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Why x,y,z?

by Darken-Sol
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circlemaker
#19
Apr13-11, 11:45 PM
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Quote Quote by DaveC426913 View Post
That is apparent. Leading to being unable to communicate your ideas to others.

You've used a phrase of your own making without describing what it means. No one knows what a twisted circle is except you.

I still don't know. Your diagram seems to show four circles mapped onto a square.

Or, I suppose if I get imaginative, it could be two figure eights. A figure eight could be a twisted circle I suppose...
Lol, sorry should have clarified. Twisted circle is my shorthand for a circle twisted 180 degrees. It looks like 2 circles from one angle, and a square from another. I might've said figure-8 but I've seen different geometries represent that.
DaveC426913
#20
Apr14-11, 08:25 AM
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Quote Quote by circlemaker View Post
Lol, sorry should have clarified. Twisted circle is my shorthand for a circle twisted 180 degrees. It looks like 2 circles from one angle, and a square from another. I might've said figure-8 but I've seen different geometries represent that.
Frankly, I still don't follow. I can see 'circle twisted 180 degrees' being a figure 8, but I don't get the 'looks like a square from one angle'.

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.)
circlemaker
#21
Apr14-11, 10:35 AM
P: 5
Quote Quote by DaveC426913 View Post
Frankly, I still don't follow. I can see 'circle twisted 180 degrees' being a figure 8, but I don't get the 'looks like a square from one angle'.
There must be a "standard" name for this shape...



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.
Darken-Sol
#22
Apr28-11, 03:27 PM
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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?
sfs01
#23
Apr28-11, 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.
klackity
#24
Apr28-11, 10:35 PM
P: 65
Darken-Sol: 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 n-dimensional space, the n-simplex is the simplest polytope. It will always have n+1 vertices.

Now, the n-simplex (with n+1 vertices) actually does define a coordinate system. Let v1, v2, ... vn+1 denote the vertices of some n-simplex.

Consider ordered (n+1)-tuples of non-negative real numbers (a1,a2, ... an+1) such that a1 + a2 + ... + an+1 = 1.

Then for every point p inside the simplex, there is a UNIQUE such (n+1)-tuple such that p = a1v1 + a2v2 + ... an+1vn+1.

Now, there is one interpretation where these v1, ... vn+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 ai are just representing how much you are moving in the direction of the point vi. If ai = 1, then we are choosing a point as close to vi as possible - namely vi itself. If ai = 0, then we are choosing a point far away from vi - namely, one lying in the face opposite vi.

Just notice how we had the condition that a1 + a2 + ... + an+1 = 1. This condition means that once we know a1, a2, ... an, we can determine an+1 by an+1 = 1 - a1 - a2 - ... - an.

In other words, we really only need n "numbers" to describe a point inside this n-simplex (in n-dimensional space).

But contrary to what DaveC is saying, we really DO need n+1 points to make sense of a coordinate system for n-dimensional space.

I'll give you the intuitive idea. Suppose we are looking at just a line (1-dimensional 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.
Darken-Sol
#25
Apr29-11, 04:17 AM
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Quote Quote by sfs01 View Post
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.
with x y z you need also -x -y -z. unless 3D space begins in a corner somwhere. just to prove a point you could lable -x -y -z as a b c.
DaveC426913
#26
Apr29-11, 09:03 AM
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Quote Quote by Darken-Sol View Post
with x y z you need also -x -y -z.
No. x y and z extend infinitely in both directions. They can have negative values.
Darken-Sol
#27
Apr29-11, 01:48 PM
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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.
DaveC426913
#28
Apr29-11, 02:21 PM
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Quote Quote by Darken-Sol View Post
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.
??

Wait. Why do you need more than 3?
Darken-Sol
#29
Apr29-11, 03:12 PM
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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.
DaveC426913
#30
Apr29-11, 04:30 PM
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Quote Quote by Darken-Sol View Post
i am saying that giving the coordinates a negative value is creating 3 more directions to travel in.
No it isn't.


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.
klackity
#31
Apr29-11, 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.

Darken-Sol: 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 up-right, diagonally down-right, diagonally forward-right, 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 half-way 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?
DaveC426913
#32
Apr29-11, 05:10 PM
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Quote Quote by klackity View Post
DaveC, you are trying to uphold a particular standard, but really aren't providing any good reasons for it.
The only standard I'm trying to uphold is using the minimum number of properties to uniquely describe something. When using more than the minimum number, you get redundant or conflicting results. Other than that, I'm good.
Darken-Sol
#33
Apr30-11, 03:58 AM
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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.
micromass
#34
Apr30-11, 08:23 AM
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Quote Quote by Darken-Sol View Post
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.
Uuh, can you give a reference of this? I searched for tetrahedral coordinate systems and I found nothing interesting...
klackity
#35
Apr30-11, 08:26 AM
P: 65
Darken-Sol: 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.
Darken-Sol
#36
Apr30-11, 02:42 PM
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Quote Quote by klackity View Post
Darken-Sol: 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.
i checked out a few links and bookmarked them. i only get about four hours a day to study, but now that its the weekend i'll go over them. this last one seems to be exactly what i was liiking for. i just read the first couple paragraphs then came back to express my appreciation for your time. thanks for the refocus.


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