Establishing Chirality For a 4D Person?

Question

The Ordians are 4-dimensional people with three legs and three arms spanning the sideways plane, with a vertical spine like ours holding their heads and shoulders above the ground and two eyes facing forward.

The idea of "left" vs. "right" doesn't make any sense to them; they can rotate their bodies to any arbitrary phase in the sideways plane without ever changing the direction in which they are facing.

Now, each individual Ordian can obviously tell the difference between all of their three arms, and use whichever one they want at a given time. And they probably have internal chirality, just like we do: there's no need for all of their organs to be arranged with radial symmetry. But when trying to communicate with others, where the disposition of internal organs isn't immediately apparent, how could they establish egocentric directions? I.e., how would an Ordian be able to communicate the equivalent of "to my left", meaning "in the direction of a particular one of my limbs which you can identify by name"?

Answer 1

Just because there's an extra dimension in the Ordians' spacetime doesn't mean that they can have no concept of relative directions and rotations.

Here on Earth, in 3 dimensions, we can have North, South, East, West, Up and Down, which are relative to Earth, or Forward, Backward, Left, Right, Up and Down, which are (mostly) relative to a person. We can also have one axis of rotation, for which we also have words: pitch (rotation around the left-right axis), yaw (rotation around the vertical axis) and so-on.

In 4 dimensions, we have the usual 3 spatial dimensions (x, y, z) plus one more: w. This gives 8 directions at 90° to one-another rather than the 6 we have. In four dimensions, it is also possible to have two completely independent axes of rotation.

So, Ordians are going to be able to define a vertical axis (say, ±z) and a direction at right-angles to that, from one shoulder to the other (call it ±w).

It's hard for a mere 3D creature like myself to say what the language of a 4D creature would be like, but it's highly likely that they'll have words for '+w' and '-w', as well as all the other possible cardinal directions and rotations.

The real difficulty for us comes with imagining them all and putting names to them.

Answer 2

Break their symmetry

The reason why we can look at a 3D person and immediately tell which of their sides is left, is that people cannot physically mirror themselves. With rotational symmetry, this doesn't work, because turning is possible. You can't tell if the 4D person you are looking at turned 120 degrees one way or the other just before you looked. You're just out of luck.

However, if they wear a red ribbon on one of their hands by custom, they can say "on my red side" and you will know. I think you can then extrapolate which side is "green" and which is "blue" without any more ribbons.

Change the paradigm

If you don't want to force any ribbons on them, just let them point directions relative to the surroundings rather than themselves. Sailors say "port" and "starboard" relative to their ship, no matter which way they face. Outside, you can say "east", "west", etc. Indoors, maybe some feng-shui based terms? I think it is reasonable they would develop a system like this when the first option is not naturally available.

Answer 3

You can add another coordinate name

Left and right in 3 space implies the x+ and x-. It could be any of the axes, but assume x for simplicity. Stretching your arms out would overlay them with the x axis. Now, if you raise them up, you imply an x component and a z+ component.

Your answer describes a hyper surface orthogonal to z. So on that plane (which to our 3d minds is an infinite cube), you would have to introduce another variable, let's call it j. The hyper surface would then vary along the x, y, and j axes. Their vernacular would create names for each: x is left and right, y is backwards and forwards, and j we'll make it up. Say weewards and wonwards. (Z is still up and down).

If the beings are four dimensional, I will assume that any description of direction will not be a single line which can be described by 2 points. In our 3d example being the origin, which is in this case your chest, and lets simplify your outstretched arm to a single point around your hand. Rather, it would be a surface in 4space. To describe a surface you need 3 points.

So your species would describe their surface "arms" in 4 space with their "chests" as the origin, but with the added j axis: left-weewards, left-wonwards, right-weewards, and right-wonwards.

Answer 4

Reducing the problem

Since up/down is set by the direction of gravity, we don't really have to worry about it; if an Ordian needs to look upwards 20deg, that's the same as on Earth. So the challenge is 3D, not 4D, which is good news for us 3D creatures. (If you've played 4d Golf you're familiar with how playing golf along flat ground in 4D is the same as putting in 3d with no gravity.)

