1. VisualizingFour Dimensions -How do we envisage a 4-dimensional space? Not easy is it?

Letus take a simple approach and identify how the fourth dimension wouldconnect or interface with the three dimensions we are so familiarwith. The first principle that most people would identify is that itmust be at right angles (normal) to our three existing dimensions,which are usually viewed as the axes of a Cartesian system ofcoordinates; not easy is it to imagine an axis normal to the existingthree, certainly not without involving 4 dimensional geometry - or isit?

Fig.15 Four Dimensional axes

Theonly way that I can envisage it is by a sphere centred on the originof three equally scaled axes that cuts each axis at the equivalentpoints, for then the surface of the sphere will be to all intents andpurposes, in the limiting case, a flat surface normal to each axis.

Butwhere then is the fourth axis? Which direction does it lie in?

Wellit doesn't; because it cannot lie in any mapped orientation withinour three existing axes.

Solet us say that it has no direction but lies in all directions, andthat therefore, it may be represented by any line drawn from theorigin to the surface of our sphere. And that if we say itscoordinate scale is ct, light-seconds for example, then we have madetime our fourth dimension, which fits well, as time can/cannot haveany orientation with respect to the three spatial dimensions.

Howthough can we mark the passage of time, our movement along thisfourth dimension, or even denote a specific point on that coordinatein relation to our other three coordinates?

Inthe first place, if we draw only 1 or two spatial dimensions then wecan, as is the current practice, draw time as one of the dimensionsrepresented in that diagram. For then, any particular time isrepresented by a line or a plane that denotes that time at everypoint on the other dimension(s).

Thedifficulty comes with trying to envisage how to draw it as the fourthdimension, for then we have to be able to represent a point in timeacross the whole of a 3D space. We cannot merely add more lines asthey would merely represent vectors in the 3D space. No it has to besomething that defines all the 3D space at a particular time. I wouldsuggest that colour could fit the bill.

Letus say that as time passes it is represented by a changing colour ofour three dimensional drawing of space, so a particular time and theassociated spatial 3D diagram would be given a specific colour.

Thenwe would have the time axis that could be drawn anywhere on thediagram as a line from the origin to a particular point in aparticular colour and we would have:

c²t²=x²+y²+z² or
c²t²-x²-y²-z²=0

2.

3. How do we envisage a 4-dimensional space?
You shouldn't even try, because ultimately it is impossible. At best such an attempt will give you only half the story, at worst it leads to gross misunderstandings.
In this case it is best to let the maths speak

4. What you have done is good spirit.but i have to stay with markus...how do u envisage the fourth dimension with d others? The answer is that the math will tell you...time is abstract,we just cannot envisage it.

5. Originally Posted by space at the centre
How do we envisage a 4-dimensional space? Not easy is it?
What is poetry ? "An extension of vision - and music is an extension of hearing." -Khalil Gibran

6. The closest I can come to envisioning 4 dimensions is to imagine 3 dimensions evolution over time. You can in a way picture four dimensional objects by picturing the 3 dimensions at some end point of the object in the fourth dimension and then watching it change in your head as you sort of run through the fourth direction as if it was time. I don't know how helpful or accurate of a picture this gives but its fun to think about.

7. Poets will tell you that science dimishes the beauty of stars by reducing them to super heated balls of gas...poets extend their vision but what they get is far from what it is--merumario

8. Originally Posted by TheObserver
The closest I can come to envisioning 4 dimensions is to imagine 3 dimensions evolution over time. You can in a way picture four dimensional objects by picturing the 3 dimensions at some end point of the object in the fourth dimension and then watching it change in your head as you sort of run through the fourth direction as if it was time. I don't know how helpful or accurate of a picture this gives but its fun to think about.
i believe you saw the thread..as explained there time has no just one point in this view...it could be seen or viewed from any point drawn from the centre point to the edge...but you cannot draw this many lines as time in the gometry of this four dimension...now how do you see the fourth dimension? Obviously not your view of a direction playing into the gometric dimension.

9. Originally Posted by merumario
Poets will tell you that science dimishes the beauty of stars by reducing them to super heated balls of gas...poets extend their vision but what they get is far from what it is--merumario
Maybe science is an extension of poetry...

10. What I am envisaging is that the axis of the sphere is ANY radius whose direction is only apparent in that fourth dimension, and the only expression of that in our three dimensions is as a magnitude. A sphere whose radius expanding represents time passing, where everything within that sphere has the same time value. where the whole of the volume of the globe is a point in time . . .

11. Does 4 dimensional space actually exist or is it created during quantum?
A stationary (static) 3 dimensional point in space would be 3 dimensional, but time is created during the dynamic changes in spacetime coordinates.

Just a musing.

12. I was thinking of a way to "see" objects in 4 dimensions. The idea is to use an intensity of grey to represent the dimension that cannot be represented geometricaly.

