That s a really crappy representation of a 4-d hypercube. I know they probably followed a mathematical formula to get that projection, but still, what we are seeing is a 2-d representation of a 4-d object. That is like, by analogy, representing a cube (3-d) on a one dimensional graph. Do you know how that would look? It is simply two points, a certain distance apart, or a line segment. the space between the points represents the 3-space inside the cube, the part of the line outside the segment represents 3-space outside the cube. That is how CRAPPY this mapping is!If you want to get an idea of a 4-d cube, or a hypercube, there is a better way, though it is difficult to picture. Begin with the tesseract, which is a 3-d representation of the 4-d 'hypercube'Note that it looks like a cube inside another cube, with the corresponding corners connected by lines. This appears as eight 'sort-of' cubes, the inside one, the outside one, and the six 'transitional cubes, that have one face of the inner cube, and one face of the outer cube. we must allow that the other faces of each of these six 'transitional cubes' , though not appearing square, are ultimately square, if seen in an actual 4-dimensional space(were we can't go).The only other stipulation to make this a true hypercube is that....those eight cubes are arranged in such a way that they are all the same size, they don't share any intersecting space, and no one of them has a unique position tat you can say is 'inside' or outside' of the other ones. Like the square faces of a cube, they are equal, with not one standing out in a unique position as being inside or outside. Thus, as a square has 4 line segments with no unique positions, and a cube has 6 squares, the hypercube has 8 cubes.the eight cubes are the same size, and none is inside or outside another, or they ALL are inside or outside, so that not one is unique in position. Note also, that when looking at a cube (imagine one made of wire), there is a way to look at t so that one square face appears smaller than, and inside, another square face. e.g. look straight at a side, and note the opposite, parallel side --looks like a smaller square, inside it. in the same way, there is a way to look at a hypercube, so that one of the eight cubes appears to be smaller than, and inside another cube.That APPEARANCE is what we call the tesseract...a mere picture of a hypercube, from one direction. and ere, we are speaking of directions infinitely more than the directions contained in 3-space. Thus, there is another way of looking at the hypercube so that you feel you are inside a cube, surrounded by the other cubes, one of them being parallel to the one you are in. This is one of the creepiest views of the hypercube, as you seem to be completely entrapped. Of course, you can just turn around and look behind you, and find you are immediately outside the hypercube, free of all the component cubes.By analogy in 3-d, you were in the middle of a square, on a cube, and looking toward the center, seeming to be surrounded by all the other squares. But just turn around, and you are looking to the space outside the cube.Haa ha. If you want to have a fair viewing of higher dimensions, witin this limited, non-Pythagorean, pseudo-mathematical sense, based on all the Cartesian crap you are conditioned with, you have to learn to think by analogies!Or, you can forget all of that-- and begin to learn real mathematics. It's still a good mental exercise, though, for some people.;-)Digg, for cute story anyhow.
@biff198Yes that's basically true, as in the example of the tesseract and the hypercube....the number of vertices, as dimensionality increases, follows the progression 2,4,6,8,etc. There are many such progressions, regarding number of edges, faces, etc., easily discovered from a graph. ( the book "Flatland" (Edwin. Abbott) is a nice little story about all this. But what distresses me, and i wanted to post as a followup to my above rant, is that....one can also follow a series where the number of vertices, as dimensionality incereases, uses the sequence 1,2,3,4,5 etc. This , in fact, uses the simplest objects of each number of dimensions, the most trivial objects. Thus, the 1-d world ("Lineland") still has the line segment, with two vertices, or endpoints, as its defining points. (2 points, 1 'edge').But, as we move to two dimensions ( "Flatland"), the figure is defined by three points, and is the triangle, thee simplest 2-d regular figure.Moving into three dimensions,we simply , again, add ONE MORE point, and get the tetrahedron.(like a 4 faced 'pyramid', with triangular base)Thus, using the simplest. I don't know the name of it...i call it a hyper-hedron.This seems a much simpler series than the cubic series. to get the hyperhedron-equivalent of a tesseract.....jst take the tetraedron,and put a point at its center, inside it, and connect with the four vertices of the original figure....recognizing that this inner point is also a 'shell', surrounding the original figure, since the closest we cancome to that 'other' direction of 4-d is to say 'simultansously toward the center and the outside.' There will be a test on all this in ten minutes...but it will be in a 4-dimiensional classroom. You are already there, and are leaving, and have not yet arrived!Kurt Vonnegut was right!
I hate when all of you try to be smart, even not reading my article. The image has nothing to do with 4D cube, its just interesing gif animation Ive put next to my article... the math behind n-deimensional spaces and calculating and predicitng beahviours is course I study for my Master work at M.I.T. and this is by-the-way article aside I worte and added some kiddie stuff animations and projections and link sto stuff I found VISUALY interesting. The more you know something, the more you will realize you know nothing....It makes me mad when people come with kindergarden stuff like: "4th dimension is Time dude, blablabla" ... not managing to thikn a centimeter out of the box! ...or "4th dimension doesn't exist!! you are full of crap..."cool down and if you find this uninteresting pass it by end its the end of discussion...
mongooseFeb 25, 2006
Trippy.
tyninFeb 25, 2006
i enjoyed cube 3 the most, even if they made the 'bad guy' a little over board.
