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How To See In Four Dimensions

timothy posted more than 6 years ago | from the wrinkle-in-time-time dept.

Math 227

An anonymous reader writes "Think it's impossible to see four-dimensional objects? These videos will show you otherwise. Some mathematicians work with four-dimensional objects all the time, and they've developed some clever tricks to get a feeling for what they're like. The techniques begin by imagining how two-dimensional creatures, like those in Edwin Abbot's 'Flatland,' could get a feeling for three-dimensional objects. When those techniques are transferred up a dimension, the results are gorgeous."

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Tagit (1, Insightful)

religious freak (1005821) | more than 6 years ago | (#24724829)

AnaKata

first post (-1, Offtopic)

Anonymous Coward | more than 6 years ago | (#24724835)

first post

Re:first post (-1, Offtopic)

Dan541 (1032000) | more than 6 years ago | (#24724929)

You Fail!

Easy to see in four dimensions (4, Funny)

religious freak (1005821) | more than 6 years ago | (#24724837)

I'm looking at my monitor in three dimensions ... wait one second ... okay, I just saw it in four :)

Re:Easy to see in four dimensions (4, Interesting)

MrNaz (730548) | more than 6 years ago | (#24725115)

I "visualize" four dimensions and more often, when programming and setting up multi-dimensional arrays of more than three dimensions.

All one has to do is acknowledge that adding a dimension simply adds a range of points that map to every single point in the (n-1) dimensional range. So, the easiest way to visualize a four dimensional cube is to simply imagine multiple identical cubes, side by side, for as many as the range has been specified. Five dimensions is a flat square arrangement, six is a cube arranged array of cubes, and so on. This way, an infinite number of dimensions can be visualized. Perhaps the term "mental addressing" is more appropriate a name for this mental method.

The limit is, of course, this only works directly for finite and discrete arrays. I find it can be extrapolated to use non-discrete spectra, but describing the way that works in my head will not be possible using this clumsy tool we call "language".

Re:Easy to see in four dimensions (1, Insightful)

Anonymous Coward | more than 6 years ago | (#24725251)

We know what dimensions are, we just can't "see" them. Enumerating orthogonal slices is a very limiting view of a higher dimensional space. That's the whole point of the exercise: To find other visualizations which better convey the relations in that space.

It is one thing for the 2-dim beings to know that us 3-dim beings can see their innards, which they themselves can't. They can certainly formulate "closedness" in higher dimensions, but it is quite another thing to have an intuition to the same effect. A "multiple 2D-spaces next to eachother" representation of 3D doesn't produce that intuition (mostly because neighboring points in one dimension appear farther away than in other dimensions.)

Re:Easy to see in four dimensions (5, Insightful)

MrNaz (730548) | more than 6 years ago | (#24725263)

After thinking about this some more, I find that the animations in the article are not at all four dimensional, as the so called "fourth" dimension they are representing exists in the same physical space as the third.

This breaks the dimensional relationship. Imagine, if you will, a single point with no dimensions. Then extrapolate that into a line to get one dimension, imagine that line them extrapolating perpendicular to the line to form a square, and then imagine that square extruding into a cube. So far, no physical overlap has occurred. The fourth dimention as represented in these videos, does nothing but add more "balls and sticks", which is not adding another dimension, it's simply adding detail to the existing dimension.

Likewise, those 2D imaginings of a 3D object are not visualizations of a 3D object in 2d, they are the visualization of a changing 2D object, with the simulated third dimension being time.

In other words, the method that they have used does not actually visualize a fourth dimension in any mathematical or logical sense, they are really just optical illusions. Personally, my method of visualization that I described in my previous post is far superior, and more accurate from a logical and mathematical point of view, as it truly does represent a 1:M maping of every dimensional unit in the (n-1) dimensional space.

P.S., I've always wanted to start a sentence with "Imagine, if you will...".

Re:Easy to see in four dimensions (1, Funny)

Anonymous Coward | more than 6 years ago | (#24725303)

Imagine, if you will, that you're ignorant. That shouldn't be too hard. Do you complain that your 3D graphics card just adds more 2D pixels, where it should instead show hundreds of 2D pictures next to each other in order to represent 3D space?

