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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."
Take LSD and sure you'll see 4th dimension.
then set N = 4....
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.
http://shambala.net/3d/tess3d1.gif
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
Here you go. It was Cosmo's take on "flatland":
http://www.youtube.com/watch?v=KIadtFJYWhw
I played around with this 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.
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.
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.
Damn straight.
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.
Cool story bro