4th dimension

[ddanbe]

My math professor in the past explained it this way:
Consider yourself a 2D creature(some sort of pancake). If placed in a circle, you can't escape in a 2D world.
But in a 3D world this is simple. You could just jump up and disappear and reappear before all your 2D friends outside of the circle.
Now consider yourself a 3D creature(some sort of meatball). If placed in a sphere(football?), you can't escape in a 3D world.
In 4D, just "jump" and reappear on the outside!
Anyone any thoughts about this?
Or a better explanation?

[pritaeas]

http://en.wikipedia.org/wiki/Flatland

A funny book on the subject.

[Hiroshe]

I like that question!

The idea your trying to explain is a geometrical interpretation of a 4th dimention. There are plenty of other interpretations of it. Perhaps it would be better to first explain what dimentions are in mathemtatics before adding he symantics of space.

You already know that a vector is just an ordered list of "things" like numbers.

The ordered linearly independant set of "things" that you used are called a basis. For example, the basis for cartesian coordinates are [(1, 0), (0, 1)]. Then the linear combination of that set is what forms vectors. This "basis" can be anything from numbers, to sound, to colour, to physical space, to a combination of whatever. That's how vectors are related to the real world.

For a geometric example, a circle can be described as the set of vectors (cos(t), sin(t)) (for 0 < t < 2pi).

For a sound example, you know the mixers you see dj's use? You know, the one's that allow you to turn up the base. That's a vector! And the basis is [1, sin, cos, sin^2, cos^2, ...]. The vector is found quickly with the fast fourier transform. The strange thing here is it's an infinately sized basis (infinite dimentions!), though in practice it's limited.

A dimention is the length of the linearly independant basis. What that actually mean in english is it's the "number of properties we're interested in, where the properties can be anything."

Going back to space, if we have the vectors (x, y), we're interested in 2 properties (the coordinates). If we have the vectors (x, y, z), we're interested in 3 properties (3D space). If we have the vectors (x, y, z, t), then we might be interested in time, or even temperature for that matter (fluid dynamics for example). The fourth dimention can also be used for charge, colour, pressure, etc...

Now, trying to pin that 4th dimention on space itself is hard. Strictly speaking, it's non-sense. Since we need to make assumptions about space to generalise it, and we have no other example of another form of space.

We can still make assumptions and try to generalise it, but it doesn't need to be consistent from one person to another. It could be interpreted as you said (where the 4th coordinate is held constant for us in the 3D world, and someone existing in the 4th dimention can "jump" to increase the value of the coordinate and disappears from us in order to re-appear somewhere else). It could also be a gradual thing where space is bending (ie, we all exist in 4 dimentions, but we have no control over the 4th coordinate) alowing the possibility of the shape of space to be something else. Ie, like the 2D space on a shpere where space seems to "wrap around" even though it's 2D.

So, yes, the interpretation you mentioned is a consistant one, but it's not the only one we can imagine.

[mHealth]

4th dimension is "time" and we can not jump it.
If a one starts running his vehicle at the speed of light and one
starts chasing him after 2 hours at the speed of light, there will
always be a gap of 2 hours.

[Reverend Jim]

It is possible to portray higher dimensional objects in lower dimiensions but the result always contains distortions. Just look at maps. Most maps (in the projection which is most commonly used) show Greenland as inordinately large.

In the same way we can portray a cube in two dimensions

and a tesseract (hypercube or four-dimensional cube) in three dimensions (putty and straws work well), or even two dimensions.

It is also possible that we cannot perceive other dimensions because of scale. When standing on the prairies we perceive the world as flat, not spherical, but from space the actual shape is apparent. If you were reduced to the size of an ant and were standing on a soda straw you could berceive the surface as cyllindrical, but if you were the size of a microbe your perception would be substantially different.

[ddanbe]

4th dimension is "time" and we can not jump it.

You seam to know alot, so tell me why the dimension of Koch's curve is log(4)/log(3)?

[Hiroshe]

To add onto what was said about the projection of a hypercube, look up an animation of a hypercube projection rotating. Strangely enough, it looks like the "cube" would be bending, yet it is indeed a solid object. The cube itself is still unrenderable, but it does demonstrate the "bending of space" idea of the 4th coordinate.

[Gribouillis]

tell me why the dimension of Koch's curve is log(4)/log(3)?

Suppose your Koch's curve is contained in a disk of radius 1, then by construction, it can be covered by 4 disks of radius 1/3. By induction, the curve can be covered by 4^n
disks of radius r=(1/3)^n. We then apply the definition of the Hausdorff dimension. Take a number d and sum the radius at the power d. It gives a sum of

s = (4^n)(3^(-n*d) = exp(n(log(4)-d*log(3)))

You see that if d is slightly above log(4)/log(3), this sum tends to zero when n tends to infinity. It proves that the Hausdorff dimension of the curve is smaller or equal to log(4)/log(3). Some other argument is necessary to prove equality, but I think it should give you an idea.

[almostbob]

The time dimension

4th dimension is "time" and we can not jump it.

Yet:

have to be careful we don't travel through occupied spaces

[jwenting]

You seam to know alot, so tell me why the dimension of Koch's curve is log(4)/log(3)?

mathematically dimensions can be broken numbers. Fractal geometry explains how those are calculated.

http://en.wikipedia.org/wiki/Fractal_dimension
http://davis.wpi.edu/~matt/courses/fractals/intro.html
http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html
http://www.wahl.org/fe/HTML_version/link/FE4W/c4.htm

It's a mathematical construct, not physical dimensions.

