
#19
Apr1311, 11:45 PM

P: 5





#20
Apr1411, 08:25 AM

P: 15,325

Also, does this twisted circle cross at a tangent or does it cross at 90 degrees? (I guess it must be the latter, since the circumference of the former is trivial.) 



#21
Apr1411, 10:35 AM

P: 5

Here's an animation I made a few years ago showing a series of these "twisted circles": link. Here you can see the transition from circle to square. I discovered it while playing with a 3D modeling program, hence my lack of a textbook explanation. There must be one though. I doubt this is something new. 



#22
Apr2811, 03:27 PM

P: 154

what about the quadray coordinate system? also ifound some information on 4D space by buckminster fuller. on a different note, 3D space only describes how it is viewed with our current coordinate system. if the same space is described with different coordinate systems,3D,4D or10D , doesnt that mean we are misunderstanding what a dimension is?




#23
Apr2811, 09:09 PM

P: 24

X, Y, Z are just one possible coordinate system and it's possible to define others e.g. Polar coordinates, although they will all have 3 components if you ar trying to describe 3d space without redundancy. Using more than 3 coordinates to describe a location in everyday/3D space means redundant information because there is only enough 'space' for 3 orthogonal coordinates, so any extra fourth coordinate direction could be represented as a linear combination of the other two.




#24
Apr2811, 10:35 PM

P: 65

DarkenSol: what you are thinking is actually kind of on the right track. See if you can understand the following:
The generalization of a triangle (2d) and a tetrahedron (3d) is the simplex: http://en.wikipedia.org/wiki/Simplex In ndimensional space, the nsimplex is the simplest polytope. It will always have n+1 vertices. Now, the nsimplex (with n+1 vertices) actually does define a coordinate system. Let v_{1}, v_{2}, ... v_{n+1} denote the vertices of some nsimplex. Consider ordered (n+1)tuples of nonnegative real numbers (a_{1},a_{2}, ... a_{n+1}) such that a_{1} + a_{2} + ... + a_{n+1} = 1. Then for every point p inside the simplex, there is a UNIQUE such (n+1)tuple such that p = a_{1}v_{1} + a_{2}v_{2} + ... a_{n+1}v_{n+1}. Now, there is one interpretation where these v_{1}, ... v_{n+1} are considered "vectors", but don't worry about that for now if you don't know what vectors are. It's just a way of expressing geometric ideas more algebraically. Really, these numbers a_{i} are just representing how much you are moving in the direction of the point v_{i}. If a_{i} = 1, then we are choosing a point as close to v_{i} as possible  namely v_{i} itself. If a_{i} = 0, then we are choosing a point far away from v_{i}  namely, one lying in the face opposite v_{i}. Just notice how we had the condition that a_{1} + a_{2} + ... + a_{n+1} = 1. This condition means that once we know a_{1}, a_{2}, ... a_{n}, we can determine a_{n+1} by a_{n+1} = 1  a_{1}  a_{2}  ...  a_{n}. In other words, we really only need n "numbers" to describe a point inside this nsimplex (in ndimensional space). But contrary to what DaveC is saying, we really DO need n+1 points to make sense of a coordinate system for ndimensional space. I'll give you the intuitive idea. Suppose we are looking at just a line (1dimensional space). Suppose I give you the task of being able to uniquely locate every point on this line from its number by some mechanical method. e.g., I give you the number "12.6" and you have to pick out the unique point on the line corresponding to that number. The problem is, every time you turn your back, I keep moving the line about, or scaling the line by 1/2. So your first problem is, "How do I know, each time I look at the line, that I am looking at the same section of the line?" You solve this by putting a big red "0" on the line, to mark the origin. Then if I shift the line left or right, you won't get confused. But this is not enough. If I were to rescale the line, you would have no idea where "12.6" is. Maybe it's very close to the origin, or maybe it's very far from the origin, depending on whether I scaled the line longer or shorter. But if you add another point and call it 1, then you will know exactly how much I've rescaled it. If 1 is half the distance to 0 as it was before, then you know that "12.6" is also half as far as it was from 0 as it was before. And this is all you need to know! For higher dimensions, you still need an origin, and then one point for each direction you can travel away from the origin. So overall, you need n+1 points when you count the origin. And if you think about it, this is really just the same as the simplex case, where you designate one of the vertices of the simplex as the origin. You always need n+1 points, but people just forget it when they do all this stuff with linear algebra. 