Re:Easy to see in four dimensions (5, Informative)

alexj33 (968322) | more than 6 years ago | (#24725637)

I find that the animations in the article are not at all four dimensional

Duh. That's because our screens are two dimensional, and you and I are three dimensional. Certainly you can't fault them for this? (Please tell me that I'm somehow misunderstanding this objection..)

In other words, the method that they have used does not actually visualize a fourth dimension in any mathematical or logical sense

That's nonsense. Their videos show the edges of the object (although distorted) as well as the interconnections of each of the vertices. What would qualify to you as a "real" mathematical or logical way of viewing these objects in a 3-D world?

As for your previous post:

So, the easiest way to visualize a four dimensional cube is to simply imagine multiple identical cubes, side by side, for as many as the range has been specified. Five dimensions is a flat square arrangement, six is a cube arranged array of cubes, and so on. This way, an infinite number of dimensions can be visualized. Perhaps the term "mental addressing" is more appropriate a name for this mental method.

Okay, when you get down to it, this is stuff that any programmer knows when working with arrays. (ie- int[][][][][], etc.) Now your task is to *draw* your example for us in 3-D space.

Re:Easy to see in four dimensions (4, Interesting)

cheater512 (783349) | more than 6 years ago | (#24725339)

Yeah I've had arrays with double digit dimensions.
I think my record is 16 or so.

I dont know why but I work with them incredibly easily.
Without them its like programming with a hand tied behind your back.

Cant visualize them at all, I can work with them though.

Re:Easy to see in four dimensions (5, Funny)

Anonymous Coward | more than 6 years ago | (#24725737)

Cool story bro

Re:Easy to see in four dimensions (1)

Yvanhoe (564877) | more than 6 years ago | (#24725535)

Yes, you can visualize a 4D arrangement of objects but I think this article is more about visualizing a shape in 4D and making deductions on it like : is it closed ? can it be projected as a cube in a given angle ? How many edges does it have ? etc...

Re:Easy to see in four dimensions (0)

Anonymous Coward | more than 6 years ago | (#24725563)

Hey, Buckaroo Banzai, don't you have a singing engagement to attend to?

-John Bigboote

Re:Easy to see in four dimensions (2, Funny)

Hal_Porter (817932) | more than 6 years ago | (#24725839)

"The third dimension is a theoretical realm of space and time in which the particles of dark matter of this parallel alternate reality bends light to collide with the electrical charges of the subconscious mind to create the illusion of movement where what is dark becomes light, what is light becomes dark. Some look at the third dimension and see nothingness. Others believe they see the very face of God."

http://uk.youtube.com/watch?v=6T0UQfKTcQw [youtube.com]

 

it see all time (5, Funny)

extirpater (132500) | more than 6 years ago | (#24724841)

Take LSD and sure you'll see 4th dimension.

Try Salvia (4, Interesting)

Nick Ives (317) | more than 6 years ago | (#24725357)

One of the most common sensations (along with the sense of absolute terror at being ripped into a void in space/time) is the feeling of moving through between more than 3 dimensions of space. In my travels I usually feel like I'm spinning and being folded in about 7 different dimensions before my visions start to settle.

To anyone who decides to take me seriously, make sure you have a sober sitter :)

Re:Try Salvia (0)

Anonymous Coward | more than 6 years ago | (#24725745)

worst. drug. ever. (imho)

Re:it see all time (4, Funny)

Eli Gottlieb (917758) | more than 6 years ago | (#24725889)

I prefer melange.

Scientology? (4, Interesting)

hansraj (458504) | more than 6 years ago | (#24724843)

Why is the story tagged scientology?

Re:Scientology? (4, Funny)

Draconix (653959) | more than 6 years ago | (#24724859)

Obviously, Lord Xenu has a Slashdot account.

Re:Scientology? (5, Funny)

elronxenu (117773) | more than 6 years ago | (#24725587)

Damn straight.

Re:Scientology? (4, Interesting)

Ilgaz (86384) | more than 6 years ago | (#24724877)

I see a flash scientology , rather very big ad at bottom of article and in "videos", there are Google Adsense ads mentioning scientology youtube channel.

It could be related to people who sees those ads (must be scientific terms used triggering them) and think the site is Scientology supported. It could be possible but it could be the adsense only too.

BTW Google Adsense advertising Scientology Youtube channel is not really a good, pretty sight. What next? Doubleclick ads too?

