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.

However, I think your example was illogical. Here's why:

Let's start with the 1st dimension and go upwards:

1st: A line. Length, but no width or depth. Can be flipped along it's length.

2nd: A shape. Length and Width. Can be flipped along the x or y axis.

Now what these two have in common, is that although they exist, their volume equals 0, therefore they can be "inverted".

Starting with the 3rd dimension, objects have volume and mass.

3rd: An object. Length, Width and Height. Can be flipped along the x, y or z axis.

4th: A freaky object. Length, Width, Height, and what on pictures seem to be shapes inside the original objects. Can be flipped along either axis, and also spin in some freaky way.

Okay, here's where I say you've stumped. First of all, us 3-dimensional beings can't "flip" any 2-dimensional shape, because we are unable to see the second dimension. Everything we see, have a Length, Width and Height above 0. Therefore, if we can't observe or interact with 2-dimensional shapes, "4-spacers" can't see us, or interact with us in any way.

Crooked Paul wrote:
something really long

you took the words right out of my mouth.

I was really hoping noone would quote that giant text!

Crooked Paul wrote:

Edit: Sorry, I couldn't resist. Edited now. (For everyone else: I posted a quote of my chirality post with no new content, for espen's benefit. )

espen180 wrote:
Okay, here's where I say you've stumped. First of all, us 3-dimensional beings can't "flip" any 2-dimensional shape, because we are unable to see the second dimension. Everything we see, have a Length, Width and Height above 0. Therefore, if we can't observe or interact with 2-dimensional shapes, "4-spacers" can't see us, or interact with us in any way.

Purely for the sake of example and illustration, we assume that awareness and interaction is possible between dimensions, or, at least, from higher to lower.

http://en.wikipedia.org/wiki/Fourth_dimension

To be honest, the fourth spacial dimension is pretty freaky.

hehe first slightly productive post outta 5 posts in this thread...

You make a lot of errors here. (No offense -- this is counterintuitive stuff, so that's natural.) Let's examine them in order.

espen180 wrote:
Let's start with the 1st dimension and go upwards:
1st: A line. Length, but no width or depth. Can be flipped along its length.

Actually, a line can be rotated around two axes. I'll explain. Start with a normal Cartesian coordinate system in Euclidean space (which is to say, a "normal" 3-space like our universe with a numbering/measuring system that starts at the origin (0,0,0) -- note the format (x, y, z,)). You're a fucking great mapper, so this is cake for you. This is what Hammer uses.

Okay, suppose you draw a line segment along the X axis (the horizontal one perpendicular to the viewer's POV). Now you can rotate it around the Z, axis, the one that extends up/down from the "table" of the XY plane. This would make, for instance, the right endpoint of the line move closer to the viewer while the left endpoint moved away. Another way to say this is it would rotate clockwise from the POV of any observer above our XY "tabletop."

Now the part you missed is this: You can also take that line segment you drew -- which coincides with the X axis (X, 0, 0) in our cartesian system -- and rotate it around the Y axis (the horizontal one that is parallel to the viewer's line of sight). So if you wanted to rotate it counter-clockwise, the left endpoint would fall as the right endpoint rises, again from the POV of an observer looking "forward" at the XZ plane (the vertical plane which is facing the viewer).

To put it in everyday language: If you hold a baton in your hand, you can decide whether to spin it horizontally or vertically or some mixture of the two. This is often called Pitch and Yaw, which is actually how we control our aim in FPS games like Portal. (We are essentially steering around the endpoint of our POV line, which is at our crosshairs.) Doom, which faked 3D but really wasn't, only let you yaw (rotate horizontally), not pitch (aim up and down -- it had cheesy autoaim for this).

It is EXTREMELY important to note here that rotating it around either axis does not deform the baton. Remember this.

Quote:
2nd: A shape. Length and Width. Can be flipped along the x or y axis.

Again, you missed one. A 2D shape, let's say in the plane of XY which is the surface of a table, can be rotated around all three axes of our 3-space system. You can rotate it clockwise or counterclockwise, leaving it on the surface of the table (rotation around the Z axis), or you can "flip" it front-to-back (rotation around the X axis), or you can "flip" it left-to-right (rotation around the Y axis).

These three kinds of rotation, along with the three possible directions of motion, are collectively referred to as six degrees of freedom. Just to reiterate, they are:

1. Movement along X axis.
2. Movement along Y axis.
3. Movement along Z axis.
4. Rotation around X axis. (Pitch.)
5. Rotation around Y axis. (Roll.)
6. Rotation around Z axis. (Yaw.)
   (Note that I'm calling the horizontal "tabletop" plane XY, to keep it consistent with previous examples.)

Again, let me use ALL CAPS here to stress that no matter how much moving and rotating of an object you do, it does not change its shape/topology.

Quote:
Now what these two have in common, is that although they exist, their volume equals 0, therefore they can be "inverted".

This is totally unclear. What you mean to say is this: If a line/shape/object is completely symmetrical across a given axis (or across a planar cross-section for 3D objects), no matter how you rotate it around a perpendicular axis, then rotation around that particular axis has no effect and can be ignored.

Examples of this: Spinning an ideal cylinder along its long axis, like a barber's pole. The pole stays in the same place as it rotates.

Try/visualize this: take a piece of paper and cut it into an ideal circle, coloring one side white and the other black, and set it on a table. No amount of rotating it (clockwise or counter) makes it look any different. Also, you can flip it front-to-back or left-to-right in order to see the different-colored side, but once the operation is completed it doesn't matter which axis you used. What you see is the same. And again, you haven't deformed the object, just changed its orientation.