[vegaseat]

Time could potentially repeat itself.

[vegaseat]

Left and right, forwards and backwards, up and down, and into and out of.

[GrimJack]

Time is not a dimension it is a coordinate; there is a 4th dimension and it is the space occupied by a hypercube,hypersphere, and so on. The mathematics of lower dimension space has some interesting 'tiling' properties - I especially like Kellor's Conjecture that state that tiling an N-dimensional space with N-dimensional hypercubes of equal sides yield an arangement in which at least 2 hypercubes have an entire (N-1)-dimensional "side" in common.

It has been proven true in dimensions 6 or less and false in dimensions 8, 10, and 12.

[GrimJack]

There is an interesting twist on the Flatland named "the Planiverse" - Flatland was a 2d world seen from above; Planiverse is 2d world seen from the side (ie a computer screen).

[Reverend Jim]

I think I have that book in my archives. I recall reading it years ago.

[almostbob]

I dont have that book, I'll buy it tomorrow, and read it yesterday

[Reverend Jim]

You must have really aced that speed reading course.

[vmanes]

4th dimension, time travel....they give me headaches!

[Ketsuekiame]

In terms of geometry isn't a dimension simply an axis that is perpendicular to all other axes?

Quaternion mathematics for rotation for example.

Also I don't believe in time as a dimension but as an artifact of something else. If time were a spatial dimension, why do we get relativity issues when approaching the speed of light? Seeing as even at the speed of light we're still travelling in 3 dimensional space.

[Reverend Jim]

Experiments suggest that time is an emergent property of the universe that is not observable from the outside.

[GrimJack]

vmanes: 4th dimension, time travel....they give me headaches!

I read a time travel novel years ago that started with the point of view of someone being rescued and ended with that same point of view being the rescuer - my head hurt for weeks. Then I read "All You Zombies" and nearly had a migraine.

[kplcjl]

It's been 45 years or so since I read flatland. My memory is fuzzy, so if you want to challenge me, I won't argue. Flatlanders never looked up or down because that dimension didn't exist to them. They were all circular in shape, their buildings were rectangular lines. They were visited by a 3-dimentional sphere which caused great consternation because it suddenly appeared as a point and got to be a bigger and bigger circle (person) in their world, and then it got smaller and smaller until it just disappeared.
It never explained how these circular people moved. I wondered if they had pathways, how you got on and off a path. (Doors like buildings had?)
Time makes a wonderful fourth dimension. We appear from nothing, we get bigger and bigger, then smaller and smaller until we disappear. So we are fourth dimensional creatures visiting a three dimensional world. We just aren’t true fourth dimensional creatures because we don’t control movement through all 4 dimensions, we just use one to change where we are in the other three dimensions. So flatlanders really were 3 dimensional creatures.

[kplcjl]

@Grimjack, your novel had two separate points of view, because two people can't have the same point of view. In fact everyone experiences changing points of view as they go through life. Since I didn't read it, for all I know, that is two points of view from the same person.
Can the person who time travels ever get back to where they started? (Since a person died in the past, if you go back in the past to save them, there is no reason to go back in the past because you saved them, so how could you get back to where you did go back, since you wouldn't need to go back in the first place.) They've got some great movies like that already, so that wasn't my idea. (I think.)

[ddanbe]

Time travel? What if you go back in time and killed your father or mother or both, when they were 10 years old?

[GrimJack]

ddanbe:

Time travel? What if you go back in time and killed your father or mother or both, when they were 10 years old?

Read All You Zombies Every single person in the short story is the same person - she started out as a teen who was seduced by an older man, became pregnant, her child was stolen, she had a sex change and became a writer using the pseudonym of "unwed mother". The bartender at his local hangout sent him into a time machine back into the past where he seduced himself (and so on - probably the single best time travel story ever told).

[GrimJack]

@kplcjl: no it didn't - the story started from the point of view of the stranded traveler and followed that point of view for a number of years and landed to rescue the stranded traveler - consider it Worm Ourobores eating its way through the earth until it finds another like itself and asks it to marry him but the answer was "I can't silly, I am your other end".

[kplcjl]

One of the things that has always amazed me about most stories of time traveling machines is that you always end up at the same location where you started. (Except for the more "realistic" spaceships that travel in time.)
The earth at its equator is spinning at around a 1000 miles per hour, everyone on earth is traveling around 4000 miles per hour around the sun. We're moving even faster than that in our galaxy. And galaxies are pulling away from each other at nearly the speed of light. Yet, we have people go outside the century old ruins of a castle, and poof, the castle is 100 yards away and a few years old.

[ddanbe]

you always end up at the same location where you started
In a story you can assume anything. And if time travel should happen through a wormhole created at your departure, it is even very plausible that you arrive back where you started.

[Stuugie]

My perception of time travel is that if you could actually do it, it wouldn't necessarily mean that you would/could land exactly on your own historical timeline so killing your parents wouldn't result in you fizzling into the ethos. It would be like looking at your timeline as just that, a line, jumping back and hoping to land exactly on that line. Now, if you don't land on your timeline, you'll infact create a new looped version of a different timeline.......this is all crazy talk. You all need Jesus! LoL.

[Reverend Jim]

Sort of like forking a software project, right Stuugie?