Re:Scientology? (1)

elguillelmo (1242866) | more than 6 years ago | (#24725103)

Google Adsense advertising Scientology Youtube channel is not really a good, pretty sight

Yeah, I agree. Adblock Plus spares me the view, though!

Re:Scientology? (0)

Anonymous Coward | more than 6 years ago | (#24725649)

Google just bought Doubleclick, like a year ago, i think.

Re:Scientology? (0)

Anonymous Coward | more than 6 years ago | (#24725905)

I saw a Co$ add on a Google search months a go... I'm pretty sure its just Co$ doing what all businesses do: advertise on Google. In conclusion, I'm not overly worried about Google helping Co$. Can you imagine the outcry there'd be? "Don't be evil! You Hippocrates!" Also, major antiCo$ videos were still on YouTube last time I checked.

BTW, the site's Slashdotted.

Simply imagine a space defined on R^N.... (5, Funny)

Anonymous Coward | more than 6 years ago | (#24724845)

then set N = 4....

not (5, Insightful)

Holi (250190) | more than 6 years ago | (#24724849)

Sorry it's on my screen, so it's a 2 dimensional representation of a 4 dimensional idea in 3 dimensional space.

Re:not (4, Funny)

node 3 (115640) | more than 6 years ago | (#24725041)

Close one eye.

Re:not (0, Offtopic)

syousef (465911) | more than 6 years ago | (#24725043)

Sorry it's on my screen, so it's a 2 dimensional representation of a 4 dimensional idea in 3 dimensional space.

Einstein syas it's a 3 dimensional representation of a 5 dimensional idea in 4 dimensional space. Unless you know how to freeze time?

Re:not (3, Funny)

Anonymous Coward | more than 6 years ago | (#24725187)

Yeah, well he is dead!
Can't defend things when you're dead.

Re:not (3, Funny)

jacquesm (154384) | more than 6 years ago | (#24725219)

good point. What's your home address ?

Re:not (0)

Anonymous Coward | more than 6 years ago | (#24725363)

flamebait ? mods without humor !

Re:not (1, Funny)

BronsCon (927697) | more than 6 years ago | (#24725385)

He;s afraid to tell you, but it's

1600 Pennsylvania Ave NW
Washington, DC 20500

Re:not (0)

Anonymous Coward | more than 6 years ago | (#24725509)

Unless you know how to freeze time?

In Flash just use the pause button.

Re:not (as in do not understand) (1)

tsjaikdus (940791) | more than 6 years ago | (#24725231)

>> Sorry it's on my screen, so it's a 2 dimensional
>> representation of a 4 dimensional idea in 3
>> dimensional space

They do not try to display 4D images on a 2D screen. They show how 3D images that you can see in 3D space are related to a 4D object.

Then you should have no trouble creating a 3D image from a 2D screen (http://en.wikipedia.org/wiki/Engineering_drawing).

Begin? (1)

mrboyd (1211932) | more than 6 years ago | (#24724855)

The techniques begin by imagining how two-dimensional creatures, [...] could get a feeling for three-dimensional objects.

I guess that's already way past my abilities.

Re:Begin? (0)

Anonymous Coward | more than 6 years ago | (#24725065)

As Holl [slashdot.org] mentions, images on your screen (or sheets of paper, etc.) are in two dimensions. You're already used to seeing 3D projected onto 2d and without the contrived method used in TFA. For a second I thought perhaps they're trying to take the 2D lizards POV differently (ie., as a projection), but in that case the lizards would see in 1D and their solution wouldn't work. And for fuck's sake, the only reason determining the 3D shape in the 2D (+ time) slicing vid was difficult was because they were unnecessarily spinning the slices without remarking on it - making it look like a more complex shape than it was.

If anything, I find this article interesting because it uses a complex method to solve a problem I didn't have in a field in which I have some experience. I'd like to figure out why there's such a difference in mental modeling, and whether there are good applications for their solution. After all, I deal with things that could be modeled as multi-dimensional spaces quite often and new ways of displaying or modeling that data could be helpful to customers.

did this years ago... (5, Interesting)

TheSHAD0W (258774) | more than 6 years ago | (#24724865)

Re:did this years ago... (2, Informative)

Zeussy (868062) | more than 6 years ago | (#24725059)

I was going to say something similar, but along the lines of Adanaxis [mushware.com] its a 4D Space shooter I played a couple of years ago.

