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How To See In Four Dimensions
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."
Sorry it's on my screen, so it's a 2 dimensional representation of a 4 dimensional idea in 3 dimensional space.
Sorry, teleporters just kill you and then make a copy. A perfect, soul-less copy.
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.
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
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?
Why OpalCalc is the best Windows calc
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...".
I hate printers.
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.
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.
If a job's not worth doing, it's not worth doing right.
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.
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