If a 3D object has 4 points. why use x,y,z to describe 3D?

[Darken-Sol]

if the first 3D object has 4 points. why use x,y,z to describe 3D? am i confused about one or more concepts? does a tetrahedron skip 3D? a plane plus another point some where off that plane would immediately provide 4 directions to move. one for each point. could these directions not be used to describe all 3D space?

 

[Darken-Sol]

as of now 52 people have viewed this thread. not one could be bothered to give me the slightest hint. someone could atleast tell me these questions are goofy. it seems pretty clear i have no background in geometry. i just want to know why we use six directions instead of four, or if my reasoning is off.

 

[DaveC426913]

I think no one can understand your assumptions.

if the first 3D object has 4 points.

What does this mean? A 3D object has an infinite number of points. Even a 2D or 1D object has an infinite number of points.

It requires three axes of measurement define a point's location in 3D space: x, y and z.
It also requires three axes of measurement to define a direction movement within 3D space (plus another datum to define magnitude of movement.)

Or simply two sets of coordinates in 3D space to define a transpostition from one point to another. to wit: A point at x=1, y=-1, z=-2 is distinct from a point at x=0, y=0, z=0.

 

[micromass]

I really don't understand what you're trying to say... Maybe the best thing I can do is to translate your statement to the 2D case (where it also applies), I'll let you figure out where your misconception lies:

if the first 2D object has 3 points. why use x,y to describe 2D? am i confused about one or more concepts? does a triangle skip 2D? a line plus another point some where off that line would immediately provide 3 directions to move. one for each point. could these directions not be used to describe all 2D space?

In the above, I translated your statement to 2D. Do you still agree with the above quote?

 

[DaveC426913]

Upon rereading your post several times it sounds like you're using a coordinate system that defines a plane with 3 points, then a fourth point off that plane. That can be done but it is excessive. You have one more dimension that necessary.

Any point on your plane can be uniquely identified with two coordinates (let's call them x and y.), then you need only one more point (let's call it z) to define how far from that plane your point is.

 

[Darken-Sol]

i was under the impression a line was defined by two points and a plane three. way back in high school this was the case. i seem to have been misinformed. the first object with hight width and depth derived by points i can fathom is a tetrahedron. each point would mark a direction and moving in any direction u could map 3D. i apologise for my ignorance.

 

[micromass]

i was under the impression a line was defined by two points and a plane three. way back in high school this was the case. i seem to have been misinformed. the first object with hight width and depth derived by points i can fathom is a tetrahedron. each point would mark a direction and moving in any direction u could map 3D. i apologise for my ignorance.

Ah no, that makes perfect sense. Indeed, a line is defined by two points. However, you can only move in 2 directions. A plane is defined by three points, and a 3D space is defined by 4 points. However, the last statement doesn't make much sense in our intuition.

In fact, a n-dimensional space is defined by n+1 points. These n+1 points are called the affine basis of the space. However, it is a curious fact, that we only need n coordinates to represent any point in n-dimensional space. So for 3D-space, we need 4 points to describe the space, but we can give any point by just three coordinates. It's something you need to get used too. If you ever take linear algebra, then this will become very clear!

 

[DaveC426913]

i was under the impression a line was defined by two points and a plane three. way back in high school this was the case. i seem to have been misinformed.

You are correct. Who said otherwise?

the first object with hight width and depth derived by points i can fathom is a tetrahedron. each point would mark a direction and moving in any direction u could map 3D.

No. That's excessive.

Let's plot the points of a tetrahedron. A is the top, then B C and D are around the base.

I could move in the direction of A by, say, three feet, right? B, C and D stay at zero. But look what happens when I try to move in the direction of B: While I am moving away from the B apex, I'm actually heading somewhat downward. I find myself also moving a short distacne in the direction of -A. (To exaggerate: a movement of a mile in the B direction would carry me several hundred yards in the -A direction at the same time.) A single movement in one direction of your coordinate system overlaps with movement in other coordinates. I cannot, if I choose to, move only in the B direction.

So, to move in the B direction also affects my movement in the A direction. And that indicates that your system has more coordinates than it needs to have.


Contrarily, in a 3 coordinate system, I am free to move in any of the three directions while having absolutely no effect on the other two. I could travel 10 miles in direction X while my Y and Z movement remains perfectly zero.

