Quote from Crooked Paul on December 4, 2007, 2:40 am

Okay, so I didn't find time today to put together those diagrams or a new explanation of 4-space, but I'm going to make up for it by introducing a concept that might blow your mind.

Chirality - An asymmetry property important in several branches of science. An object or a system is called chiral if it differs from its mirror image. A non-chiral object is called achiral (sometimes also amphichiral). A simpler (but negative) way to define chirality is: If an object is symmetrical along at least one axis, it is non-chiral. Otherwise it is chiral.
(from the Greek for "handedness," root word chir, hand)

Chirality is actually really useful in several applied sciences, notably protein folding and molecular biology.

The best example of this property is your own hands (and indeed, the word chiral means "like a hand," while non-chiral means "not like a hand"). There is no way you can rotate your right hand in 3-space to make it identical to your left, and vice versa. Therefore hands are said to be chiral (which is tautological).

Your hands are symmetrical relative to each other, which is a long way of saying "complementary," but they are not identical, and it would be gross if they were. It would be just so wrong to have a left hand in place of the right, even though it's a very very similar (complementary) shape. It would seem obviously wrong to any observer because the two hands would be identical where they are expected to be mirror opposites. The exact and only property that differentiates the shape of your (idealized) right hand from your left is its chirality.

Again, a concise way to say this is: An object is non-chiral if it is symmetrical to itself along at least one axis. If there is no axis of symmetry to the object, it is chiral. For 3D objects, if you can take a plane (a 2-space manifold) and bisect the object from any angle at all such that the left and right halves are complementary (mirror images), then it is non-chiral.

Some people really get their head around this concept with the aid of a kind of topological koan: No matter how you try, you cannot get a left glove to fit on your right hand.

That's because gloves are chiral. However, you may have seen those oven mitts with the thumb pocket in the center, designed to be used by either hand. Those oven mitts are non-chiral.

So, to take another example, a beer bottle is non-chiral. And so is a coffee mug. And so is a normal pitcher. (Because they look like identical shapes viewed head-on and when viewed in a mirror.) But have you seen those cream (or sometimes syrup) pitchers with the spout on the side relative to the handle? That's a chiral object. If you look at them in a mirror, they look complementary to the original object but not identical. Therefore they are chiral.

Any shape that is symmetrical to itself across any axis is non-chiral. This is because if it is symmetrical on any one axis, all you have to do is rotate it 180 degrees around that axis, and it has taken on the orientation it appears to adopt in a mirror.

One more example (now with 100% more personification!) about this to drive the point home. Anything that can be considered symmetrical is off the Chiral Allstars Non-Ambidextrous Baseball Team. Cut on the first day of practice: circle, ellipse, square, rhombus, rectangle, equilateral and isoceles triangles, and every regular polyhedron (pentagon, octagon, decagon...). But (non-rhombus, non-rectangle) parallelograms, which are hella chiral, are the starters at 1st base, 2nd base, 3rd base, and catcher (of course). Non-isoceles trapezoids are chiral; they're manning the outfield. Six-point pinwheel -- known to fans as "ninja star" -- is chiral (on his mother's side); he's the team's formidable lefty pitcher and team captain. Any triangle that is neither equilateral or isoceles is chiral, too. Right triangle is the shortstop.

These shapes are chiral, because no amount of rotating them relative to a mirror will make the object and its image identical. The simplest way to see this is with an actual triangle. Try this experiment: Get a bit of cardboard or some paper and cut yourself a nice right triangle that isn't also isoceles (it should look like a shallow ramp). Color one side black, leave the other white. Go look at it in a mirror. There's no way you can rotate it such that you wouldn't be able to distinguish the orignal from its mirror image, even and especially if you could "reach into" the mirror and handle both objects at once. The would look like bookends -- complementary -- not the same triangle.

Now, the really interesting property about chirality from a topological perspective is that chiral objects in n-space can be turned into their complementary versions if they are rotated in (n+1)-space. Right?

For a very simple example, let's return to the right triangle. No amount of Flatlander engineering can turn

|...
|...
|__

into

.../|
./..|
/__|

But for us 3-spacers, that's dead easy. You just grab it and flip it through the Z dimension, which doesn't exist in Flatland. You just performed a miracle for all those poor 2D motherfuckers.

In the same way, if a 4-spacer were to come here, she could easily grab, say, a Frank Lloyd Wright building and by rotating it around the one axis we can't perceive, turn it into its mirror opposite.

If a 4-spacer picked you up and rotated you through that fourth spatial dimension and then put you back down, your handedness would be reversed. Note that you would not be able to feel or detect the change while it was happening. You wouldn't be harmed or "turned inside out" or any nonsense like that. You're just being moved through a dimension you can't see. So, say you got flipped by a 4-spacer. If you are normally right-handed and this happened, you would be left-handed from then on, unless you could convince the being from 4-space to flip you back.

Think on that.

Many edits for clarity.