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Typically if a body of say, four dimensions is to be shown in 3D space, it's shown as a "slice" of the full body and one of the dimensions is bypassed entirely.

But has anyone ever tried to instead represent a 4D body with the each dimensions corresponding to different axis?

How could that be done? There are just 3 axii in 3D, right?

Well, think about how the six-sided cube can be said to defined 3D space. Each dimension is represented by two opposing faces.

What if we use the four pairs of opposing sides of the octahedron to define 4 different dimensions, that are all equal regarding rotation and such?

Of course, the locations would be ambigous, since points that have different vectors would have the same actual location in the 3D space, but it's at least one way to show all 4 dimensions without "flattening" any of them, right?

What if this is used to show the 4D "julibrot" or other 4D fractal bodies? Has anyone tried or proposed this before?

Also the dodecahedron could be used as a base for showing 6 dimensions, and so on....

I like the idea. I don't think it would accurately represent four dimensions because points could overlap, but I bet it would make some very cool looking images. Maybe this could be a method of finding the next "Holy Grail".

I think that is the idea behind this thread...
http://www.fractalforums.com/3d-fractal-generation/platonic-dimensions/

A simple approach is to bump up the dimensions like this:
vec4 z=length(p)*normalize(abs(vec4(p.x+p.y+p.z,-p.x-p.y+p.z,-p.x+p.y-p.z,p.x-p.y-p.z)));

Then proceed with a 4d formula.

Main problem with 3D mandelbrot is it being revolution surface, and that is becouse its Y and Z dimensions are equal.

Here is cannonical representation of four dimensional cube, tesseract:
(http://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Tesseract.gif/240px-Tesseract.gif)

It sounds that you suggest something like truncated octahedron coordinate sistem, with one plane having 3 axis, something like davis star.
(http://upload.wikimedia.org/wikipedia/commons/thumb/2/20/Truncatedoctahedron.jpg/240px-Truncatedoctahedron.jpg)
There must be angle at wich each 4 axis would stand equaly to each other.

I don´t think it´s possible to represent 4 dimensions in a 3D space, this has no sense at all since 3D space means 3 dimensions.
A four dimensional cube it´s simply a 3D cube extended in time.
But the main problem it´s that we don´t know what we are talking about when we say "time".
 

That cube Asdam posted is being rotated, that's why it is moving. The fourth dimension it uses is not time. I think matsoijare knows that is impossible, what he is suggesting is a way to overlap information in 3D space.

A monitor is only 2D, but it is certainly possible to display 3D images - typically through perspective projection. And yes, points will overlap (two 3D coordinates may project to the same 2D point, but we always only show the nearest one), but we are still able to get a good feeling of 3D objects.

It is also possible to do (euclidean) 4D to 2D projections, see for instance here: http://eusebeia.dyndns.org/4d/vis/01-intro

You still need to find a proper algebra for the 4D space, though. Quaternions seems to be somewhat boring, but perhaps they would be more interesting if drawn using some kind of projection, instead of slicing.

Besides Aexion's transformations, there is also Knighty's work on stereographic projection of (4D) polychora: http://www.fractalforums.com/general-discussion-b77/solids-many-many-solids/15/

A tetradimensional object, that is, an object extended in four dimensions, will show to us as an event, something developing in time, with a start and an end, a storm, for example, or  a war, or a life.
 Of course, these tetradimensional things can be represented in our three dimensional world, but only trough time, and even then this is only a re-presentation, the true reality has more dimensions.
 

So does that make a 3D Animation a 4 Dimensional object?

I think there is some confusion here- time is a dimension, but it could easily be the 5th Dimension if discussing 4 Dimensional Geometry. 

There are different ways to model a mathematical four dimensional space. For instance, Einstein's special relativity uses four dimensions, three spatial and one temporal. The dimensions here are not treated equally: the temporal dimension has a different sign in the metric, meaning you can get negative and zero distances between points, which here corresponds to events. This is called a Minkowski space.

But it is also possible to work with objects in the simpler 4D (and higher) Euclidean spaces, where each dimension is spatial-like and equal to all others. The Tesseract image in Asdam's post is an example of this.