Re:did this years ago... (4, Interesting)

doombringerltx (1109389) | more than 6 years ago | (#24725107)

The article linked to in TFS fairly crummy, but following through leads to the full videos [dimensions-math.org] which are really good. I even sent it to a non-math nerd friend. Its worth a look for anyone who had little trouble imagining geometric shapes in Rn. God knows that was me when I had classes that delt with that. Eventually I was like "Fuck it. It doesn't have to make sense, just get to where you can pull it off on the final." Plus its doing a good job of showing multiple methods to represent it, past what your gif shows. Right now I'm only a few chapters in, so I hope it keeps up the quality.

Re:did this years ago... (2, Funny)

Kent Recal (714863) | more than 6 years ago | (#24725209)

oO [is.gd]

Dupe! (4, Funny)

Xfacter (1075973) | more than 6 years ago | (#24724875)

Learned to do this on Tralfamadore.

Just imagine (3, Insightful)

Joe Jordan (453607) | more than 6 years ago | (#24724885)

Just imagine how incredible the true nature of the universe must be if current theories hold true that 10, 11, or even possibly 26 dimensions exist in our universe.
To think about it is mind bending, awe-inspiring, and dream provoking.

Re:Just imagine (0)

Anonymous Coward | more than 6 years ago | (#24724911)

They aren't necessarily spatial dimensions, which is what TFA is talking about.

Seeing in 4 Dimensions? (4, Funny)

EdIII (1114411) | more than 6 years ago | (#24724895)

Just go to any Burning Man concert and eat the multi colored Brownies.

Carl Sagan (3, Interesting)

Ilgaz (86384) | more than 6 years ago | (#24724899)

Does anyone remember in how a good way Carl Sagan explained the problem if there are more or less than 3 dimensions exist?

I remember he was explaining the imaginary 2d creatures not being able to see 3d creatures and so on. It was on a TV documentary. Sorry if I remember it all wrong. I was like 13 ;)

It must be an episode of "Cosmos" http://www.imdb.com/name/nm0755981/filmoseries#tt0081846 [imdb.com]

Re:Carl Sagan (2, Informative)

Vectronic (1221470) | more than 6 years ago | (#24724957)

Indeed it was...

You can watch them (except part 5) on Guba [guba.com] (change query to find the rest)

I think it was part 8 specifically. I got the DVD but its been awhile since I wandered through it... But its fairly brief, everything on the difference between 2, 3 and 4 Dimensions is basically described as it was here, and just as brief.

However, its still a good way to spend 13 hours because of everything else he covered in that series.

Rod Serling (0)

Anonymous Coward | more than 6 years ago | (#24725163)

I can imagine a dimension not only of sight and sound but of mind.

Re:Rod Serling (2, Funny)

BronsCon (927697) | more than 6 years ago | (#24725395)

Dimented mind?

(Forgive the spelling, the pun dies without it)

Just so we are clear... (5, Insightful)

Anonymous Coward | more than 6 years ago | (#24724907)

A 4D object is mathematically projected to a 3D representation, that is then projected into a 2D representation for display on the monitor, that is then transformed by my brain back into a 3D representation, and then further needs to be transformed into a 4D object... /looks for his linear algebra textbook //begins drinking

Re:Just so we are clear... (5, Insightful)

Drinking Bleach (975757) | more than 6 years ago | (#24725079)

In a way, it's also projected into a 1-dimensional stream of bits.

Re:Just so we are clear... (1, Interesting)

Anonymous Coward | more than 6 years ago | (#24725149)

Thought of that as well...and I also didn't include that the 2D monitor image travels through 3D space for a "2D" projection at the back of my eye, where my brain then translates it's 3D position of a 2D image and then we return to the stream of the previous musing...

/still drinking
//isn't helping

Re:Just so we are clear... (1)

DDLKermit007 (911046) | more than 6 years ago | (#24725453)

STOP!!! My brain it hurts!!!
I like to think I'm smart, but I so can't wrap my head around this right now.

Interacting is the easiest way to learn (5, Interesting)

Eighty7 (1130057) | more than 6 years ago | (#24724933)

I played around with this [vanderwal.eu] applet a few months ago. After some practice, getting out & hitting the ball becomes easy. Getting back in is only slightly harder & I still can't hit the point reliably.

