h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Hi

I'm not a scientist .. or even a mathematician ... but I've been looking at the large number of representations available of the tesseract ... a 4 dimensional hyper-cube (there is even a playable version of a 4-dimensinal Rubik's cube) but I'm now beginning to think that there exist large holes in the basic premise for the concept.

In trying to build one for myself (using Flash) I've happened across some features which put doubts into my mind as to whether the entire basis of the construction of the tesseract is correct.

If you have any views please feel free to air them .... most posts on this subject seem to be 3 or 4 years old ....

If you are a mathematician or physicist and you are interested in modelling some of the consequences of 4 and more dimensions (on something simple like a cube) then please join the convesation.

Hi TechNoiZ and welome to h2g2

It's not my specialist subject, but if you don't get a good response here you may find a couple of experts if you post your query on the Maths forum (A895205)

Have you read the Edited Guide Entry on tesseracts at A510986? I don't think any of its authors are still around, unfortunately.

Icy

It does all work out in the end! But not always how you'd expect.

Do you know how to project a 3D cube into 2D?

If so, projecting a 4D cube into 2D is simply an extension of that

It basically comes down to two things: distance, which extends in the obvious way into four dimensions, so sqrt(x^2 + y^2 + z^2) becomes sqrt(w ^ 2 + x^2 + y^2 + z^2), and the fact that if something is twice as far away it looks half the size.

So work out how far away everything is in four dimensions, and divide by every distance to work out the 2D position.

Along those lines, anyway

Thanks for the replies ..


Icy North - I read that entry, but it is not very exact when it comes to definitions - there was a sentence which merely said something about this all being possible in the fourth dimension .... which is not really an answer at all ... I went to the Maths forum but was unsure which box to fill in .... I will go back and pick one ...


26199..

Thanks ... I understand the theory - but whenever I see an explanation of how the tesseract works, it makes no mention of the coordinates of the second cube within the fourth dimension. In fact, I saw one explanation that said the two cubes should exist in the same dimensional plane .... Mmmmmmmm!

That's OK. I've just posted on the Maths Forum on your behalf. Anyone who sees it has been directed to here.

Icy

Ah, well I can answer that one!

For a square you have points:

(0, 0) (0, 1)
(1, 0) (1, 1)

For a cube it's:

(0, 0, 0) (0, 0, 1)
(0, 1, 0) (0, 1, 1) <-- first square

(1, 0, 0) (1, 0, 1)
(1, 0, 1) (1, 1, 1) <-- second square

And a hypercube is:

(0, 0, 0, 0) (0, 0, 0, 1)
(0, 0, 1, 0) (0, 0, 1, 1)
(0, 1, 0, 0) (0, 1, 0, 1)
(0, 1, 0, 1) (0, 1, 1, 1) <-- first cube

(1, 0, 0, 0) (1, 0, 0, 1)
(1, 0, 1, 0) (1, 0, 1, 1)
(1, 1, 0, 0) (1, 1, 0, 1)
(1, 1, 0, 1) (1, 1, 1, 1) <-- second cube

Notice that the second cube has the same (x, y, z) as the first cube. In 3D space it's in exactly the same place! It's the fourth coordinate, w, that differs. When you project to 2D the difference in w will usually give you a different x and y position ... if it doesn't, that means you're looking at the 4D cube end-on and will see it as a square or a 3D cube!

When you come to draw lines between them you need to know which points to connect. Fortunately there's a simple rule: two points are connected if and only if they differ in exactly one coordinate value.

So, for example, (0, 0, 0, 0) and (1, 0, 0, 0) are connected because they differ in just the first coordinate. (1, 1, 0, 0) and (1, 1, 1, 1) aren't connected because they differ in two values. This rule works for any number of dimensions!

Hope that helps (somewhat)

Thank you .. Thank you .. (again) ..

This I understand also ..... But part of me thinks that this is really just a trick of the eye moving two connected cubes around eachother and assuming the lines can be seen as bounding areas to internal volumes. Whether they overlap is presumably not seen important - and therfore of no consequence for us in our dimension.