noodhoogFeb 25, 2006
Fascinating subject, but what a s**te little blog entry.. I won't even call it an article. Wikipedia has a much better article here: <a class="user" href="http://en.wikipedia.org/wiki/Hypercube">http://en.wikipedia.org/wiki/Hypercube</a>
waterdragonFeb 25, 2006
That s a really crappy representation of a 4-d hypercube. I know they probably followed a mathematical formula to get that projection, but still, what we are seeing is a 2-d representation of a 4-d object. That is like, by analogy, representing a cube (3-d) on a one dimensional graph. Do you know how that would look? It is simply two points, a certain distance apart, or a line segment. the space between the points represents the 3-space inside the cube, the part of the line outside the segment represents 3-space outside the cube. That is how CRAPPY this mapping is!If you want to get an idea of a 4-d cube, or a hypercube, there is a better way, though it is difficult to picture. Begin with the tesseract, which is a 3-d representation of the 4-d 'hypercube'Note that it looks like a cube inside another cube, with the corresponding corners connected by lines. This appears as eight 'sort-of' cubes, the inside one, the outside one, and the six 'transitional cubes, that have one face of the inner cube, and one face of the outer cube. we must allow that the other faces of each of these six 'transitional cubes' , though not appearing square, are ultimately square, if seen in an actual 4-dimensional space(were we can't go).The only other stipulation to make this a true hypercube is that....those eight cubes are arranged in such a way that they are all the same size, they don't share any intersecting space, and no one of them has a unique position tat you can say is 'inside' or outside' of the other ones. Like the square faces of a cube, they are equal, with not one standing out in a unique position as being inside or outside. Thus, as a square has 4 line segments with no unique positions, and a cube has 6 squares, the hypercube has 8 cubes.the eight cubes are the same size, and none is inside or outside another, or they ALL are inside or outside, so that not one is unique in position. Note also, that when looking at a cube (imagine one made of wire), there is a way to look at t so that one square face appears smaller than, and inside, another square face. e.g. look straight at a side, and note the opposite, parallel side --looks like a smaller square, inside it. in the same way, there is a way to look at a hypercube, so that one of the eight cubes appears to be smaller than, and inside another cube.That APPEARANCE is what we call the tesseract...a mere picture of a hypercube, from one direction. and ere, we are speaking of directions infinitely more than the directions contained in 3-space. Thus, there is another way of looking at the hypercube so that you feel you are inside a cube, surrounded by the other cubes, one of them being parallel to the one you are in. This is one of the creepiest views of the hypercube, as you seem to be completely entrapped. Of course, you can just turn around and look behind you, and find you are immediately outside the hypercube, free of all the component cubes.By analogy in 3-d, you were in the middle of a square, on a cube, and looking toward the center, seeming to be surrounded by all the other squares. But just turn around, and you are looking to the space outside the cube.Haa ha. If you want to have a fair viewing of higher dimensions, witin this limited, non-Pythagorean, pseudo-mathematical sense, based on all the Cartesian crap you are conditioned with, you have to learn to think by analogies!Or, you can forget all of that-- and begin to learn real mathematics. It's still a good mental exercise, though, for some people.;-)Digg, for cute story anyhow.
waterdragonFeb 25, 2006
@biff198Yes that's basically true, as in the example of the tesseract and the hypercube....the number of vertices, as dimensionality increases, follows the progression 2,4,6,8,etc. There are many such progressions, regarding number of edges, faces, etc., easily discovered from a graph. ( the book "Flatland" (Edwin. Abbott) is a nice little story about all this. But what distresses me, and i wanted to post as a followup to my above rant, is that....one can also follow a series where the number of vertices, as dimensionality incereases, uses the sequence 1,2,3,4,5 etc. This , in fact, uses the simplest objects of each number of dimensions, the most trivial objects. Thus, the 1-d world ("Lineland") still has the line segment, with two vertices, or endpoints, as its defining points. (2 points, 1 'edge').But, as we move to two dimensions ( "Flatland"), the figure is defined by three points, and is the triangle, thee simplest 2-d regular figure.Moving into three dimensions,we simply , again, add ONE MORE point, and get the tetrahedron.(like a 4 faced 'pyramid', with triangular base)Thus, using the simplest. I don't know the name of it...i call it a hyper-hedron.This seems a much simpler series than the cubic series. to get the hyperhedron-equivalent of a tesseract.....jst take the tetraedron,and put a point at its center, inside it, and connect with the four vertices of the original figure....recognizing that this inner point is also a 'shell', surrounding the original figure, since the closest we cancome to that 'other' direction of 4-d is to say 'simultansously toward the center and the outside.' There will be a test on all this in ten minutes...but it will be in a 4-dimiensional classroom. You are already there, and are leaving, and have not yet arrived!Kurt Vonnegut was right!
vurdlakFeb 26, 2006Submitter
I hate when all of you try to be smart, even not reading my article. The image has nothing to do with 4D cube, its just interesing gif animation Ive put next to my article... the math behind n-deimensional spaces and calculating and predicitng beahviours is course I study for my Master work at M.I.T. and this is by-the-way article aside I worte and added some kiddie stuff animations and projections and link sto stuff I found VISUALY interesting. The more you know something, the more you will realize you know nothing....It makes me mad when people come with kindergarden stuff like: "4th dimension is Time dude, blablabla" ... not managing to thikn a centimeter out of the box! ...or "4th dimension doesn't exist!! you are full of crap..."cool down and if you find this uninteresting pass it by end its the end of discussion...
osiriscky3Feb 26, 2006
the 4th dimension is time....all this is, is a nice illusion so no digg
raknightFeb 26, 2006
Total BS posted by a guy that can't spell the word "crook". Don't waste your time if you have 10 brain cells left.
stillboredJun 12, 2006
Did you just try to capitalize the number "3"?