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chevron_right04-05-2003
Hypercubes and hyperspheres
Cubes, along with perfect spheres, have the property that they are among the simplest geometric shapes to draw or imagine in n dimensions. For example, one can "convert" a sphere to lower dimensions relatively easily by subtracting one dimension from the surface of the sphere. For example, there exists the idea of a four-dimensional hypersphere whose three-dimensional surface is our universe. If one subtracts one dimension each from this set, one receives a three-dimensional sphere with a two-dimensional surface - this surface can be still considered to be our universe, one must keep in mind though that one dimension is missing now. Regularities are valid in these cases, however, also for the missing dimension, if it is clear that they are valid for the remaining two as well.
So much for the excellent property of spheres. Arbitrary-dimensional cubes have the advantage that one can show by a relatively simple procedure what an (n+1)-dimensional cube looks like if one has a cube with n dimensions available. This has the disadvantage that such an n-dimensional cube becomes totally confusing with far more than 3 dimensions, since we must convert/shadow any additional dimension into our own amount of dimensions. But even then you can still show what a cube with an additional dimension would look like: namely exactly twice as confusing. :o)
In order to illustrate how to add another dimension to an n-dimensional cube object, let's start with zero dimensions. Any object with zero dimensions is just a dot.
In order to get from zero to one dimension, we just add another dot and connect both dots with a line. Using this line you can get to two dimensions now, simply by duplicating them and by connecting the unconnected dots with two lines.
Now we have a square, more or less :-). You can now repeat the same procedure to get to three dimensions: you double what you already have, and you connect the same dots of each part. In other words: duplicate the squares - then connect the equal dots, like for example: connect the top left dot of the first square with the top left dot of the second square. And so on.
Although we are limited to the spatial view, which has only three dimensions, we can now make a hypercube out of the cube. A hypercube is a cube with four dimensions. We simply copy again what we already have to some arbitrary other place and then we connect the same dots with each other.
I have now colored the connecting lines differently because of the amount of dimensions, which is greater than 3 from now on. Now we have a hypercube - the original shape consists of two cubes, which are simply connected at every point. This creates more cubes, just as 2D-->3D creates more faces when you connect the still unconnected dots of two squares. Now there is a picture for hypercubes which is just as admissible for us as my totally high quality illustration with the light blue connecting lines - this one looks like a small cube in a big one.
This is a somewhat more professionally made version of a hypercube - you can see that inside a large cube is a slightly smaller one, from our limited three-dimensional view at least. This hypercube now has a surface consisting of exactly 8 cubes, each with three dimensions. For this purpose, here is a small GIF animation, to which it should be added that the large cube and the small cube represent two completely separate cubes of this surface - the large cube does indeed completely encompass the entire hypercube in one of the images of this GIF animation, but it is completely independent and exists parallel to the small cube and the 6 truncated pyramids. Since it is a property of the hypercube that each side of each cube is connected to a side of some other cube, this outer cube is necessary, so to speak, because otherwise the outer sides of the truncated pyramids would have no connection to any other cube on the outside - as if an ordinary cube were missing a face somewhere.
So much for the hypercube. Now you can extend it - but from now on it becomes much more confusing. One can still simply double the existing and connect all the same dots with each other. In this case, the result would be something like this:
In the right picture we see a five-dimensional cube, drawn in three dimensions and "pressed down" to two dimensions, because your monitor seems to show a rather two-dimensional picture :). The two four-dimensional cubes (cubes connected with light blue lines) are connected by orange lines here. Of course you can extend this jumble even further...
To simplify things a bit, let's now color this image black and white and move the right 5D hypercube down a bit.
Afterwards, we can connect both of these five-dimensional hypercubes relatively neatly with colored lines - though I take a different color to connect the respective 8 dots of each 3D cube within the 5D cube, so you can still distinguish it a bit....
So this is how we can imagine a 6D hypercube - actually reason enough to drop it right away, isn't it? :-)
One would hardly believe on the basis of this picture that the six-dimensional hypercube still has a right angle at each corner - however, this is only deceptive because we must represent 6 dimensions somehow on 2 dimensions, which falsifies some things a bit. At EACH corner there is a 90° angle, if one could view this cube in 6 dimensions.
Let's get back to 4 dimensions for now, because imagining them is difficult enough. Apart from this, there is still a little bit to say to these, which could surely interest a few hobby mathematicians who might not know this yet.
Here is a view of a hypercube rotating in the fourth dimension.
Staring at the animation for ages doesn't help that much in this case, though. I don't remember where I got this animation from, but the creator took the liberty of choosing a viewing angle from which it is relatively difficult to understand how the cube rotates. Nevertheless, it shows relatively clearly that this cube contradicts our three-dimensional understanding.
Now, as we have seen above, our common conception of a hypercube is "a small cube within a large cube" - but why? Here is an illustration of what a three-dimensional cube would look like if you saw its shadow in two dimensions.
A small square in a larger square - our idea of a hypercube looks quite similar, so it is only the shadow of a real hypercube: a small cube inside a larger one. However, this is by no means bad or even useless - just as a two-dimensional living being could imagine how it functions logically and physically on the basis of the shadow of a 3D cube, we can imagine how it functions physically/logically in four dimensions on the basis of the three-dimensional shadow of a hypercube. The inner cube represents, so to speak, the part of the hypercube that is closer to the "screen" - just as the inner square in the picture represents the end of the cube that is closest to the screen. Of course, we still can't get a picture from this shadow of what a real hypercube looks like now - but with enough training, you can learn to mentally imagine any hypercube rotation.
A hypercube can be unfolded just like a normal cube. This can be done by the cube itself as well as by its shadow. Since we cannot imagine four-dimensional hypercubes directly, we have to resort to their shadows. Therefore, below is a video that shows the unfolding of a hypercube.
