Dinosaur pointed out that I do not know what a dimension is. So what is a dimension and what isn't. x,y,z and time are dimensions. yes? spin and weirdness are dimensions. yes? Are smell and colour dimensions? Can you convert 10 dimensions to 4 dimensions, how? ignore or cancel 6 dimensions? Can you convert 4 dimensions to 10 dimensions? how? Simultaneous equations? If a dimension is something you can put a mathematical quantity to then is everything dimensionalable?
Good question. I always thought that spatial dimensions above three were nothing more than mathematical abstracts and had no place in reality. Following from that question... What different "types" of dimensions have been postulated (or are universally accepted)? 1.) Spatial 2.) Temporal What else IS there?
From linear algebra The number of dimensions in a vector space is the number of linearly independant vectors required to span the space i.e. the set of vectors are able to generate any point on the space. Now this is a mathematical object, and in our orinary 3D world, we require 6 dimensions to describe the motion, or time evolution of a single particle, namely x,y,z(spacial) and px,py,pz(momentum in the x,y,z directions). 2 particles require a 12 dimensional "phase" space and N particles require a 6N dimensional phase space. However this is convenient only mathematically, the fact is that graphical interpretations of vectors end at 3 dimensions, and thus although we cannot visualise the solution to a problem in 4,5,6..dimensions on a piece of paper(not including time, which would be a series of pictures changing in time to depict the time evolution of hte system) we can still use the concept of higher dimensionality mathematical objects. Please don't try to visualise 4D objects, it's impossible. If any one tells you that they can visualise 4 spatial dimensions, ask them if you can smoke some of there stuff, because they are full of shit.Please Register or Log in to view the hidden image! Although Tom may try to convince you he can.
Ryans: You are correct: It is impossible to visualize 4D and higher dimensional objects. Many years ago, I spent a lot of time trying. I also encountered others who tried. You cannot get a perceptual image of even a simple 4D object like the corner of a tesseract. However, it is possible to develop formulae for volume, distances, curvatures, et cetera in higher dimensional spaces. For example: It is known that the longest diagonal of a unit hyper-cube is the SquareRoot(of the number of dimensions). In 4D, a unit hyper-cube has a 2-unit diagonal. In 9D it has a 3-unit diagonal. In 441 dimensions, it has a 21-unit diagonal. The inscribed hyper-sphere only encloses one unit of that 441D diagonal and is ten units from any vertex of the 441D hyper-cube. This indicates that the volume of the inscribed hyper-sphere is very small compared to the volume of the hyper-cube. There are some nice analogies which provide a little insight. There are known formulae for the volume of hyper-spheres well beyond 4D ones. Differential Geometry is a mathematical discipline which deals with the geometric properties of higher dimensional surfaces. While we cannot visualize much, we are far from ignorant about geometry in 4D and beyond.
The word 'dimension' can be used in several ways. There are the usual 'spatial' dimensions, of which there are 3 (on a macroscopic scale). However, think of a graph of, say, force F against extension e (for a spring). You wouldn't normally think of 'force' as a dimension but the graph is drawn on a plane surface and so the force and extension axes could be thought of as 'dimensions' in a more abstract sense. Each point (F,e) is a point in a 2-D space. Cheers, ron.
A profesor for whom I have a great deal of respect once told me the following: The 9th dimension looks flat to the 10th dimension... and so on and so forth. I liked that explanation.
But they have a LARGE place in understanding reality. For instance one of the best applications of "higher dimensional" problem solving that I'm aware of is the optimization of linear functions. If you can model a system from a set of linear variables (and the contraints on said variables) you can thusly employ a bunch of cool algorithms and stuff to find the optimal value of each variable to maximize or minimize the value of the model. Please Register or Log in to view the hidden image! Further, dimensions can be thought of as "degrees of freedom". In that context there are many applications in the "real world" that exceed the typical three dimensional perspective.
I think that dimension can be defined as "independent variable". The number of the independent vars makes the dimension of the space.
Thanks guys, now i'm really confused! I'll have to go and read some advanced maths to get an idea of what a 4D 'cube' is, I don't get what the 4th dimension is. Is it like simply replacing x,y,z with a,b,c,d. As for visualizing a 4D object, isn't that possible, if you can view a 3d object in 2d then surely you could view a 4D object in 3D, you would have to be within the 3d environment, maybe something like an IMAX planetarium. Can I conclude that there are different types of dimensions, geometric dimensions and quantative dimensions? One final question for the mathematicians, can you have curved vectors, that is in the 6D motion of a particle can you give the vector a curvature for each x,y,z and can you allocate functions to these dimensions to decribe complex curvatures eg closed spiral motions. I'm thinking maybe you can remove the particle and describe the curvature of space-time.