Let us consider a curve such as y = cos(x) in a space in 2 dimensions. It is possible to project this curve on the x axis. Will, it is not very exciting because you will get only a straight line. But if you modulate the drawing of this line varying the shade of grey between white for y = 1 and black for y = -1, you will get a representation in one dimension of the cos function. With a little imagination, you will "see" the missing dimension. I used this trick in order to represent dizains of curves of a single small screen.

You can also project a 3-dimensions object into a plane. For instance, we can project a sphere in the plane x, y ; z being a shade of grey. We will get a serie of concentratric circles with shade of grey from white in the center to black in the peripheral circle.

The proposal is to do the same, projecting a 4d object into a 3d space. For instance, let us consider the 4d object such as
and . We can project this object in the x, y, z space, using the shading of grey for s. It will appear as a sphere with a black spot suggesting some hole at the place where the x axis crosses the surface of the sphere.

It would not be very difficult to write a software to do this. This program could also apply rotations to the object in order to change the point of view.

Do you know if it has be done ? I guess I saw something similar representing a function , they used a colored representation for complex numbers.

Actually, he same idea is used with IR pictures where the temperature of an object is represented by colors varying from blue to red or yellow.

13. After all you have done,did you envisage the 4th dimension? No.

14. After all you have done,did you envisage the 4th dimension? No.
Me ? I am one-eye blind since childhood, so I see the universe totally flat, in 2d only, but, with some efforts, my brain can conceives that the universe has 3 dimensions, even if I can perceive only 2d projections.
And I can also conceive (to some extend) a 4d space that could be perceived only thru 3d projections.
Here is a little game :
Suppose you visit some new office building of twenty floors or so. The building in not used yet, not even furnished. Let us start the visit with one of the floors, let say the10th. We see this floor as a slice of 3d space.
Now, let us take the lift and go to another floor. At first glance, you may think you are in the same place because this floor is a replica of the first one. You can imagine that you didn't change of position and that you see the same slice of 3d space than in the first floor. Now, if you look better, you will see that this floor in not exactly identical to the first one : a large meeting room that exists in the first floor has been turned here into two distinct offices by adding a partition and a door.
So you see the same 3d slice of space but it is different. You may imagine that the lift had made you travel in a 4th dimension rather than in altitude axis. Of course, it is not true, just a game of imagination.
Remark that this is similar to consider the time as a 4th dimension. The main difference is that the lift can travel up and down, according to your will.
One may object that the example is not convincing because this move in the 4th dimension is discontinuous, changing from one floor to another one. If the changes between to consecutive floors are not too important, it is easy to imagine a continuous transformation from one state to the other. Some softwares are very good at this (morphing) .
Congratulations, you just succeed an imaginary travel in a 4d space.

15. Originally Posted by caKus
I was thinking of a way to "see" objects in 4 dimensions. The idea is to use an intensity of grey to represent the dimension that cannot be represented geometricaly.
As I envisage colour for the 4th dimension?

16. Originally Posted by space at the centre
Originally Posted by caKus
I was thinking of a way to "see" objects in 4 dimensions. The idea is to use an intensity of grey to represent the dimension that cannot be represented geometricaly.
As I envisage colour for the 4th dimension?
Not exactly. Your proposal, if I understand it correctly, is to draw the same object with different colors in order to represent this object at different times.

In the representation I suggested, the color of a point represents the value of the variable in the dimension that can not be spatially represented.
For instance, if I want to draw, in a plane, the function , I will do it the classical way with an axis horizontal for t and a vertical axis for x.
Now, if I want to draw (always in a plane) a function , I will draw a planar curve ; the color of each point t, x of this curve will be a function .
This was an example with 3 dimensions (t, x, y), easier to explain, but it can be extended trivially to 4 dimensions.
As I wrote this answer, I realize that this representation can be done only if the variable that is represented as a color is a function of the other variables. This is a strong limitation, but I think can be overstepped.

17. [QUOTE=space at the centre;371495]VisualizingFour Dimensions -How do we envisage a 4-dimensional space? Not easy is it?

thinking: Consider Escher as a catalyst

18. are you overstepping it with your one eye or would you have others help as well?

19. Originally Posted by Markus Hanke
How do we envisage a 4-dimensional space?
You shouldn't even try, because ultimately is impossible. At best such an attempt will give you only half the story, at worst it leads to gross misunderstandings.
In this case it is best to let the maths speak
Traveling in unchartered territory requires one to travel without a map. In exploration often math dosnt serve discovery but is needed to verify it Math isnt required to apply scientfic observation to our surroundings or understanding of concepts. But it is something you can rely upon.

20. so the story is inevitable,4th dimesion exist yeah,but cannot be viewed.....nice story,great stories never end well.

21. Originally Posted by merumario
so the story is inevitable,4th dimesion exist yeah,but cannot be viewed.....nice story,great stories never end well.
Except that all the evidence seems to point toward 4 dimensions. The problem is in our lack of imagination, but might well be a real physical limitation.
Perhaps it is impossible to view all dimensions, due to the very structure of these dimensions which forbid trespass of all 4 at the same Time.

22. thats exactly why i said its inevitable,you get it with math and theories but cannot observe it...thesame old story!