More than one way to skin a cat (1)

Nymz (905908) | more than 6 years ago | (#24725083)

Seeing a forth dimension doesn't have to mean actually seeing it visually or pictorially. Since this is /. how about imagining a multi-dimensional array, with 3 indexes (x, y, & z) and then adding another (4th), Presto. Another way physical space could be defined is by relationships. When playing blind chess I don't actually see a board in my head, I just remember how all the pieces relate to each other.

Re:Interacting is the easiest way to learn (1)

DDLKermit007 (911046) | more than 6 years ago | (#24725483)

I feel stupid, I can't figure this out. All I can get it to do is spin.

Re:Interacting is the easiest way to learn (1)

eric-x (1348097) | more than 6 years ago | (#24725707)

Use SHIFT to move the cursor forward/backward.

Awesome. (2, Interesting)

buchner.johannes (1139593) | more than 6 years ago | (#24724939)

Awesome. However, mathematicians and physicist usually don't try to "see" or "get a feeling" of higher (or infinite) dimensional objects.
They familiarize themselves with mathematic properties of two and three-dimensional objects and space and what they mean, and then just use these properties in higher dimensional spaces.

Trying to see these spaces or getting a feeling on how these objects would look like most likely confuses for calculations (our brain wasn't really made for this).

Nice and interesting videos though!

Depends on the mathematicians (2, Interesting)

Cigaes (714444) | more than 6 years ago | (#24725023)

That completely depends on the mathematicians, and the kind of mathematics they do. For proofs that rely only on calculations, you do not need even to understand the low dimension case, just do the computations right.

But proofs with computations are rarely elegant. Some mathematicians prefer a more geometric approach, and for that, they need to see, un to a certain level, the objects in higher dimensions.

Furthermore, the 2D or 3D spaces we have direct access to are really limited. There are lots of phenomenas that only happen starting with dimension 4 or 5. For example, think of this 2D property: "two lines perpendicular to a common third line are parallel"; if you try to take it as is in higher dimensions, you get something false; fortunately, you can think in 3D and see that it is false. There are similar examples in higher dimensions. Curvature, for example: curvature of 2D surfaces in 3D spaces is misleadingly simple, compared to curvature of higher dimensional spaces.

Sometimes, there just is not space enough to build the objects you need in 3D space. For example, if you want to study circles drawn on a sphere, the object you need to make the properties apparent is a 3D hyperboloid in a 4D space. If you settle for a 2D hyperboloid in a 3D space, you end up studying pairs of points on a circle, which is rather boring.

Re:Awesome. (1)

stephanruby (542433) | more than 6 years ago | (#24725313)

However, mathematicians and physicist usually don't try to "see" or "get a feeling" of higher (or infinite) dimensional objects. They familiarize themselves with mathematic properties of two and three-dimensional objects and space and what they mean, and then just use these properties in higher dimensional spaces.

Is that you Mr. Spock? I'm sure that Albert Einstein (not a mathematician for sure), Richard Feynman, and Stephen Hawking, would beg to differ.

Trying to see these spaces or getting a feeling on how these objects would look like most likely confuses for calculations (our brain wasn't really made for this).

Our brain wasn't made for exact calculations either. Physics and mathematics are usually much sloppier, at least initially, than what most physicists and most mathematicians would care to admit. It's very much like the sausage industry. The end result look perfect and nice, that's the illusion that's being promoted at least, but *only* the real sausage-makers know how the sausage usually gets made.

I can visualize 11 dimensions (4, Funny)

harlows_monkeys (106428) | more than 6 years ago | (#24724941)

Four? Trivial! I can visualize 11 dimensions...but 8 of them are very very small.

Re:I can visualize 11 dimensions (1)

ya really (1257084) | more than 6 years ago | (#24725069)

I see someone knows their string theory and quantum physics, lol. Great comment though, gave me a good laugh.

Re:I can visualize 11 dimensions (4, Funny)

Ibag (101144) | more than 6 years ago | (#24725183)

That reminds me of a joke:

An engineer, physicist, and a mathematician are sitting at a bar, and the bartender asks, "Can any of you guys think about four dimensions?"

"Sorry, not me," the engineer replies.

The physicist chimes in, "I suppose I can, if the fourth dimension is time."

The mathematician starts laughing. "Oh, you guys, this is easy! Picture n-dimensional space. Now, let n be equal to four..."

