How To See In Four Dimensions 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."
Tagit (Score:1, Insightful)
not (Score:5, Insightful)
Sorry it's on my screen, so it's a 2 dimensional representation of a 4 dimensional idea in 3 dimensional space.
Just imagine (Score:4, Insightful)
To think about it is mind bending, awe-inspiring, and dream provoking.
Just so we are clear... (Score:5, Insightful)
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
How I visualize 4-D objects is with color (Score:1, Insightful)
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.
Re:Just so we are clear... (Score:5, Insightful)
In a way, it's also projected into a 1-dimensional stream of bits.
We see in 2D not 3D (Score:4, Insightful)
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?
Re:Easy to see in four dimensions (Score:1, Insightful)
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 (Score:5, Insightful)
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...".
Haven't seen the video yet. (Score:4, Insightful)
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 (Score:3, Insightful)
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.
Re:Easy to see in four dimensions (Score:5, Insightful)
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?
Imagine, if you will, that you're also ignorant (or perhaps a member of congress). That shouldn't be too hard...
Do you think that humans actually see in three dimensions? We don't. We see in two dimensions. The retina is a plane. By using two planar sensory arrays, our brains use parallax to calculate depth. This is 2D vision with depth cues. Actual 3D vision would have us able to see the back side of the TV while watching a show on the front. When we talk about "visualizing" dimensions beyond the third, we're not talking about actually seeing things with our eyes. We're talking about mental pictures. We can "visualize" the back of the TV because our sensory system is accustomed to using a series of depth-cued 2D images to construct a model of the 3D world. Pushing that up to four dimensions isn't even remotely the same as drawing a ray traced 2D picture on a fucking computer monitor.
Re:Easy to see in four dimensions (Score:4, Insightful)
I am interested in what problem space you are working with.
In some very extreme cases, I can see it being a requirement to work the way you are, but in most real-world code, what you suggest would be far simpler to maintain (for you AND others) if you would just take a few extra minutes to think about what your data structures need to be.
Just because you CAN, doesn't mean you SHOULD. If it is a one-off script to solve a complex problem, then you have my respect. If anyone else EVER has to grok your code, for any reason, then you are just incompetent :)
BTW, this is probably an incredibly stupid question, but I just want to clarify. "The fourth dimension" is such an incredibly loaded term. In the context of this article, it is referring to time, correct?
Assuming I am correct, I have always had a very simple theory I use to wrap my mind around it. Bear in mind I am a high-level programmer, not a quantum physicist. I think that we (humans) exist within the first three dimensions while we travel along the fourth. Hence we are aware of, and can, to some extent, measure the fourth, but it is very difficult to perceive it in any concrete manner.
Re:Easy to see in four dimensions (Score:3, Insightful)
Re:Easy to see in four dimensions (Score:3, Insightful)
Re:Easy to see in four dimensions (Score:2, Insightful)
It's like a database table; each column is potentially a "dimension". Thus, if you have a 50-column table, you have 50 virtual dimensions. If you use something like Query-By-Example, then you can potentially filter by any of these dimensions.
This allows you to find all green customers living next to orange lakes who order blue products who've also ordered red products within the last 3 months while skydiving and eating sushi.
Perhaps a bit of an odd example, but I've been involved in queries almost like that for marketing research.
Know your dealer, know your dose (Score:1, Insightful)
Calm down and educate rather than this idiotic fearmongering, please. A small dose of knowledge may just save your life and a large dose will keep your mind at ease. Fear is stupid.
Ahem.
With any mind-altering substance from caffeine to alcohol to marijuana to LSD to DMT to mescaline to crack cocaine to salvia and beyond:
1. Know the source: know the purity of the substance and that it is safe to ingest, inhale, inject, etc.
2. Know the drug: read the common histories and medical/scientific research of these substances. If they are natural substances, pay particular attention to the ways that native peoples used them.
3. Know your setting: Take pains to be free of encumbrances when you will be intoxicated. Make sure you have a plan in place in the event you are in mental or physical distress. Make sure there are at least a few sober people on hand to corroborate the details of your experience later. Make sure you will be in a safe place and will not have to leave that place for the duration.
4. Start with a small dose: it does not make you a pussy to be cautious. Start off slow. You can always take more next time. With certain drugs (I'm looking at you, cocaine, crack, and heroin) if you can get them in their raw forms and try those first you will gain a greater appreciation for them. The profitability of raw opium and raw coca leaves in the black market is not great so in many places your only access to these substances in in highly concentrated form. In this case there is all the more reason for a cautious beginning.
5. Take a break: so you had a fun time with recreational use of a drug. Now go do something else for awhile and later you can always try it again under safe conditions. Using too often and too much are half of the reason for drugs being demonized by society. The other half is the massive profits that can be generated by limiting access to the medicine chest (and then letting some quantities of contraband through in the black market).
Re:Easy to see in four dimensions (Score:2, Insightful)
chances are we've all worked with structures of more dimensions that 16.
such as trees, which are equivalent to jagged arrays with dimension = max depth.
i bet slashdot's comment page markup template has dimension > 16.
i've got a couple maven poms that definitely hsve dimension > 16.
Re:Easy to see in four dimensions (Score:2, Insightful)
...as the so called "fourth" dimension they are representing exists in the same physical space as the third. This breaks the dimensional relationship.
Actually, it doesn't. Imagine the shadow of a 3 dimensional object (a 2 dimensional representation of said 3 dimensional object). As the object rotates (imagine a wireframe cube for example), the faces of the cube seem to intersect and split eachother. This is because we are merely looking at a second dimensional representation of that third dimensional object; we cannot see the third dimension that prevents those planes from intersecting.
The animations of the fourth dimensional objects, as projected on the third dimension, also suffer from the appears of intersecting planes. This is because our third dimensional display (which is, in reality, a second dimensional display that is playing some lighting tricks to fool our brains into interpreting it as third dimension) is incapable of displaying the *fourth* dimensional distance which prevents those planes from intersecting. Remember, we are dealing with *fourth* dimensional objects here.