#25
Apr2911, 04:17 AM

P: 154





#26
Apr2911, 09:03 AM

P: 15,325





#27
Apr2911, 01:48 PM

P: 154

i was reading about descartes and how he figured out he could find pisition in a room by only 3 axes. what i couldnt understand was how then find position outside the room. i understand we assign negative values to the coordinates so it extends in all directions. we could have labled these a b c instead. then with a combination of all six directions find position anywhere in 3D space.




#29
Apr2911, 03:12 PM

P: 154

i am sorry i lack the background to convey what i am trying to say. i am saying that giving the coordinates a negative value is creating 3 more directions to travel in. traverse the x axis 5 units then move 5 more. you will always move away from 0. now move toward 0. you need a different direction to do this. i was pointing this movement could be labled as moving in the (a) direction just as easily as assigning a negative value to the (x) direction.




#30
Apr2911, 04:30 PM

P: 15,325

If you did that, it wouldn't make sense. To define a point in your space, you'd either 1] express redundant coordinates: x=1 (or a=1) y=47 (or b=+47) z= 2 (or c=2) or 2] express some coordinates as not having values at certain times (because they're mutually exclusive). x=1 y=undefined z=2 a=undefined b=47 c=undefined Either way, you can see that you're using more coordinates than necessary. 



#31
Apr2911, 05:06 PM

P: 65

Look, there's something ridiculous going on. There are standards to mathematics. These standards were established for reasons that most people have forgotten  some going back to the ancient Greeks.
DaveC, you are trying to uphold a particular standard, but really aren't providing any good reasons for it. DarkenSol: You really need to think about what is meant by "direction". If you start out at the origin, there are an infinite number of directions you could travel. (Up, down, left, right, forward, backwards, diagonally upright, diagonally downright, diagonally forwardright, etc...). We don't want to have to name all these directions individually, so we come up with a systematic way of naming them. This involves using axes x,y, and z. The idea is that any particular direction you might want to go can be described using only these three elementary directions. We don't want to give the direction halfway between up and left a special name. We can already describe it as x/2 + y/2. To give it an entirely new name would be redundant. Similarly, your directions a,b, and c are redundant, because we can describe them as x, y, and z. Why should we give new names to things we can already describe using x, y, and z? 



#32
Apr2911, 05:10 PM

P: 15,325





#33
Apr3011, 03:58 AM

P: 154

damn i was hoping someone would look into it and see if there was anything to it. when i was researching a tetrahedral coordinate system, i came across fuller, he apparently had the same idea. some kid in breckinridge, mn also thought of it. he asked for help developing the math processes to make it useful. while traversing links about it i come across fibiannaci numbers alot. and also i dont see alot of cubic structures in nature. i am curious if it was passed over for a reason or if it was overlooked. perhaps it would work, but we are already dependant on xyz. i just dont know enough yet to come to a conclusion myself.




#34
Apr3011, 08:23 AM

Mentor
P: 16,703





#35
Apr3011, 08:26 AM

P: 65

DarkenSol: Did you not read my post about simplexes (the generalization of tetrahedrons)?
It's been done before. Here's a wikipedia article on exactly what I told you about: http://en.wikipedia.org/wiki/Barycen..._(mathematics) The wikipedia article uses linear algebra, but you can define Barycentric coordinates using compass and straightedge constructions (I think). But I don't think anyone here is going to bother explaining the geometry to you. 



#36
Apr3011, 02:42 PM

P: 154




Register to reply 