It has no real meaning I guess unless further dimensions are assumed from which a vantage point can be taken to view the results ... But if this is so .... there is surely a consequence for how our 3 dimensional world may be percieved from that viewpoint, for in the fourth dimension, it has no size .....

The difficulty is in you (or me or anyone else for that matter) being a 3-D being and dealing in 4 dimensions. Very easy to draw the first two dimensions on a piece of paper, not too hard to project 3 dimensions on a piece of paper. But the 2-dimensional drawing starts to sag a bit when you add the 4th dimension.

For visualization purposes, you could find two similar cardboard boxes and attach a piece of yarn from each corner of one box to the corresponding corner of the other box. Look at one corner and imagine each line leaving that corner being 90-degrees from the other three.

We deal in 4 and more dimensions all the time. For example a marketing professional may wish to track the demographics of h2g2 users. For each h2g2 researcher who responded to the survey, the marketing professional could track age bracket, male/female, income bracket, Douglas Adams fan yes/no, and already there are 4 pieces of information about each survey respondent.

Linear algebra teaches us how to develop matrices that track as many pieces of information as necessary.


Happy Nerd

isn't time the 4th dimension

or was that just einstien

You can visualize four dimensions using time by imagining how a three-dimensional object changes over time (how it changes as you move from one end to the other of the fourth dimension). For instance, a 4D cube would be visualized as a 3D cube that pops into existence, remains for (say) one second, and then disappears.

A 4D sphere would be a little different--you'd picture a 3D sphere that grows from a single point all the way up to its full radius, then shrinks back down to nothing. (If you were to graph the radius as a function of time you'd get a semi-circle).

>>> isn't time the 4th dimension

As MuseSusan says.



Another wonderful exposition is Flatland.

Believe me, I have no trouble visualising it - I have modelled the tesseract in 3D - and there is NO difference in producing it using just 3 dimensions and disregarding any overlap of the cubes to doing the very same using 4 dimensions - unless you try to view the hypercube from the perspective of the extra dimension, in which case the 3 dimensional cube has zero size ... the 3 dimensions we know and love were, of course, defined by just one 4d coordinate - well, one for each cube.

One could I suppose define the extra dimension as 'time' - simply a line joining two points within one dimension - but I don't believe that to be wholly correct.

Interestingly .. when modelling the tesseract I did indeed start off with two cubes occupying the same (x,y,z) coordinates, but instead of placing the links between them directly, I crossed them over within the cube producing a vanishing point in the centre, which I then used as an origin for all 4 dimensions. I then, as was pointed out earlier, assigned each cube its proper coordinate within the fourth dimension - equidistant from the origin. Of course this was breaking the rules slightly because the length between the corners was longer than the dimensions of the cube. But it did create a logical framework for the cube to exist: as one cube moves away from the origin so it drags the second cube through - reversing and inverting it in the process to create the classic tesseract formation (which gets bigger the further from the origin it moves). All well and good .... except for the less than sneaking suspicion that in creating a central origin I had in fact made use of 2 extra dimensions instead of one ... LoL .. both being at 45 degrees to our three ....

But, thinking about it, it makes more sense that way .... If the fourth dimension is not alone but forms part of a (higher) two dimensional plane, the one dimesional line of our cube could at least move in a circle back to the origin ... and could also have size or thickness. Kind of fits in with black holes and expanding universe type philosophy.

Flatland indeed .... unless there are more .... Mind you, I don't actually think they are somewhere we can go, although their influence can shape and bend our three, but they may be something we can take advantage of if their actual effect on us can be modelled and predicted.

My real interest in the hypercube is probably that the 4-dimensional theory doesn't really explain it, not enough for me anyway, especially in relation to what we see around us in the universe .... whereas a more-dimensional answer might.

Time being the 4th dimension is still a statement which I don't quite understand. I mean, I know that you can just 'add a coordinate' and get a 4th dimension, but then why not use, say, temperature as the 4th? That might be because time ans space are not independent as space and temperature, 'cos of quantum. That's where I don't really understand anymore.



