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

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Discussion Overview

The discussion revolves around the question of why three coordinates (x, y, z) are used to describe three-dimensional space, particularly in the context of a 3D object defined by four points, such as a tetrahedron. Participants explore concepts related to geometry, dimensionality, and coordinate systems.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the necessity of using three coordinates to describe 3D space when a tetrahedron has four points, suggesting that these points could define directions in 3D space.
  • Another participant clarifies that a 3D object has an infinite number of points and that three axes are required to define a point's location and movement in 3D space.
  • A participant attempts to relate the original question to a 2D case, questioning whether the reasoning holds in that context.
  • Some participants express confusion about the definitions of lines, planes, and the dimensionality of objects, stating that a line is defined by two points and a plane by three.
  • There is a discussion about the concept of affine bases, with one participant noting that while four points can define a 3D space, only three coordinates are needed to represent any point within that space.
  • Another participant argues that moving in one direction in a coordinate system affects movement in other directions, suggesting that the original reasoning may be flawed.
  • Some participants share their experiences of learning geometry, indicating that their understanding of dimensionality may have been misinformed.
  • There are mentions of alternative geometric shapes, such as octahedrons and twisted circles, and how they relate to the discussion of points and dimensions.

Areas of Agreement / Disagreement

Participants express a range of views, with some agreeing on the necessity of three coordinates for 3D space while others challenge the initial reasoning about the use of four points. The discussion remains unresolved with multiple competing views on the topic.

Contextual Notes

Participants highlight limitations in their understanding of geometric concepts, and there are references to personal educational experiences that may influence their reasoning.

  • #31
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?
 
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  • #32
klackity said:
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.
 
  • #33
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 a lot. 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 dependent on xyz. i just don't know enough yet to come to a conclusion myself.
 
  • #34
Darken-Sol said:
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 a lot. 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 dependent 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...
 
  • #35
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.
 
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  • #36
klackity said:
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.
 

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