OT but another mathematics joke (5, Funny)

Sycraft-fu (314770) | more than 6 years ago | (#24725479)

A physicist, and engineer, and a mathematician are sleeping in a hotel when fires break out in all their rooms. The physicist get up, does some quick calculations, and then gets the exact amount of water required to put the fire out, not a drop wasted. The engineer also does some calculations to work out the amount needed, then proceeds to flood most of the floor, to ensure that there is a sufficient tolerance for error. The mathematician wakes up, and does some extremely complex calculations but does them much quicker than the other two. He then exclaims "I have proven I can put the fire out!" and goes back to bed.

Buddhabrot (3, Interesting)

Xelios (822510) | more than 6 years ago | (#24724983)

Buddhabrot in 4D (in 3D, in 2D). [youtube.com] The Mandelbrot fractal never looked so good.

Bloat (1)

Tablizer (95088) | more than 6 years ago | (#24725003)

Shapes with gas. I wonder what he ate to inspire that?

Infinite dimensions... (1)

Greyor (714722) | more than 6 years ago | (#24725013)

Perhaps obscure, but perhaps also obligatory: what about using the Resonator [photobucket.com] to see these dimensions? Humans are such easy prey...

Pfffft, is that all ? (3, Funny)

BlueParrot (965239) | more than 6 years ago | (#24725017)

Here is a one dimensional projection of a 5 billion dimensional sphere: _

Re:Pfffft, is that all ? (1)

smellotron (1039250) | more than 6 years ago | (#24725893)

Here is a one dimensional projection of a 5 billion dimensional sphere: .

Fixed that for you.

How I visualize 4-D objects is with color (1, Insightful)

Anonymous Coward | more than 6 years ago | (#24725031)

Take, for example, a hypercube. Imagine a regular 3D cube, hollow, but the outside surfaces are one color and the inside surfaces are another. The fourth dimension is the spectrum of colors in between. If you were to rotate the hypercube in four dimensions, any or all of the three physical dimensions could expand or contract (so the cube could grow or shrink or change into a rectangular block) as the fourth dimension rotated into the other three, while at the same time the inside and outside colors would also change (with a larger or smaller spectral width) as the three physical dimensions rotated into the fourth. At the "reddest" end of the spectrum is the moment of the big bang.

rotating tesseracts (2, Interesting)

xPsi (851544) | more than 6 years ago | (#24725071)

Definitely enjoyable stuff. Of course, you could just play Portal. Oh, sorry, that's just an ordinary 3D space which happens to be multiply disconnected and topologically unsettling. For more (Euclidian!) 4D visualization tools, here [uiuc.edu] are a couple nice (but old) clips of rotating cubes and tesseracts through higher dimensions. For example, it gives you the (x,y,z) view of a cube then a simultaneous projection of that object in the (w,x) plane where w is a 4th orthogonal direction. It then proceeds to rotate the (w,x) projection in a circle to see what the 3D "shadow" in (x,y,z) space is doing. Rather than getting bigger and smaller (simulating perspective) as it moves back and forth in the 4th direction, the faces are color coded (I personally think this makes it easier to visualize). Run the simulation back and forth slowly a couple times and your brain locks in pretty well.

Old maths joke... (4, Funny)

SoVeryTired (967875) | more than 6 years ago | (#24725097)

Pff. Real mathematicians just picture N dimensions, then set N = 4.

image overlays (0)

Anonymous Coward | more than 6 years ago | (#24725111)

Is it just me, or are there more people that take issue with the way these new popups appear as overlays within the page? I usually ctrl-click on a link to see the target (image, flash) in a new page, and return to the original page when I want to.

This is for me a reason to start using greasemonkey. Anyone knows if there is a script or other plugin for FF that kills these misfeatures?

Re:image overlays (0)

Anonymous Coward | more than 6 years ago | (#24725137)

ok, just looked at the page source. I'll clarify my question: I'm talking about shadowbox links. I've searched for greasemonkey+shadowbox but can't find any references to it. Is it easy to write something like that myself?

Seeing four dimensions. (4, Funny)

os2fan (254461) | more than 6 years ago | (#24725127)

You can teach yourself to see in four dimensions, by using analogy and other things.