Kaduseus: In 2D, you can graph functions like x<sup>2</sup> + y<sup>2</sup> = r<sup>2</sup>, which is a circle of radius r. You can also graph a function like x = Distance<sup>2</sup> y = Distance The above is a parabola. The circle could also be defined as two functions of distance. These are called plane curves. In 3D, you can graph functions like x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = r<sup>2</sup>, which is a 2D curved space called a sphere. You can also graph a function like x = r*cos(Distance) y = r*sin(Distance) z = k*Distance) The above is a 3D curve called a circular helix. It looks like a spring or a shock absorber. In 4D, you can graph a function like x<sub>1</sub><sup>2</sup> + x<sub>2</sub><sup>2</sup> + x<sub>3</sub><sup>2</sup> + x<sub>4</sub><sup>2</sup> = r<sup>2</sup>, which is a 3D curved space called a 4D hyper sphere. You can also graph a 4D space curve using four functions of distance, or a 2D surface using 4 functions of two distance variables. Note that when you get beyond 3D, you start numbering the variables instead of using extra letters. In plane geometry, a curve has an important property called curvature. If the curvature can be expressed as a function, it indicates the radius of a circle with the same curvature. I think the curvature of a circle is 1/r. In 3D geometry, a curve can have another property called Torsion. Imagine a point moving along a 3D curve. At each point on the 3D curve, you can establish three lines perpendicular to each other. One line is directed along the tangent to the curve, and indicates the instantaneous direction of motion for the point. One line is directed toward the center of curvature, and can be proven to be perpendicular to the tangent line. For a circle, this second line is a radius from the center to the tangent point. These two lines determine a plane called the osculating plane. The third line is perpendicular to the other two, and obviously perpendicular to the osculating plane. Using calculus, it can be shown that as a point moves along the curve, the third perpendicular is the instantaneous axis of rotation for the tangent. It can also be shown that the torsion property is a measure indicating the instantaneous rotation of the third perpendicular about the tangent line. For a 2D or plane curve, the torsion is zero, indicating that the osculating plane is always the same plane (it contains the 2D curve), and that the third perpendicular does not rotate about the tangent as a point moves along the curve. The torsion indicates how much the curve deviates from a 2D or plane curve. It indicates how far the moving point will move from the current osculating plane for a fixed (small) amount of movement along the curve. You can think of each osculating plane for a 3D curve as containing a short segment of a plane curve which approximates the 3D curve. For a 2D curve, you have one osculating plane. For 4D, there is another property which indicates how much the curve deviates from a 3D curve. Note that the three perpendiculars described above establish a 3D coordinate system and define a 3D osculating space. For a 3D curve, that space is always the same 3D space. For a 4D curve, those three perpendiculars define a different 3D space as a point moves a short distance along the curve. This extra property of a 4D curve indicates how much the curve deviates from a 3D curve. It indicates how far the moving point will move from the current osculating 3D space for a fixed (small) amount of movement along the curve. You can think of each osculating 3D space for a 4D curve as containing a short segment of a 3D curve which approximates the 4D curve. Just as a 3D curve like a helix cannot be fit into a plane, a 4D curve cannot be fit into a 3D space. The above generalizes for higher dimensional spaces. I hope the above gives you some hints about how to start thinking about 4D & higher spaces. The mathematical details require concentration and effort, but the concepts are simple extensions of 2D and 3D concepts. Visualization is impossible beyond 3D.
Four-Cubes Here is a simple analogy to constructing higher dimensional objects. In this instance I will be referred to the cube. 0-Cube A 0-Cube is obviously a singularity, as is 0-spheres etc... This is because a 0-cube has no dimensions, ie. it does has zero length, zero breadth and zero height (in our 3D visualisation that is!) 1-Cube As we progressively add dimensions to our cube, we can systematically construct a 1-cube by drawing a 'line' between two 0-cubes. Thus, by this definition, a 1-cube is a straight line! (1 dimension) 2-Cube A 2-cube then would be two 1-cube's joined together by their ends by a staight line. And we all know this to be a square and we can draw it on paper but theoretically impossible to construct it with matter. 3-Cube Guess how we would construct a 3-cube. If you guessed (or perhaps you already knew) that it would be to draw two 2-cubes and join each of their vertices with their corresponding vertex. This is where it gets a little hard to work out. Remember all I am saying is how to 'draw' higher dimensional cubes on paper. That is, drawing n-dimensional objects on 2-dimensional paper! When you draw your 2-cube (square) label the vertices 00,01,10,11 There are 4 vertices on a square. Drawing two 2-cubes you would label the 2nd 2-cube the same way. Now join these points up (makes it easier if you orient the two squares the same way so you get a nice looking 3-cube). 4-cube Here is where it gets fun! Draw two 3-cubes and label the 8 vertices on each 3-cube 000,001,010,011,100,101,110,111. Now join these up with straight lines. (NOTE: for simplicity, do not draw the two 3-cubes exactly in line with each other. As you would do when constructing a 3-cube from 2 2-cubes.) When all 16 lines are drawn you have made a 4-cube (Or Hypercube) It looks weird doesn't it! That is because you essentially have 2 3-dimensional object in one, each trying to jump out at you. This is exactly what is happening when you draw a 3-cube on paper. Notice how you can look at it one way and see the hidden face in front, another way it is behind! Now imagine a 4-cube in 3-dimensions, totally impossible. There would be several 3-cubes trying to jump out at you and you would have a very hard time trying to work out how it could geometrically exist! (Especially if it is solid and not see-though) Anyway, now that you have a 4-dimensional cube drawn in front of you, you can go ahead and draw 5,6 or 7 dimensional cubes and impress your friends. Just remember, this is only a representation in a 2-plane and very, very different it real life (space-time). Cheers. Ben.
