It's true: The directions left and right are really hard to define in terms of other spatial relationships.

Other directions aren't so tough. Take up and down, for instance. "Up" is away from the ground, out into space, the opposite of the direction gravity pulls in. "Down" is towards the Earth, away from space, and it's the direction things go in when they fall.

Or, consider forwards and backwards. "Forwards" is the direction your face points, the direction it's easiest to walk in, and the most comfortable direction to look in. "Backwards" is the direction your butt points in and when you walk a ways, the place you were earlier is in that direction.

What can we say about right and left? Well, right is the opposite of left, and left is the opposite of right. It's a wonder any of us remember which is which.

If you are facing North, left is to the West, right is to the East.

It requires a different set of parameters, but if all you're looking for is a definition, that works.

The way that anglophones can remember their "lefts" from their "rights" is to hold both hands out from the body, with the palm facing away, to close the hands into fists, then to lift the pointing finger(2) and the thumb(1) of each hand. Looking at these hands, only one forms the letter "L". That is the "L"eft hand.

For a majority, too, the right hand is the one with which we right. Rather, the write hand is the...

Also, it is possible, as long as we can remember that reading most anything in roman characters, to remember that one scans the page from the left to the right. In this sense, one can then remember that the smaller numbers (closer to negative infinity) are at the left, and the larger (closer to infinity) are at the right. At this point, one can allow the X-axis of the (in?)famous Cartesian plane to manifest itself in the mind. This allows any relative graph to have left and right as perpendicular to our established "ups" and "downs".

Further, the brain can be divided into left and right to define function. Politics can be divided into left and right to express ideology. You have many mnemonics. One not to use is the shoe; putting on the right (correct) shoe means that the other is left (behind).

Sinister and dexter are another story however.

In my fairly-well-reasoned opinion it's impossible to define left and right in a mathematical sense without invoking preexisting conventions that are typically described using those concepts. The statement

left = up x forward

(where x denotes the cross product, and "up" and "forward" are perpendicular but otherwise arbitrary) doesn't help. The cross product itself is typically defined as a vector operation in a right-handed coordinate system, which is usually defined visually: the y-axis points up, z points out of the page, and x points right. Alternatively, the cross product is defined using the "right hand rule" and a right-handed coordinate system is one where X x Y = Z if X, Y, and Z are the unit vectors. Either the cross product or the unit vectors are defined using the preexisting concepts of left and right.

Elementary particle physics evidently can help - in the standard model there are some things which can distinguish left from right. For more knowledge on this topic than I can furnish see pealco's writeup in Parity and Miles_Dirac's in Chirality.

More basic physics seems as though it should show a difference, but does not. The magnetic field, for instance, would define left and right, if only it was directly empirically observable: if a positive charge moves "up" then the magnetic field in the "forward" region points to the left. But the actual measurable value is not the field but the force on a particle in a field, and that force is always perpendicular to the magnetic field. The convention that says the magnetic field curls right-handedly about the direction of motion of a positive charge dictates that the force on another positive-charged particle is in the direction of the right-handed cross product of the velocity with the magnetic field: if you substituted left for right in this rule, the field direction would change but the force on the second particle would remain the same. The force is coplanar with the velocity of the first particle and therefore doesn't distinguish left from right.

Take this from the view of trying to tell a completely alien race what we look like. For instance, tell them that our left side of our bodies is symmetric to our right side. "What's Left? What's Right?" Well, if you face north, then your-- "What's north?"

The truth is that the only way we can differenciate left from right is by particle physics, as snol already mentioned. It has something to do with beta-decay.

However, the wierd thing is, that every particle has an anti-particle. When the two are combined, they anniliate each other and turn into pure energy. The other wierd thing is, everything about anti-particles is opposite of the regular particles.

When you explain to this alien about beta-decay and how left is different than right, if he carries out some experiments with beta-decay, he can know what we are talking about when we say "left." However, if his entire world is made of anti-particles, his "left" will end up being our "right". The particle physics only works if we assume both parties are using our "regular" particles, and not the anti-particles. (If his world is made of anti-particles, he wouldn't know it, they would be normal to him, and he would think we were the ones made of the anti-particles!) Which means, after explaining our left, and our right, and our cultures, if we ever rush out in space to meet each other and you put our your right hand to shake his hand, and he put's out his left hand instead of his right, you better watch out!

I only know this after reading some Richard P. Feynman lectures. Amazing guy that Feynman is...
In the February 5 issue of Physical Review Letters, it is reported that there's now experimental evidence of "handed" nuclei.

By shooting ions on certain elements, new nuclei were created. These had 75 neutrons, and 55, 57, 59 or 61 protons. The scientists measured the spin states for the nuclei, noting especially the energy and angular momentum of each state. The results showed doublets for the energy states - seemingly identical with the same angular momentum, results in two closely spaced energy states.

