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Resu
2003-Feb-21, 01:26 PM
So used to living in the three-dimensional world, that I cannot imagine what a 4D world would look like.

I couldn't even imagine what a projection of a 4D object into the 3D space would look like.

Pictures are 3D objects projecting into the 2D space, so we ought to be able to make a 3D projection of a 4D object. Has anyone seen one?

Glom
2003-Feb-21, 01:32 PM
In truth, the univese is described as four-dimensional. Three dimensions of space and one dimension of time.

Current thinking in theoretical physics is that there are 11 dimensions, but the remaining seven are curled up so we don't have very far to go in them, hence we don't notice them.

Try reading Stephen Hawking's book The Universe in a Nutshell. It talks about 11 dimensional super gravity and brane theory and all that stuff.

_________________
"Sulphur drops keep fallin' on my head, they keep fallin'. The apocalypse ain't botherin' me." -BeelzeBob Masters Encyclopedia Satanica now in paperback

<font size=-1>[ This Message was edited by: Glom on 2003-02-21 08:33 ]</font>

traztx
2003-Feb-21, 04:01 PM
I believe our brains do this automatically in many cases.

For example: When you turn on a faucet and see a line of water from the faucet to the sink. Turn out the lights and view it again under a strobe light and it is no longer a line, but a series of droplets.

Mainframes
2003-Feb-21, 04:27 PM
On 2003-02-21 08:32, Glom wrote:
In truth, the univese is described as four-dimensional. Three dimensions of space and one dimension of time.

Current thinking in theoretical physics is that there are 11 dimensions, but the remaining seven are curled up so we don't have very far to go in them, hence we don't notice them.

Try reading Stephen Hawking's book The Universe in a Nutshell. It talks about 11 dimensional super gravity and brane theory and all that stuff.

_________________
"Sulphur drops keep fallin' on my head, they keep fallin'. The apocalypse ain't botherin' me." -BeelzeBob Masters Encyclopedia Satanica now in paperback

<font size=-1>[ This Message was edited by: Glom on 2003-02-21 08:33 ]</font>

The Elegant Universe by Brian Greene is also very good on the subject of multiple dimensions, also goes into a lot of string theory.

sacrelicious
2003-Feb-21, 04:57 PM
clearly the question is regarding a fourth physical dimension, so Resu, here is the help you need:

http://dogfeathers.com/java/hyprcube.html

now one thing that needs to be clear is that you only see in two dimensions. now I expect that you will protest that claim by saying "but I have two eyes, so if I only saw in 2D I would have no depth perception". well two eyes give you a perception of your 3D enviroment, and motion helps refine that even more, so that you can manipulate and conceptualize the enviroment without much embarrassment, but were you to truly see in 3D you would be able to see all sides of an object at the same time (conversly, a 2D person would only be able to really see in 1D, and a 1D creature wouldn't be able to see at all. well, you know what I mean...)

to those much smarter than myself: is that last statment consistant with the best theories on extra dimensions, ie. does it follow that if a 4D creature were to exist it could in fact see all side of a 3D object at the same time?

copabera
2003-Feb-21, 05:00 PM
Resu: Pictures are 3D objects projecting into the 2D space, so we ought to be able to make a 3D projection of a 4D object. Has anyone seen one?

Draw two squares on a piece of paper and then join corresponding corners with straight line segments. You have drawn a cube, a 3D object projected onto 2 dimensions. You will get the same effect if you take an actual wire-frame cube, back-lit it and project it onto a flat surface. Note that the projection is necessarily skewed. Whereas the line segments of a cube join at 90 degree angles, you won't get the 90 degree angles on your projection, except in the trivial cases.

Now get two wire-frame cubes and join their corresponding vertices with straight line segments. What you are then holding is the projection of a 4 dimensional cube (a tesseract) into 3 dimensional space. This wire-frame cage is the "shadow" of a tesseract. The tesseract line segments join at 90 degree angles, but its projection is skewed as in the 2D case.

