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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."
I'm looking at my monitor in three dimensions ... wait one second ... okay, I just saw it in four :)
If you can read this... 01110101 01110010 00100000 01100001 00100000 01100111 01100101 01100101 01101011
Take LSD and sure you'll see 4th dimension.
Why is the story tagged scientology?
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
Learned to do this on Tralfamadore.
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.
Just go to any Burning Man concert and eat the multi colored Brownies.
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
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
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.
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!
NB: The message above might reflect my opinion right now, but not necessarily tomorrow or next year.
Four? Trivial! I can visualize 11 dimensions...but 8 of them are very very small.
Buddhabrot in 4D (in 3D, in 2D). The Mandelbrot fractal never looked so good.
Murphey's fighting Occam, and we're in the stands.
Here is a one dimensional projection of a 5 billion dimensional sphere: _
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.
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 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.
i\hbar\dot{\psi}=\hat{H}\psi
Pff. Real mathematicians just picture N dimensions, then set N = 4.
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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
OS/2 - because choice is a terrible thing to waste.
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.
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
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.
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.
"I may be synthetic, but I'm not stupid." -- Bishop 341-B
To download any of the videos directly, go here:
http://www.sciencenews.org/pictures/mathtrek/082208/
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 :)
Nick
Dimented mind?
(Forgive the spelling, the pun dies without it)
APK quotes people (including myself) without context and should not be trusted. Just thought you should know.
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.
Anybody interested in visualizing hyperspace should learn about Alicia Boole Stott and her amazing story. 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.
Your retinas are, even together, a 2 dimensional array. You never "saw" anything but what your brain constructed from 2 dimensional arrays. Turns out your brain is very, very good at visualizing a 3d object based on this input. Would you say you can't visualize an actor's physical body because the screen is 2 dimensional?
If you click through the sciencenews.org article, you can get to the actual website of the people who made these videos. From there you'll find that these videos are Creative Commons licensed, and available for download as high res MOVs. I tried the torrents, but they all stalled, so I just used wget to grab them from the US mirror.
Give me Classic Slashdot or give me death!
http://bittornado.com/torrents/Dimensions-English.torrent
BitTorrent download for all the (English) movie files on the source website.
Thanks for the great torrent link. And here is a link to the full website full of clickable movies that you can forward to your friends, some of whom might not know what a torrent is. :-)
Currently hooked on AMP
Edwin Abbot Abbot wrote Flatland, not Edwin Abbot, who was Edwin Abbot Abbot's father.