Now, the best example is a perfect sphere. It is absolutely symmetrical across any plane you care to imagine through its center. No amount of rotating a sphere around any of our 3 axes will alter its appearance/shape/topology.

Quote:
4th: A freaky object. Length, Width, Height, and what on pictures seem to be shapes inside the original objects. Can be flipped along either axis, and also spin in some freaky way.

I think this quote contains the heart of your misapprehension. The 4th spatial dimension is not "freaky." You must try to think of it as completely boring and normal. Its relationship to all three of our axes is identical (perpendicular to all), the same way that the Z axis of our 3-space is perpendicular to both X and Y axes.

[Also, be careful saying "flipped" when you mean "rotated," because "flipped" implies that some change in the object's shape has occured, relative to some n-D observer. This is not always the case. Specifically, non-chiral (symmetrical) objects will not appear to change to us. Chiral (hand-like) objects will appear to change, but in fact their topology will remain identical to how it began. The object has just been rotated to a new orientation we don't understand, and it freaks us out.]

So 4-spacers can pick "up" (in the 4th spatial direction) 3D objects and rotate them around any axis including the fourth, which we'll call W and which is mutually and simultaneously perpendicular to axes X, Y, and Z. The W is no different from the 3 we can perceive, it is just one greater, the same way the number 4 is no different from 3 (both integers), just greater.

What would happen if you took a sphere and rotated it around the W axis? Nothing. Fucking nothing at all, because a sphere is completely symmetrical no matter how you slice it (as long as you go through the middle ). That is the essence of non-chirality.

But say you took some irregular shape, like a prism with one right angle in its triangular cross-section. If you rotate -- notice I'm using the verb rotate -- that shape around the W axis, our perception of its shape changes, while the object's shape itself does not.

This is exactly analogous to our miracle in Flatland, where we turned one right triangle into its complementary opposite, which amazed and terrified the 2D natives. From their point of view we turned the shape "inside out and backward." But we know we didn't change the shape/topology of the triangle at all in order to accomplish that feat. All we did was rotate it on an axis Flatlanders don't have.

Quote:
Okay, here's where I say you've stumped. First of all, us 3-dimensional beings can't "flip" any 2-dimensional shape, because we are unable to see the second dimension. Everything we see, have a Length, Width and Height above 0. Therefore, if we can't observe or interact with 2-dimensional shapes, "4-spacers" can't see us, or interact with us in any way.

This is semantic tomfoolery. I call shenanigans! Of course we can see the second dimension. The monitor you're looking at right now contains any number of 2D shapes. And of course we can rotate them any way we want.

Let's imagine that the plane of your monitor is sentient (not the whole machine, just the display). It is aware of what is being displayed on it at all times. Because it is a 2D plane, it understands 2D shapes very well, but it can't quite get a handle on 3D shapes.

Now, when you play Portal, your monitor is basically having an acid trip. For example: You're looking head-on at a square window in someone's Portal map. You turn like 30 degrees to your left.

The window is no longer square! It's a trapezoid. Because we're 3-spacers and we have seen this happen a million times per day every day since we were born, we understand that is only perspective in action, and we know that the window is still square. We don't freak out.

Your monitor Monty, however, is losing his shit in a major way. Why the hell did that square just change its shape?! How did it do that? Why? Squares and trapezoids are completely different and unrelated! Bscly: what the fuck?!**

Now we take the analogy up one dimension. We would freak out if we saw a Frank Lloyd Wright building turn into its complementary version, but to a 4-spacer this is totally normal and usual and goddamn boring, if you want to know the truth. All they did was rotate it 180 degrees. Chill out, 3-spacers.

Is this starting to make more sense to you?

It would be a billion times easier to understand if you included diagrams and/or illustrations.

espen180 wrote:
I'm clearing up, yes, but I kinda snuffed a little when you said my monitor display is a 2d object. It simulates 2d, yes. But in reality it consists of thousands of tiny 3d lights.

Actually it is 2D and simulates 3D. Your monitor cannot display holograms. If you think about it rigorously, you will see how unavoidably true this is. In "reality" (that is, 3-space), the monitor is a 3D object and so is its screen, and so are the liquid crystals (if LCD) or cathode-sensitive cells (if CRT) which display the image. But the image itself is just light rays interpreted by your eyes/brain, and in a very literal way the image is an idealized 2D object, because it has length and width but 0 depth.

Another way to drive this home: Look at this smilie image and tell me how deep it is. You cannot and can never perceive this depth, because it isn't there. You are conflating the depth of the display device with the depth of the image.

Yet another way to think about this: The retinas in our eyes are 3D objects. They are curved around the back surface of our (mostly) spherical eyes, plus they are made up of rods and cones that have distinct 3D shapes. But the image they send to the brain is 2D simulating 3D. We're just so used to the simulation that we take it for granted.

Quote:
It would be a billion times easier to understand if you included diagrams and/or illustrations.

Fair enough. I'll do some googling and post some links.

1. A scanned copy of the original Flatland novel in its entirety.
2. A scanned portion of Flatland with scholarly notes (preview section only).
3. Some other interesting links, including Flatterland: Like Flatland, Only More So.

Also it might be helpful just to read up on the terms:
topology
manifold
tesseract (aka hypercube, 4-cube) -- cool animations on this one!
penteract (5-cube) -- brace yourself
hypersphere (aka n-sphere)

That is all.

In my experience working with more than three dimensions isn't really that hard. As long as you don't try to make geometrical sense of it.

But of course that's what this thread and discussion is all about.

Soo...

What if portals weren't planar entities but 4-dimensional entities. Ie. connecting hype-volumes of space PLUS THE FREAKY FOURTH DIMENSION.

Meh...

Troll failed...

Rebooting!