Hope this helps!
Corey

sacrelicious
2003-Feb-21, 05:04 PM
yes, refer to the link I posted (and break out an old pair of 3D glasses) for a view of one.

JS Princeton
2003-Feb-21, 05:07 PM
The fact that we perceive in 2D is a result of our projection screens (the retina in our eyes) being 2D. In a well ordered universe, in principle, there's nothing that would prevent a 3D animal from having a 3D projection screen. As you pointed out, we kinda have one already by looking at angular differences on two projection screens, stereoscopically. We could just as easily have a 1D projection screen (a slit)... then we would be struggling to break into perception of the 2D all the time.

Other senses are interesting to consider too. The sense of touch, for example, is an excellent judge of the dimensionality of our space. Hearing can also get at three dimensions, although it uses a different coordinate system (basically spherical coordinates). Tasting and smell aren't very good at space delineation at all.

Basically, if you wanted to have an analog to sight in the 4D spatial universe, you could consider the fact that you'd have a 3D projection screen, but you might also have a 2D one. It's not clear. The details of "perception" are more artifacts of biology than they are mathematical.

For example, I know at least one person who claimed to have really understood the hypercube when she went on an acid trip in the sixties. I don't know if I believe her, but perception is basically what you make of it.

Resu
2003-Feb-21, 05:15 PM
thank you sacreliciou and copabela.

yes, our brain can only see in 2D.

anyway, I was sitting there watching those two cubes in a cross-eyed fashion (like those disney magic books where you cross your eyes and see the patterns, you will perceive 3d objects), and it was pretty fascinating.

it is kind of odd to see a 3D object morph like that.

SiriMurthy
2003-Feb-21, 06:24 PM
http://dogfeathers.com/java/hyprcube.html

Ooh, my eyes hurt looking at the stereo.

Pretty cool. /phpBB/images/smiles/icon_eek.gif

Zap
2003-Feb-21, 08:09 PM
Is there such thing as 1D? Looking at it geometrically I always thought it was simply a line, but I may (and probably am) wrong.

Just imagine what 11D would look like. /phpBB/images/smiles/icon_eek.gif

Resu
2003-Feb-21, 08:33 PM
Zap, you are correct. 1D is just a line.

But 11D? my brain will probably go schizophrenic if I step into a 11D world.

David Hall
2003-Feb-22, 07:16 AM
I can understand a hypercube in my mind, to an extent. The idea is that you have to expand everything by one dimension. Here's how I see it.

Take a regular cube. In fact, take a six-sided die. Look at the side with one pip. Now, expand it and stretch it out towards you until it's a cube also. Notice how the edges of the original square have to also expand to become 2D planes.

Look now at the adjacent two-pip side of the die and do the same thing. See how the edges of that side also stretch to two-dimensions. But wait, one of those edges is also an edge of the one-pip side. The new plane therefore must cojoin both of the new cubes. Now do the same thing with all six sides of the original cube and you have a 4D tesseract.

The only problem with human perception is that we can't do it and keep them all as perfect cubes. Our minds have to distort them to keep everything connected. In the above example you can do it only if you distort the cubes into rhomboids with the outer side larger than the inner side. But in true 4D space they can maintain their 3D cube shapes and still remain attached to each other.

So that's what happens. The 2D planes become 3D cubes, and the 1D lines connecting them become 2D planes. In addition, the 0D corners become 1D lines, if you hadn't figured that out already. /phpBB/images/smiles/icon_smile.gif

That little java applet linked to above is cool in this way. After a while I could really see what was going on, as long as I kept the above in mind. But without this understanding, it's hard to see what it's really showing.

xriso
2003-Feb-22, 09:54 AM
On 2003-02-22 02:16, David Hall wrote:
I can understand a hypercube in my mind, to an extent. The idea is that you have to expand everything by one dimension. Here's how I see it.

Take a regular cube. In fact, take a six-sided die. Look at the side with one pip. Now, expand it and stretch it out towards you until it's a cube also. Notice how the edges of the original square have to also expand to become 2D planes.