For some reason I've always found the Klein bottle useful for visualising something in 4 dimensions - you make a self-intersecting representation of the surface and then, hop, I can almost visualise it as something non-intersecting.



>>>For each h2g2 researcher who responded to the survey, the marketing professional could track age bracket, male/female, income bracket, Douglas Adams fan yes/no, and already there are 4 pieces of information about each survey respondent. <<<

But you have 2 piece of 0-dimensional information (male/female, DNA fan yes/no), so you actually get a 2-dimensional information space here (with 4 connected components).

Ha ... GOT IT

I needed a way to visualise what I was seeing ... and I DO .. lol ..... 'tis just a sliver of the tesseract we would see - as one would see the toppest end of a piece of rock held up against a window ... each slice - no matter ...how thin - is a cube, rotating in its own 3 dimensions ....

If you arrange five touching squares as a cross and add one more square at the bottom, then you get what might be called the "net" of a cube, and you could fold the squares up to make a cube.

One way I've found to try to imagine a tesseract (well really just the "surface" of the tesseract) is to take a cube and place a touching cube at each face, with one more cube (that's eight in all) at the base of the assemblage. Then you can, just about, imagine folding this set of cubes into a tesseract in four dimensions, and you can certainly label all the corners to see where they are supposed to touch.

This may or may not help.

>>> Time being the 4th dimension is still a statement which I don't quite understand.

>>> But you have 2 piece of 0-dimensional information (male/female, DNA fan yes/no), so you actually get a 2-dimensional information space here (with 4 connected components).

Good points, I could have written a much better transition there. Okay, I'll try again.

Let's imagine that we are just talking about information, not necessarily attached to a position on any axis. Thus the vector (0,0,1) is not necessarily a dot on a graph or a point in 3-space. But it is a piece of information, actually 3 pieces. And the vector (212, 2pi/3, e, -5) is also information, but now there are 4 pieces of information.

That 4th piece of information could as easily be temperature as time. Either can be valid. Calling the 4th piece time can be valuable when the other three pieces of information take on different values at different times. Calling the 4th piece temperature makes sense when the other three piece of information change at different temperatures. The 4th piece of information could as easily be anything else, or it could for the moment remain an abstraction.

As an abstraction, a single piece of information can be generally thought of as belonging to the set of Real numbers, R, but for some applications R may not be as useful. For example, time is often thought of having an origin and taking on positive values. Likewise meaningful values of temperature and pressure come from subsets of R. For some applications R may be too restrictive, and we may need the set of complex numbers.

For other applications meaningful values may only come from a rather small set, such as Y/N. Or maybe red-green-blue in a physics application. Or the marketing one.

"Time being the 4th dimension is still a statement which I don't quite understand. I mean, I know that you can just 'add a coordinate' and get a 4th dimension, but then why not use, say, temperature as the 4th? That might be because time ans space are not independent as space and temperature, 'cos of quantum. That's where I don't really understand anymore."

We imagine the 4-cube (or tesseract) as being in four spatial dimensions. This is an entirely conceptual piece of pure mathematics.

According to Einstein's Theory of General Relativity, the universe is made up of three spatial dimensions and one temporal one, which are curved and warped by gravity. This is physics.

You mentioned using temperature as the fourth dimension. That might be a little tricky to draw, but I have seen pictures of 4-d objects that use colour to represent the fourth.

what i meant there was if there are 3 spacial dimension
and 1 temporal one any real time cube is a 4 dimension object and this 4d hyper cube is realy a 5 dimension object

this is all above my head i only did o level phisics 25 years ago

>>what i meant there was if there are 3 spacial dimension
and 1 temporal one any real time cube is a 4 dimension object ...

It is a 4D object if you can hold Cube(t=yesterday) in one hand while holding Cube(t=today) in the other.

Yeah, I'd just like to reiterate my point that pure mathematics is different from physics.