Matemathical dimensions aren´t the same thing as Geometric dimensions, this is the first confussion.
But there is more, in a given moment and place, there can be several objects or beings with different timespan, that is different extension in time, moreover, as each object or being has at least four dimensions, each can show a different dimension, that is, we can see the third dimension of an object, and the fourth of another at the same time, this leads to another confusion.
Finally, while we move trough time due to cosmic motion, and without any other reference, we think that time flows.This is the greatest confusion.
     Imagine a wall in the background of a scene, a car parked in front, and a dead tree in the foreground, the wall, even if it´s part of a building, only shows two dimensions to you, look around you, most of time you can see only planes, two dimensional objects, even if they have more dimensions
     Now consider the parked car, it shows three dimensions, you see a full three-dimensional object, even if it has a fourth dimension or more.
     And finally, think about the dead tree, what you are seeing its a four dimensional path created by something´s motion trough time, something like a freezed motion made by three dimensional moments.
    We must agree wit the fact that this unknow entity should have more than four dimensions.

Probably 4D quaternionic julia set represented in 3D would be interesting, but mandelbrot set not. Becouse we will get Y(dimension)=Z(dimension)=W(dimension). But it doesn't mean, 4D fractals would not be interesrting, julia sets are nice too and there are lots of about mandelbrot formulas with interesting mandelbrot sets.

In theory, as 3D can be represented in 2 dimensions, say drawing of cube on plain paper, so 4 dimensions could be represented in 3 dimensions. And we have working 3D software engines.

There is string theory saying that there possibly be more than 3 spatial (space) dimensions. Throught none realy knows, and there are no proof od string theory. (All effects of theory of relativity is well known. But there is no known effect of string theory.)

Yes - the pure 4D Quaternion Mandelbrot is not very interesting, but if you add a rotation or reflection to the inner loop, they become more interesting, e.g.: http://www.flickr.com/photos/syntopia/5425626649/

This is still only 3D cross-sections of a 4D object. Perspective projection would be funny to try out.

Pretty nice object. Stalks looks like 3D julia sets, a feature missing in quaternion m-set. But it don't looks as I imagined 4D represantion in 3D, tesseract there looks like something inside of cube;)

The ancients viewed the 4th Dimension as depth, macrocosm/microcosm, "as above, so below".  In other words, fractals.

You probably know what i am going to say, but here goes  :embarrass:
The term dimensions in space , geometric dimensions etc are all buzzwords, created by our scientific past, and our science fiction past. Ihave no objection to that as long as one does not confuse that with what we can actually perceive.
The real analysis of space belongs to Grassmann, and he says quite tellingly that we are not restricted to 3 dimensions. We live in an n-dimensional space, and we can describe an n-dimensional space if we want to. Grassmann wanted to and he did.

The solution you seek is that of Lagrange, and it is usually called generalised coordinates. It is of course easier to represent generalised coordinated in our liin space than on a flat surface, a projection from our living space.

The book Adventures in Flatland has a lot to answer for in perpetuating this mythology! The orthogonal set of axes is in fact a set of 6 axes, and five rotational dimensions, yes dimensions, and one of the axes has to be defined as the generator of fundamental dimension of the set, in order to establish a fundamental orientation. In addition we have to establish a common formalism to avoid confusion even in this familiar framework. This reference frame is Cartesian because it developed along the lines of Descartes proposal. Nowadays most mathematical physicists work free of the Cartesian system exactly because it is such a Special system. It has 11 or 12 dimensions depending on how you want to count. 3d is and always has been a synonym for our living space, not a description of its dimensional structure.

There is another use of the word dimension in dimensional analysis. These dimensions are standard quantities of measure and relate back to Grassmann through Riemann and Gauss. The SI syatem was set up to establish a common system of measures, and tese were recognised as dimensions precisely because the dimension, that is cut or quantize space.

Quantizing space has to be understood too as dealing with our experience of it in terms of the notion of extension. Extension was the Cartesian notion that leavin god out of the picture, our reality is basically objects and things extending themselves everywhere! Of course this is a simple explanation of what he meant, but nevertheless it generated considerable philosophical debate, only really answered by Newton's Principles of Mathematics, in the sense thar everyone eventually went along with Newton. Leibniz had developed a theory of Monads, but it never really caught on. Extension had to be quantified, and consequently dimensions were created from extension, philosophically.
Of course, any artisan knows this innately, it is just certain thinkers get an idea that goes against our common experience and then we spend literally centuries straightening it out!