This is interpreted as a result of the nuclei spinning left-handed or right-handed.

###### reference: Scientific American, Physical Review Letters

When you have a space with anything more than one dimension, all senses of direction become a matter of perspective.

The only way to make a definitive statement about direction is if it is relative to yourself, and your perspective on the universe. People can, and usually do say something like "it's on my left", or "on your right."

People generally accept, incorrectly, that "Up" and "Down" are not able to be argued. Remember, that what is "Down" to people in the Arctic, is "Up" to people in the Antarctic. The reverse applies as well. It is implied earlier in this node that "Up" is away from the Earth. This is again, relative to individual perspective. "Up" at the Equator could be "left", "right", "forward", or "backward" to someone in the Arctic or the Antarctic, depending on their position.

And of course, there are the axes: X, Y, and Z. Unfortunately, these are again, relative to our planet. Since our planet is not a plane, X, Y, and Z can't be practically applied.

For the sake of argument, let's assume an alien ship is flying toward Earth. They are parallel to what we refer to as the Z axis (Up and Down, for the uninformed.) When they reach Earth, the front of their ship points at the Arctic.

Now, are they above us, pointing downward? Or are we ahead of them? Both, and neither. Each answer is correct and incorrect, depending on your perspective. If you are aboard the alien ship, this big blue ball is in front of you. If you are on Earth, this ship is pointed down, hovering above your planet.

Take a look at some works by M.C. Escher, especially his work titled "Relativity." Study it from lots of different angles, then try to decide which person is oriented correctly.

The book The Ambidextrous Universe by Martin Gardner discusses this in great detail. I put it here, because it is one of the first things he mentions to get you to think about how mirrors work. He convinces you that it's impossible to tell an alien which side is left/right and then tortures you by hinting that there's a solution to the problem, and then says there isn't, and then says there is, and then explains a little, and then ... He ended up saying something about the lines of what snol and kefabi and bigmouth_strikes said. Something about those lines because at that late in the book i was pretty sick and tired of particle physics. I would have to read the book about 2 more times to actually follow what on earth he was blathering about. In any case, if you want to start a really long conversation, mention this problem to some physicist at a party some time.

He also talks about what would happen if your parity was reversed. Bad Stuff. Drinking milk could kill you! Some stereo isomers (hopefully the word i mean) are actually parity-reversed versions of another substance. Often times rendering one distinguishing feature of a substance less apparent or potent in it's parity-reversed form. H.G. Wells also wrote a wonderful tale (called The Plattner Story) in his book 28 Science Fiction Stories. A chemist ends up in four-space and is thus able to turn himself around. (know how you can't rotate a 2 dimensional object front-to-back unless you move it in 3-space? Same idea). Martin Gardner liked this story, and so do I. The chemist ends up with his heart on his right side!

Interestingly, though right and left and forward and backward are reversed when looking in a standard bathroom mirror, up and down are not. Why is that?

Though the problem seems trivial, asking someone why up and down are so inherently different from right and left produces many strange looks and not a lot of comprehension. I have heard only two explanations (from exactly two people) that actually make a modicum of sense, though each carries with it many flaws.

• Up and down are defined relatively to the planet, whereas left and right are relative to the individual; that is to say, my right might well be your left whereas my up is always also your up. Therefore, when the mirror reverses the individual, it flips his right and left, but since its focus is not on the planet, it leaves up and down well enough alone.

• When a mirror is oriented vertically (as is the case with the standard bathroom variety), it reverses all directions save that of verticality. If one were to lay a mirror on the ground and gaze downward into it, one would behold the sky; thus, the mirror would have reversed down and up. Moreover, in such a position, the observer's right and left and forward and backward would remain in tact.

Obviously, there are flaws in both arguments. In the first explanation, how does the mirror know that we intend it only to flip the individual? Why does it not indiscriminately include the world and the stars in its irreverent flipping? In the second explanation, why are two axes (left-right and forward-backward) reversed when the mirror is vertical but only one axis (up-down) when the mirror is horizontal? Physicists will tell you all about the nature of light waves, incident angles with respect to normal vectors, and so forth, but it really misses the point. The dichotomy comes with our definitions of left and right, up and down, forward and backward, concepts which appear to be internally consistent yet are extremely difficult to explain articulately when presented with the above quandary.

Note that I do not attempt to explain why mirrors reverse (in the conventional sense of the word) left and right but not up and down; a very articulate node has already been written on that very subject: why mirrors reverse LEFT and RIGHT, but not UP and DOWN. The purpose of this write-up is to identify the flaws in the two most common explanations I am given; no more, no less.