Look now at the adjacent two-pip side of the die and do the same thing. See how the edges of that side also stretch to two-dimensions. But wait, one of those edges is also an edge of the one-pip side. The new plane therefore must cojoin both of the new cubes. Now do the same thing with all six sides of the original cube and you have a 4D tesseract.

Well, there is a bit more than that of course. You've only mentioned 7 of the 8 cube-faces of the tesseract. The remaining cube is the one on the opposite side of your starting cube.

The main thing in your method that prevents it from being a true tesseract is that you're not folding along the planes. As a result the different faces have to be constrained to a common space. It's like forming a cube from a cross-shaped piece of paper while not letting yourself lift up the edges. /phpBB/images/smiles/icon_smile.gif (and no pushing the edges together so the middle pops off the table!)

Using that analogy, here's an amusing way to make a true hypercube while staying completely in space: Take an original cube, and extend all six square-faces into cubes. Then, squeeze *really* hard on the outer square-faces until the whole middle part pops off space. Finally, take a new cube, smother it in glue, shove it in-between the outer square-faces, and then push them onto it.

Too bad nobody would believe you when you show them your amazing tesseract.

FP
2003-Feb-22, 06:41 PM
Robert heinlein wrote an excellent short story about the hypercube:
And He Built A Crooked House (http://www.wegrokit.com/upojh.htm)

Edited to fix link (I hope)

_________________
Learning without thought is labor lost; thought without learning is perilous.
Confucius, The Confucian Analects

<font size=-1>[ This Message was edited by: FP on 2003-02-22 13:43 ]</font>

sacrelicious
2003-Feb-23, 09:13 PM
On 2003-02-21 15:33, Resu wrote:
Zap, you are correct. 1D is just a line.

philisophical science question ('Sci-Phi' if you will): can something with length, but absolutely no width or hieght whatsoever, really be called a line? in other words, is a line that cannot be seen, felt, percieved, or impactful in any way really a line after all?

<font size=-1>[ This Message was edited by: sacrelicious on 2003-02-23 16:14 ]</font>

Donnie B.
2003-Feb-23, 09:50 PM
A line is, by definition, one-dimensional. The visible things we normally call "lines" in our 3-(spatial)-D world are, at best, approximations of lines - or more accurately, of line segments, since lines are infinite in length.

So really, an ideal line is an abstraction - none has ever existed except in the human mind.

JS Princeton
2003-Feb-24, 12:54 AM
The same is true of planes and points. They are also abstractions... though handy ones.

Senor Molinero
2003-Feb-24, 01:03 AM
Following up on xriso and David Hall.
The hypercube is a 4D structure made of 8 cubes as its "faces". Imagine, if you will, one wireframe cube with a second wireframe cube at its centre. Join the same corners from the inner to the outer, and voila, one internal cube (still rectilinear), 6 distorted cubes on the sides of the inner one, and, where is the eighth? It's inside out and contains the entire rest of the universe.

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<font size=-1>[ This Message was edited by: Senor Molinero on 2003-02-23 20:04 ]</font>

johnwitts
2003-Feb-24, 01:25 AM
OK, you lost me. Right after the phrase 'dimension projection'. /phpBB/images/smiles/icon_smile.gif

Zap
2003-Feb-24, 01:40 AM
OK, you lost me.

Welcome to my universe!!

Charlie in Dayton
2003-Feb-24, 04:22 AM
http://www.kleinbottle.com

and scroll down just a tad...

_________________________________
I am Dyslexia of Borg; prepare to have your *** laminated. - Lori Martin

Yours in cogno-intellectualism
Charlie in Dayton

<font size=-1>[ This Message was edited by: Charlie in Dayton on 2003-02-23 23:24 ]</font>

sacrelicious
2003-Feb-24, 03:57 PM
On 2003-02-23 16:50, Donnie B. wrote:
A line is, by definition, one-dimensional. The visible things we normally call "lines" in our 3-(spatial)-D world are, at best, approximations of lines - or more accurately, of line segments, since lines are infinite in length.