Click here to see oxymoron's explanation in action: http://www.geocities.com/CapeCanaveral/7997/hypercube.html
James R: The link you posted was interesting, but raised a question. It shows 3D slices of a 4D cube. Some of the slices look like 3-sided pyramids. I would have expected those pyramids to look like the corner of a 3D cube sitting on an equilateral triangle. They do not seem to have right angels at the apex. Is this due to the perspective of the drawing on a 2D surface, or am I missing something? I assume that some 3D slices would look like cubes.
Dinosaur, The angles need not be right angles. Think about a 3D cube sliced by a plane, and consider the 2D intersection of the cube and the plane. What kinds of 2D shapes can result? Well, you can get triangles, squares, rectangles, even hexagons, simply by slicing your cube in different ways. Now think about slicing a 4D cube by a 3D space. There are lots of ways to do it, and the results are the kinds of shapes you see on the site I linked to.
James R: The more complex 3D slices are a bit more than I can visualize. It seems to me that the simpler 3D slices should look like pieces of a 3D cube. The corner of a 3D cube looks like a pyramid with right angles at the apex. There should be some slice of a 4D cube which looks like that. Are there other pyramid-like slices which do not have right angles at the apex? I find it hard to imagine how the tesseract can be sliced to create such a piece.
So what do you call 4D space? getting my pencil out i did the euclidean point, line, plain ... thing. a point is 0D a line is 1D which is length a plane is 2d which is area a tetrahedron is a 3d object which is volume What property does a 4D object have, the best I can come up with is STRUCTURE? line has 2 points, plane has 3 lines, volume has 4 planes and the 4D object should have 5 volumes????? One weird thing I came up with was that the path of an object needs : n dimensions + (n-1)! attributes to plot the path, 3d needs 6 attributes - x,y,z,f(x,y),f(x,z),f(y,z) the f(a,b) functions would be vectors? 4D needs 10 attributes? the number of attributes is the same as the number of lines in the simplest case in each nD geometry. What would you call 5D space if 4D space is structure? 6 structures would be what? a mechanism? I know simply adding 1 point to a tetrahedron doesn't sound like the 4D geometry you are describing but some of the maths seem to work out, the simplex has 5 tetrahedral cells, 10 triangular faces, 10 edges, 5 vertices. Another question I tried to answer was if you use perspective to show 3d in a 2d plane what could you use to show 4D, it would have to be opacity wouldn't it?
many dimensions used in physics are considered *collapsed* so anything travelling in these dimensions loop around in an endless cycle, its because of these extra dimensions particles have charge, etc... 3d wouldnt seem flat to these collapsed dimensions, because they are point like themselves.
you should read Flatland. It basically explains that it is hard for a body living in 2 dimensions to visualize the third(z) dimension. This relates to how a 3 dimensional creature(human) has a hard time visualizing a 4th dimension. I guess i just told you the meaning of the book, however it is still a good read.
Reminds me of something I read when I was younger: Imagine 2D people living on the surface of a 3D sphere. They discovered trigonometry and found 3 angles = 180 deg. But later on their equipment get more accurate and they measured bigger triangles. They found the 3 angles > 180 deg. They could come up with 3D in their math, but never really be able to understand the actual 3D nature of their 2D universe.
An ant crawling around on a piece of rope thinks he's on a 2-dimensional plane. An outside observer can readily see the piece of rope has three dimensions. your computer screen is a 2-dimensional space in a 4-dimensional world. We can't comprehend dimensions other than the ones we live in. It's hard to imagine a 1-dimensional object. How can you have length with no width?