First, the large cube is divided and reassembled below so that the entire upper part of the hypercube is exposed. Next, the six truncated pyramids can be morphed into normal cubes, while the middle (initially smaller) cube also expands into a normal-sized cube. This way you get a so-called tesseract, which is the unfolded hypercube.
So much for the excellent property of spheres. Arbitrary-dimensional cubes have the advantage that one can show by a relatively simple procedure what an (n+1)-dimensional cube looks like if one has a cube with n dimensions available. This has the disadvantage that such an n-dimensional cube becomes totally confusing with far more than 3 dimensions, since we must convert/shadow any additional dimension into our own amount of dimensions. But even then you can still show what a cube with an additional dimension would look like: namely exactly twice as confusing. :o)
In order to illustrate how to add another dimension to an n-dimensional cube object, let's start with zero dimensions. Any object with zero dimensions is just a dot.
In order to get from zero to one dimension, we just add another dot and connect both dots with a line. Using this line you can get to two dimensions now, simply by duplicating them and by connecting the unconnected dots with two lines.
Now we have a square, more or less :-). You can now repeat the same procedure to get to three dimensions: you double what you already have, and you connect the same dots of each part. In other words: duplicate the squares - then connect the equal dots, like for example: connect the top left dot of the first square with the top left dot of the second square. And so on.
Although we are limited to the spatial view, which has only three dimensions, we can now make a hypercube out of the cube. A hypercube is a cube with four dimensions. We simply copy again what we already have to some arbitrary other place and then we connect the same dots with each other.
I have now colored the connecting lines differently because of the amount of dimensions, which is greater than 3 from now on. Now we have a hypercube - the original shape consists of two cubes, which are simply connected at every point. This creates more cubes, just as 2D-->3D creates more faces when you connect the still unconnected dots of two squares. Now there is a picture for hypercubes which is just as admissible for us as my totally high quality illustration with the light blue connecting lines - this one looks like a small cube in a big one.
This is a somewhat more professionally made version of a hypercube - you can see that inside a large cube is a slightly smaller one, from our limited three-dimensional view at least. This hypercube now has a surface consisting of exactly 8 cubes, each with three dimensions. For this purpose, here is a small GIF animation, to which it should be added that the large cube and the small cube represent two completely separate cubes of this surface - the large cube does indeed completely encompass the entire hypercube in one of the images of this GIF animation, but it is completely independent and exists parallel to the small cube and the 6 truncated pyramids. Since it is a property of the hypercube that each side of each cube is connected to a side of some other cube, this outer cube is necessary, so to speak, because otherwise the outer sides of the truncated pyramids would have no connection to any other cube on the outside - as if an ordinary cube were missing a face somewhere.
So much for the hypercube. Now you can extend it - but from now on it becomes much more confusing. One can still simply double the existing and connect all the same dots with each other. In this case, the result would be something like this:
In the right picture we see a five-dimensional cube, drawn in three dimensions and "pressed down" to two dimensions, because your monitor seems to show a rather two-dimensional picture :). The two four-dimensional cubes (cubes connected with light blue lines) are connected by orange lines here. Of course you can extend this jumble even further...
To simplify things a bit, let's now color this image black and white and move the right 5D hypercube down a bit.
Afterwards, we can connect both of these five-dimensional hypercubes relatively neatly with colored lines - though I take a different color to connect the respective 8 dots of each 3D cube within the 5D cube, so you can still distinguish it a bit....
So this is how we can imagine a 6D hypercube - actually reason enough to drop it right away, isn't it? :-)
One would hardly believe on the basis of this picture that the six-dimensional hypercube still has a right angle at each corner - however, this is only deceptive because we must represent 6 dimensions somehow on 2 dimensions, which falsifies some things a bit. At EACH corner there is a 90° angle, if one could view this cube in 6 dimensions.
Let's get back to 4 dimensions for now, because imagining them is difficult enough. Apart from this, there is still a little bit to say to these, which could surely interest a few hobby mathematicians who might not know this yet.
Here is a view of a hypercube rotating in the fourth dimension.
Staring at the animation for ages doesn't help that much in this case, though. I don't remember where I got this animation from, but the creator took the liberty of choosing a viewing angle from which it is relatively difficult to understand how the cube rotates. Nevertheless, it shows relatively clearly that this cube contradicts our three-dimensional understanding.
Now, as we have seen above, our common conception of a hypercube is "a small cube within a large cube" - but why? Here is an illustration of what a three-dimensional cube would look like if you saw its shadow in two dimensions.
A small square in a larger square - our idea of a hypercube looks quite similar, so it is only the shadow of a real hypercube: a small cube inside a larger one. However, this is by no means bad or even useless - just as a two-dimensional living being could imagine how it functions logically and physically on the basis of the shadow of a 3D cube, we can imagine how it functions physically/logically in four dimensions on the basis of the three-dimensional shadow of a hypercube. The inner cube represents, so to speak, the part of the hypercube that is closer to the "screen" - just as the inner square in the picture represents the end of the cube that is closest to the screen. Of course, we still can't get a picture from this shadow of what a real hypercube looks like now - but with enough training, you can learn to mentally imagine any hypercube rotation.
A hypercube can be unfolded just like a normal cube. This can be done by the cube itself as well as by its shadow. Since we cannot imagine four-dimensional hypercubes directly, we have to resort to their shadows. Therefore, below is a video that shows the unfolding of a hypercube.
First, the large cube is divided and reassembled below so that the entire upper part of the hypercube is exposed. Next, the six truncated pyramids can be morphed into normal cubes, while the middle (initially smaller) cube also expands into a normal-sized cube. This way you get a so-called tesseract, which is the unfolded hypercube.