WHY there is no scalar equation for a line in three dimensions ?

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Discussion Overview

The discussion centers around the question of why there is no scalar equation for a line in three-dimensional space. Participants explore the implications of dimensionality and the nature of equations in relation to geometric representations.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the clarity of the original question, suggesting it may need rephrasing for better understanding.
  • Another participant humorously responds to the question with a rhetorical analogy about the length of a piece of string.
  • A participant presents a mathematical perspective, stating that a line in three dimensions requires losing two dimensions, and explains that a single scalar equation would describe a two-dimensional surface instead of a one-dimensional line.
  • Further contributions reiterate the humorous notion that all pieces of string have a length of forty-two, indicating a playful tone in parts of the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the question and the mathematical reasoning behind dimensionality. There is no consensus on the original question's clarity or the implications of the mathematical arguments presented.

Contextual Notes

The discussion includes informal responses and humor, which may affect the seriousness of the mathematical arguments. The mathematical claims rely on assumptions about dimensionality and the nature of equations that are not fully explored or agreed upon.

Rosicky08
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WHY there is no scalar equation for a line in three dimensions ?

:confused:
 
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Perhaps you might like to try rephrasing that in a manner which is understandable.
 
How long is a piece of string?
 
Rosicky08 said:
:confused:
There is

ds2 = \eta_{\alpha\beta} dx\alpha dx\beta

Pete
 
Last edited:
christianjb said:
How long is a piece of string?

Forty two.
 
masudr said:
Forty two.

All pieces of string are of length 42, just in different units.
 
Rosicky08, R3 has three dimensions. A curve has one dimension so you have to "lose" 2 dimensions. If you had a single, scalar equation, in three variables, you could solve for anyone in terms of the other TWO. That gives you "two degrees of freedom": pick any two coordinates and solve for the third: that one equation allows you to "lose" on dimension. Any single scalar equation describes a two dimensional surface. In order to get one dimension you must have either 2 equations in the three space coordinates so that you could solve for two variables in terms of 1 or 3 equations in the 3 space coordinates and one parameter: 3 equations in 4 variables gives you 4-3= 1 degree of freedom or one dimension.
 
masudr said:
Forty two.
Well said. :smile:
 

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