a fair amount of mathematics, but it also has some nice visuals:
To be honest, my head is still spinning after all of that . For something like this, I think we really do need a real-world 3D model, rather than just a 2D representation on a screen!
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Moonface (in 'Woman runs 49 red lights in ex's car')' Wrote: If only she had ran another 20 lights.
(Thanks to Nilla for the avatar, and Detective Osprey for the sig!)
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