So really, an ideal line is an abstraction - none has ever existed except in the human mind.

allow me to defend my thesis by quibbling over definitions like a chatroom troll who throws out the whole of any well made argument if he finds even a single typo or spelling error therin /phpBB/images/smiles/icon_smile.gif :

I submit to you that a line is not an abstraction at all, but in fact (or much more accurately 'can be a') a physical (or at the very least visual) manifestation of a vector, and indeed it is vectors that are the abstractions.

the vector is the one dimensional thing. it is to a line what a point is to a dot, or what a plane is to a polygon. I would also argue that the concept of a line as a visually apparent item has been the concept that has prevailed for most of the history of human thought, and since definitions are relative creatures, the idea of the vector was in fact necesitated by the concept that a line must in some way exist in physical space; if indeed the line is as you define it, there would be no need for the term 'vector' to come about in the first place.

this is the long way of explaining the intent of my original post on the subject, and the reason I labeled it as 'Sci-Phi' in the first place, because all of this is really just a longform paraphrasing of "if a tree falls in the forest, does it make a sound?"

and surely this question is every bit as doomed to a fruitless nature as the original "tree falling" question /phpBB/images/smiles/icon_biggrin.gif

<font size=-1>[ This Message was edited by: sacrelicious on 2003-02-24 11:00 ]</font>

informant
2003-Feb-24, 04:28 PM
Both the terms line (or line segment) and vector have two meanings: a mathematical meaning and an informal meaning.

The mathematical concept describes an entity which cannot be observed. So we come up with graphical representations for it ('drawing a line segment'). These reprsentations give us an approximation of what 'lines' and 'vectors' really are, to help us think, but they aren't real lines and vectors themselves.

'Geometry is the art of thinking correctly about incorrect figures.'
Bertrand Russell

kilopi
2003-Feb-27, 03:06 PM
On 2003-02-21 11:01, traztx wrote:
I believe our brains do this automatically in many cases.

For example: When you turn on a faucet and see a line of water from the faucet to the sink. Turn out the lights and view it again under a strobe light and it is no longer a line, but a series of droplets.
I'm not disputing the idea of persistence of vision (isn't that why it's hard to look at a computer monitor just after you've had those opthamolic drops?), but that particular example is hard to believe. I've seen still photos of the stream from hoses, and the stream certainly doesn't look like it's composed of discrete drops.

JackC
2003-Feb-27, 05:52 PM
I've seen still photos of the stream from hoses, and the stream certainly doesn't look like it's composed of discrete drops.

With photography, you can do either. In fact, I exhibited this effect to my wife introducing her to our newest camera.

It is an interplay of speed of film and exposure, to be very simplistic about it.

In reality, for most photography, something like a stream of water or fountain or waterfall will be more "interesting" if shot with a (relatively) long exposure time, allowing the stream of water to "blend". A very fast shot showing the water as a more instantaneous occurance is perceived as "boring" - unless that is the effect you are going for.

I shot our fountain with three settings and showed her the results. You CAN get the droplets in a film shot - with right lighting, etc - but the shot is usually not visually appealing.

Of course, why someone would be trying to make a garden hose visually appealing is left as an excercise for the student....

Actually, if it is a general scene, it is most likely just an artifact of the way the picture was taken. Fast shutter shots with wide-open aperatures are really not all that common.

Jack

traztx
2003-Feb-27, 08:02 PM
On 2003-02-27 10:06, kilopi wrote:
I'm not disputing the idea of persistence of vision (isn't that why it's hard to look at a computer monitor just after you've had those opthamolic drops?), but that particular example is hard to believe. I've seen still photos of the stream from hoses, and the stream certainly doesn't look like it's composed of discrete drops.

Yes, it's hard to believe until you've seen it with your own eyes. The strobe light I used had a knob to control the rate. At the right rate, each flash would light the next drop at the position of the previous one, making it appear to float in mid-air except that the shape would fluxuate wildly. It looks unreal.