Relation between coordinate axes labels and chirality?

Question

I have a very basic question. Assume I want to setup a 3D orthogonal coordinate system - I choose an origin, and move on to pick 2 orthogonal vectors as the first two axes.

For the third axis, I'm left with two options - two opposite vectors. This is where we decide whether we want a right-handed or left-handed coordinate system. Now I haven't labeled my axes yet - so I don't know which one is x, y or z.

Most definitions of a right-hand coordinate system assume a labelling for axes which makes the definition confusing. Can you define the chirality of a coordinate system without using specific labels for the axes?

Answer 1

The following is my understanding as a physicist -- definitely not a mathematician! -- who likes to deal with problems in varying numbers of dimensions.

The short answer is No, you cannot: the handedness is determined by how you choose or label the axes.

(The rest of this answer was heavily edited on 2018-09-06, to correct factual error pointed out by mr_e_man in a comment.)

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Handedness is in my understanding an arbitrary convention codified by the right-hand rule: Using your right hand, label your extended first/pointer finger $x$, your middle finger extended at right angles to it $y$, and your extended thumb at right angles to both previous fingers $z$ (curling the rest of your fingers to a fist makes this easier), you have the 3D (and if you ignore your thumb, 2D) coordinate axis labeling that is called a right-hand coordinate system.

Essentially, chirality and handedness convey the exact same idea: an object or a set of points (or more generally, any algebraic set with distances between all members defined, i.e. any metric space, Euclidean or not) that can be mirrored (an odd number of times) so that no rotation and translation alone makes it identical to the original one, is chiral or "handed".

Furthermore, when something is chiral or "handed", there are exactly two mirror configurations. (That was surprising to me, and where I originally erred. I had to actually examine the 2D, 3D, and 4D integer unit basis vectors (i.e., with all Cartesian coordinates being $0$, $+1$, or $-1$) using a simple program to convince myself! For reference, there are 384 such orientations in 4D; 192 "right-handed" and 192 "left-handed" (that is, if "right-handed" and "left-handed" were defined for 4D, which I do not believe they are). Mirroring any two or all four axes in 4D is simply a rotation; mirroring any two axes in 2D and 3D is also a rotation.)

The difference between chirality and handedness, in my opinion, is that handedness can be understood to imply there is a recognizable or agreed-upon labeling for the "right-handed" configuration, the other being therefore the "left-handed" one; whereas chirality is completely neutral.

For further information, I warmly recommend Michel Petitjean's Symmetry, Chirality, Symmetry Measures and Chirality Measures: General Definitions at http://petitjeanmichel.free.fr/itoweb.petitjean.symmetry.html site (and his Chirality in Metric Spaces article).

Answer 2

Any statement (as i.e. a 3d computer graphic setup, if I understand your perspective correctly) regarding a specific space/specific objects in a space needs a coordinate system to be implemented.

Statements about chirality of objects, while dealing with properties partially indipendent of the choice of a coordinate system, still need a coordinate system to be implemented.

Without requiring verbose and conventional axioms regarding labels, we use the "hand convention" in an attempt to simplify this choice of a coordinate system where we require certain chirality properties to hold.

Answer 3

Speaking in general, there’s no intrinsic mathematical property that identifies one orientation as “positive” (“right-handed”) or “negative” (“left-handed”). The most that you can say is that there are two distinguishable orientations, that each basis has (belongs to) only one orientation and that change-of-basis transformations can be classed as either orientation-preserving or orientation-changing. Which orientation you choose to call positive is otherwise arbitrary.