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