To begin, consider that a 2d picture can either be a picture (things can fall), or a map (things don't fall). Since the corresponding 3d thing is a picture/map of four dimensions, we can build objects like houses, furniture, etc from plan and views.

Not all seems to be aimple. A knife cuts: literally, it makes a surface by motion, and is therefore tipped by a space of N-2 dimensions. Rivers can be either "latrous" (1d) or "hedrous" (2d). A fault lake is 2d (since faults are a break of surface).

Holes come in two types, although these are topologically the same. One can have a "bridge" or "tunnel" kind of hole: in 3d, these are the same, in 4d they are different.

The planet rotates on clifford motion. This makes every point of the 4-sphere go around the centre. One sees this by equality of energy in modes of rotation.

None the same, there can be seasons. If the sun does not follow in the year-circle any of the circles of the earth rotating, then there will be seasons. You don't just have hemispheres in summer vs winter, but season-zones to match the time-zones. That is, for example, Christmas (normally in summer), can fall in early spring, or late winter.

The poles are replaced by circles of extreme climate. One has a "equator circle", and a "polar" circle. At the tropics (a singular torus-shape thing), the sun becomes to the zenith once a year. At the artic torus, the sun hugs the horizon for the equate of the shortest day.

Because the sun is relatively still in the sky, there is no variation in the number of hours. What makes the seasons is that the the sun is lower in the horizon, even at midday.

See, eg my site http://www.geocities.com/os2fan2/gloss/index.html [geocities.com]

Re:Seeing four dimensions. (4, Funny)

ColaMan (37550) | more than 6 years ago | (#24725439)

Careful.

You read just like the timecube guy did, before he took that last hit of bad acid.

particle physics (0)

Anonymous Coward | more than 6 years ago | (#24725147)

Three dimensions are not a barrier but a consequence of measurement. This is just silly.

You need another dimension to visualize the 4th. (3, Interesting)

Anonymous Coward | more than 6 years ago | (#24725157)

For the same reasons you can't visualize a 3D object on a 1D space you can't visualize a 4D object on a 2D space.

You cannot go up 2 dimensions.

Just as we can visualize a 3D object on a 2D space we can visualize a 4D object on a 3D space.

Thus we need something like this:
http://dogfeathers.com/java/hyprcube.html

*Click the Stereo button 2 times to switch it to cross-eyed view for no glasses. Simply cross your eyes to bring both shapes together in the center and it should become clear.

Seeing in 4 Dimensions? Bah! (-1, Redundant)

Anonymous Coward | more than 6 years ago | (#24725201)

Seeing four-dimensional objects is not hard at all. First I imagine it in N dimensions, and then I take the case when N=4.

Ridiculous explanation (1)

xquark (649804) | more than 6 years ago | (#24725217)

The 2nd explanation for projection of 3D objects onto a plane so as to allow the 2d lizards to perceive the objects is simply ridiculous as it requires them to have an external view point defeating the purpose altogether. The 1st example was ok, but its nearly 100 years old, something new/unique/novel would have been more interesting to watch, also the presentation drags on for too long, it should be sped up.

Nonsense! (1, Funny)

Anonymous Coward | more than 6 years ago | (#24725237)

Three dimensions ought be enough for anybody!

We see in 2D not 3D (3, Insightful)

Twinbee (767046) | more than 6 years ago | (#24725239)

Of course, we can't really see in 3 dimensions, otherwise, we'd be able to see through stuff. The image projected onto our eyes is a 2D image, and we have 2 eyes, so it's (x*y)+(x*y), not (x*y*z). The third dimension is a cheat and is represented as 'stuff getting smaller'.

If we really could see in 3D, we can use the 'getting smaller' trick to visualize 4 dimensions much more easily.

Anyone know of some images or videos on the net using reverse perspective, where things behind get bigger instead of smaller?

Feh. (0)

Anonymous Coward | more than 6 years ago | (#24725241)

I got a screen saver that does this.

Obligatory (1)

OricAtmos48K (979353) | more than 6 years ago | (#24725269)

But my existence is in two dimensions you inclod sensitive !

Haven't seen the video yet. (3, Insightful)

Saint Stephen (19450) | more than 6 years ago | (#24725273)

But I can guess how it works. A sphere passing through a plane would look at first like a dot, then a gradually wider line, then a dot. I remember flatland saying something about brightness at ends of the line.