 

[Darken-Sol]

i was hoping some one could set me straight. my mentor said he didn't know why we chose one over the other. he has me reason out most things before he fills in the gaps. i don't see how i could have come to that conclusion on my own. thank you

 

[Darken-Sol]

i can't get the multi quote to work, but i can move 3 ft in the A direction without changing B,C, or D. how is that different than moving ten miles in the x direction without changing y or z? barring distance. make a 3d movement say xn yn zn. to do this i would have to move An -Bn -Cn -Dn or something to that effect. it is just simpler to describe with x y z. that is what i get from your explination. i am just using an inefficient coordinate system.

 

[DaveC426913]

i can't get the multi quote to work,

1] Select [ Multiquote ] from different posts.
2] Finally, select [ Quote ] from (any) one of them.

but i can move 3 ft in the A direction without changing B,C, or D. how is that different than moving ten miles in the x direction without changing y or z?

No. Moving 3 feet in the A direction does move you a short distance in all of the B, C and D directions.

If you put tick marks along the B, C and D axes, and extend those tick marks both positive and negative, you will see that, as you move along the axis, your B coordinate is also changing. So is your C and D coordinate.

 

[circlemaker]

Maybe the "first 3D object" is 1/8th of an octahedron? Then you'd have your 4 points and x,y,z axis.

Although its a lot easier to work with a whole octahedron which looks like a square from any of its axis points.

FYI, if you take a twisted circle, duplicate and rotate 90 degrees around each axis, these twisted circles form a sort of "blueprint" with which you can derive the octahedron and cube. Where they intersect you can get more shapes. Its a sequence that keeps going. Within this sequence, the tetrahedron comes in pairs. It does not appear as a standalone shape.

 

[DaveC426913]

I've looked through every textbook I have and have not come across a official definition or diagram of a twisted circle.

 

[circlemaker]

I've looked through every textbook I have and have not come across a official definition or diagram of a twisted circle.

I don't work from textbooks.

Here's a picture: [PLAIN]http://www.perspectiveinfinity.com/images/bud_side_view.png

And here's another perspective showing the platonics and how they connect to it: http://www.perspectiveinfinity.com/images/platonics.png" .

The length of a twisted circle is (pi+(pi*2sqrt))/2.

 

[DaveC426913]

I don't work from textbooks.

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...

 

[Darken-Sol]

isnt a mobius strip a twisted circle?

 

[DaveC426913]

isnt a mobius strip a twisted circle?

That was one of my guesses.

I was giving circlemaker enough credit that, if he meant moebius strip, he would have said moebius strip. Still, it has nothing to do with the diagram he posted, nor can I see how a moebius strip could be applied to our discussion.

 

[micromass]

I'm also not aware of something called a twisted circle. I did hear of a twisted cubic, though. This is a very interesting curve in projective geometry. I doubt, however, that this is what is meant with twisted circle...

So, I welcome circlemaker to clarify his ideas!

 

[circlemaker]

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]

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?

[PLAIN]http://personalweb.donet.com/~paulrace/trains/primer/watchable/figure_8.gif
(I guess it must be the latter, since the circumference of the former is trivial.)

 

[circlemaker]

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...

http://img371.imageshack.us/img371/2909/originsplitzc2.th.gif

Here's an animation I made a few years ago showing a series of these "twisted circles": http://img296.imageshack.us/i/timewave150rv9.gif/" . 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]

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 , doesn't that mean we are misunderstanding what a dimension is?

 

[sfs01]

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]

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, ... v_n+1 denote the vertices of some n-simplex.

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

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

Now, there is one interpretation where these v₁, ... 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 a₁ + a₂ + ... + a_n+1 = 1. This condition means that once we know a₁, a₂, ... a_n, we can determine a_n+1 by a_n+1 = 1 - a₁ - a₂ - ... - a_n.

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]

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]

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]

i was reading about descartes and how he figured out he could find pisition in a room by only 3 axes. what i couldn't 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]

i was reading about descartes and how he figured out he could find pisition in a room by only 3 axes. what i couldn't 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]

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]

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]

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]

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]

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 don't see a lot 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 don't know enough yet to come to a conclusion myself.

 

[micromass]

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 don't see a lot 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 don't 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]

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/Barycentric_coordinate_system_(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]

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/Barycentric_coordinate_system_(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.