The 4D horizon is a 3D sphere, the same way the 3D horizon is a 2D circle. Ordians have a "forwards" based on where they're looking, but they're radially symmetric along the other axis. So you might think of the problem as equivalent to situations like:

• Two astronauts can each see the direction the other is looking but not which way the other is oriented head/toe-wise (e.g. they're too far to see each other in detail, but they have laser pointers they can use to gesture with). How can they both orient themselves with the same "up"?
• Somebody gave me a globe with no latitude/longitude lines, but it does have countries labelled. Over the phone, how can tell me what direction is north, or tell me how to rotate the globe to find any particular country?
• I'm looking for Jupiter using a star chart, but I don't know which direction is north or what time it is.

Solutions

Alice and Bob are Ordians. She can see where Bob is looking, and where Bob is in relation to her, but not what Bob's internal sense of ana/kata/left/right is. She also knows Bob can look at her and at the direction she's looking.

"With me at 12-o'clock,"

Bob can keep facing forwards, but spin clockwise/anticlockwise in such a way that Alice is "12-o'clock" of him, and then use her to define the rest of the directions.

If Alice isn't visible, she can choose whatever other object to specify 12-o'clock, and if it isn't clear which way Bob is facing, she can pick two objects. E.g. if Alice is giving Bob directions, she might say "Alright, looking at the bank, with 12'oclock as the grocery store, now turn 90deg towards 5'oclock."

This is how we tend to learn the constellations, which is probably the closest we come to dealing with this problem regularly. To find Polaris, I find the Big Dipper and then I look in the direction pointed by the last two stars of the dipper.

"That way"

If you're trying to get somebody to look somewhere, or move in a certain direction, it's likely sufficient to point. If you need to point and indicate a 12-o'clock, you can point and then move your hand up and down to indicate an orientation.

Customary directions

We end up defining all kinds of 'customary' directions in day-to-day life. "3 blocks downtown of...", "windward", "downhill", and "towards the chemistry building" are all relative directions that, in a particular context, specify an absolute coordinate system. If you exist on more than a featureless void, there's likely tons of ways to establish a shared direction.

Answer 5

3-D creatures cannot see 4-D creatures, only their "imprint" on three dimensions.

Your aliens just show us what they want us to see. And to keep themselves away from the uncanny valley, they show us something closely related to humans.

To better illustrate this, think of 2-D creatures we are visiting. They live in a sheet of paper, for simplicity.

If you are looking at the paper without touching it, the 2-D creatures cannot perceive you. They only have X and Y axis, and cannot perceive or interact with the Z axis.

When you poke the sheet with your fingertip, they see a fingerprint on their paper-world. To them, you are the fingerprint. But you can poke the sheet with another finger, the side of your hand, your tongue, or anything, really.

(Just don't make it weird and use the wrong body part).

The 2-D creatures might even think you are several different creatures, because they can only recognize the part you are showing.

You could even move through their plane and show you the cross-section of your ankle.

So to the circlians, who are very round, we show them the tip of our pinkie, which closely resembles a circle.

To the half-moonians, we poke the 2-D plane with a fingernail, which is somewhat moon-shaped and narrow.

The 4-D aliens find whatever body part they have that resembles a human the closest, and that is what they put in our 3-D space. The rest of their bodies is safely out of our view.

Answer 6

They communicate directions relative to their face, not their arms.

The face is visible, well-defined, and a natural point of perspective for the individual in question. Humanoids can't rotate our heads far relative to our bodies, but if we could, I imagine we'd either have a similar convention or untwist our necks for clarity when referencing directions. Your creatures' rotationally symmetric body plan makes it even simpler; there's no conflicting information.

If they need to reference their hands specifically, not a point in space, they probably have a unique word for each arm relative to their organs. If someone absolutely needs to know which internally-referenced hand corresponds to one of the limbs they can see, they can wave it and say "this one"—outside of immediate spatial proximity, how often do we really need to specify one of our hands except in the context of dominance, injury, or another property not intrinsic to chirality?

I only question why they have two eyes instead of three for depth perception in all four dimensions.