So, a hyperball passing through a 3-space would look like a dot, gradually expanding to a sphere, and gradually shrinking to a dot.

Falling WAY short (2, Insightful)

DynaSoar (714234) | more than 6 years ago | (#24725283)

These 2D videos show 2D diagrams of what a 4D projection into 3D would look like if it were flat. Entirely unsatisfying.

Want a 4D-in-3D demo? Take a small balloon, blow it up then let it go flat. That's what a 4D sphere projecting into 3D would look like.

You can imagine in 4D fairly easily if you decide to ignore your senses and decide that the smaller faces on the internal cube in a tesseract are indeed the same size (an in fact coincide with) the larger, outer faces, and so the outer pseudo-cubes are in fact cubes with all 90 degree corners. You see perpective with fake apparent angles, you can use the same trick your mind uses to see more.

By the way, we do not see in 3 dimensions. We see in 2.5. We can't see the backs of things. We can feel in 3 dimensions if we can get our hands all the way around it.

We do NOT see in 2 dimensions (as a previous comment stated) unless we have no depth perception. Stereoscopic vision gives us much more than flat projection, and stereointegration in the visual cortex gives us even more. In fact, a one-eyed being with stereointegration need only moves its head around and collect visual images from different angles in order to create a successfully adequate 3D concept.

And ask the previous commenter asked, yes we do have examples of reverse perspective where things behind get bigger. Gravitational lensing of galaxies passing behind smaller, intense gravity fields (theoretically black holes or neutron stars). Can't point to any I've seen on the web offhand, but I've seen them there as well as on some astronomy shows on TV.

Subject (1)

Legion303 (97901) | more than 6 years ago | (#24725349)

"Think it's impossible to see four-dimensional objects? These videos will show you otherwise."

No they won't. Submitter needs to learn math.

I've never seen a four-dimensional object (0)

Anonymous Coward | more than 6 years ago | (#24725429)

So it goes ...

Saving throws (1)

Fryth (468689) | more than 6 years ago | (#24725471)

My dungeon master used to bring those 4D tetrahedrons out when he felt especially mean. I still hate him for it.

Improper Tag (1)

dreamchaser (49529) | more than 6 years ago | (#24725529)

This has nothing to do with string theory, so let's all do a !stringtheory up on the tagline. There is nothing about cosmology in this; it's about methods used for years by mathematicians to visualize 4D abstract objects as they move through 3D space.

It's not even new.

Alicia Boole Stott Got There First (3, Informative)

cybrpnk2 (579066) | more than 6 years ago | (#24725617)

Anybody interested in visualizing hyperspace should learn about Alicia Boole Stott [agnesscott.edu] and her amazing story [ub.rug.nl] . She was the daughter of George Boole (of boolean algebra fame) who developed a mind-boggling series of paper cutout models of four dimensional objects that won her an honorary math doctorate in 1914. Check out these extensive photos of her work [math.rug.nl] .

JWZ's xscreensaver has done this forever (0)

Anonymous Coward | more than 6 years ago | (#24725645)

Jamie Zawinski's xcreensaver has a hack that does this. It's been in the package since the late 90s, at least. This is news how?

Time!? (0)

Anonymous Coward | more than 6 years ago | (#24725713)

Isn't time the 4th dimension?

Must... not... say... (1)

dword (735428) | more than 6 years ago | (#24725787)

"keberT xelA"

slashdot is broken (1)

mestar (121800) | more than 6 years ago | (#24725865)

Please help, Slashdot seems to be totaly fucked since yesterday. New comments format is not working, I can't change the comments level, dropdown box is gone, just a number is there.

Also, I can't go to my preferences, it all comes up as a big empy box, so no way to change, or even see, anything. I hope somebody is aware of this.

This is on IE7. Also, there will be no way for me to see any replies to this. geez.

Re:slashdot is broken (1)

mestar (121800) | more than 6 years ago | (#24725901)

Oh, those things that look like sliders actually are really stupid sliders (you have to move them one fifth of the screen for them to actually move).

Still, "preferences" comes as an empty box, just the words "Discussion 2" on the top, and hint of some text on the bottom, you can only see the top of the letters. ftw?

4 dimensions? You insensitive clot (1)

Hells (1166547) | more than 6 years ago | (#24725881)

Slashdotters still cant see in more than 1.
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