Defining handedness, right-left, or clockwise-counterclockwise

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In summary: Maxwell's Equations in four dimensions can be represented by a single equation that looks superficially like this: ##\mathbf{E} = -abla_4 \cdotabla_x## However, there is a deeper level at which this equation is symmetric. Each term in the equation can be rewritten as a symmetric function of the other terms in the equation. The symmetry of the equation is preserved even when the order of the terms is changed. In summary, while there is no fundamental reason to distinguish "right-hand" from "left-hand" or clockwise from counterclockwise, two conventions for 4D coordinates can be chosen from, depending on whether
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fagricipni
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I'm wondering if there is a way to mathematically define these terms without essentially physically pointing at something.

I'm not even sure it is possible to do with physics, even if one assumes the matter dominance of the universe. One is tempted to think that one could define it electromagnetically: let the direction of the travel of an electron the same as the direction you are facing, call the direction that a magnetic field points up, and the electron will drift to the left. The problem is how does one define the direction of a magnetic field without using the terms right, left, clockwise, counterclockwise, north, or south. While east can be defined as the direction of the rotation of the Earth, there is no way to define north or south without the terms right, left, clockwise, or counterclockwise.

I was inspired to think of this by Sagan's Contact, where the alien culture has to define terms for humans in the radio message. So far, I have only thought of two ways they could make sure that we are both using the same handedness: use circular polarization in the radio carrier message itself, or make a map of the nearby stars and only using the correct convention will make them match the observed pattern.

There is another reason for wanting to know if there is a mathematical definition for right-handed and left-handed coordinate systems, extending the definition to other dimensional "spaces". In 2 dimensions, the Flatlanders would also that there are two possible conventions for coordinate axes that can not be translated and rotated on to each other; one would have to rotate one through the 3rd dimension to make one coincide with the other. Likewise, we 3-dimensional creatures have two conventions for coordinate axes that can not be translated or rotated on to the other; and again, if a 4th spacial dimension existed, a right-handed coordinate system could be rotated in to a left-handed one by rotating through the 4 dimension. Hypothetical 4-dimensional creatures, would find our distinctions of 3-dimensional coordinate systems to be useless, but have two conventions for 4-dimensional coordinates to choose from.
 
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Mathematically, there is no fundamental reason to distinguish "right-hand" from "left-hand" or clockwise from counterclockwise. Two non-parallel unit vectors, ##\hat{u_1}## and ##\hat{u_2}## can define one orientation as the direction of the smallest angle from ##\hat{u_1}## and ##\hat{u_2}##. Then there is the other orientation. Neither orientation needs to be called clockwise and there is no reason to prefer one over the other. Geometric Algebra develops those ideas very systematically in higher dimensions. A great deal of physics can be represented that way. Maxwell's Equations can be represented using Geometric Algebra in one deceptively simple-looking equation.
 
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fagricipni said:
While east can be defined as the direction of the rotation of the Earth, there is no way to define north or south without the terms right, left, clockwise, or counterclockwise.
I believe that this is correct, though (as usual) I don't see how to go about proving it. These concepts don't really exist in mathematics. One can arbitrarily say that positive is right and negative is left, but that's a convention of applied mathematics. Taking it further, up and down only have meaning relative to the surface of a planet.
 
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