Wint and Zant

[Hugh]

The directions "Wint" and "Zant" are described in the glossary on this site.

wint adverb [Jonathan Bowers] - One of the two extra turning directions of a tetronian object. In tetraspace, represented by the vector <0,0,0,1>. In realmspace there are two turning directions, left and right, but in tetraspace, there are two more. The four tetronian turning directions are located in a particular order depending on which direction you look at them. If you look at an object containing these directions and you look from the back of the object towards the front, then when you traverse them in a clockwise direction starting from the left, the order is left, wint, right, zant. Thus, wint is 90 degrees clockwise from left and 90 degrees counter-clockwise from right. Zant is the opposite: 90 degrees counter-clockwise from left and 90 degrees clockwise from right. See the chart under direction.

Just wondering how others have tried to understand and visualize this, while at the same time visualizing the up/down and forward/backward axes as well.

[wendy]

Want and Zint are much harder to stabilise, since there is no natural grain to set them. Instead, understand that Left/Right is a 2d thing.

W

[Hugh]

Just trying to visualize things from a 4d perspective, all at once. Even though left/right is a 2d thing, they are still 4d turning directions, as are wint/zant. Do these end up more as a 3d thing, on the 3d surface?

The directions of up/down and forward/backward are easy for us to visualize, wouldn't they be similar in a 4d perspective as well?

[wendy]

If you look at a picture, there is an up/down vs across. That is the usual nature of pictures. A thing in the top of a picture would fall to the bottom.

If you look at a map, there is just across in two directions. A thing at the top of the map does not fall down to the bottom.

The up/down and forward/backward have gravity, and time to lend a hand to going one way. You can clearly do not see people walking upside down, or back to front, for confusing these directions. We all stand up, and we all walk forward.

Left/right, and Wint/Zant do not have this natural grain. People do get lost turning left when they ought turn right, and there is no grain in four dimensions that suggest "this is left" in the way that "this is up".

Even if there were four hands where we have two, and up and forward become the same, there is no reason why my left ought be yours. It could be your right, or your zant.

Consider this.

Lay some clocks on the floor, face up. What you have, now is a map, showing a number of people, facing forward (where forward maps to up). Do the 12'oclocks have to point the same way? No.

So if 3, 6, 9, 12 are left, wint, right, zant respectively, you can easily preserve up/forward, and still have no common reason to make every left point the same way.

In short, left/right/zant/wint are relatively meaningless in 4d, because nothing forces the hand. The only grain is "clockwise" vs "anticlockwise".

W

[Hugh]

you can easily preserve up/forward, and still have no common reason to make every left point the same way. In short, left/right/zant/wint are relatively meaningless in 4d, because nothing forces the hand.

Okay, but still, if one is putting together 4 perpendicular axes in 4d space, there would be the up/down(y), forward/backward(z), left/right(x) and wint/zant(w) ones right? Couldn't a tetronian line up their body so that each axis had a general direction relative to their body shape?

[wendy]

Not really. Why, really, should they?

[Hugh]

Well, yes. I keep hearing that in 4d, one would be able to put 4 straight axes perpendicular to each other, isn't this right?

So then a 4d being should be able to position itself to those axes so that its front/back is aligned with z, its up/down is aligned with y, its left/right is aligned with x, and its wint/zant is aligned with w, shouldn't it?

[moonlord]

I keep hearing that in 4d, one would be able to put 4 straight axes perpendicular to each other.

This is the definition of 4D.

[Hugh]

This is the definition of 4D.

Yes, moonlord, but do you see how what Wendy is saying doesn't quite match up to this? I'd like to understand why.

Also, I'm wondering how things would look and feel from the viewpoint of a tetronian. I'm trying to get a sense of this. How it all fits together, and how a spherical viewpoint differs from a tetraspherical one.

[moonlord]

Because the rotations occur in 4D around planes, defining a rotation will allow the directions not involved to freely rotate around a point in the plane they define. This means you can never actually know where you're heading to.

[Hugh]

Because the rotations occur in 4D around planes, defining a rotation will allow the directions not involved to freely rotate around a point in the plane they define. This means you can never actually know where you're heading to.

Wow. I wish I could fully understand this.

Could you go into more detail about how this would affect a 4d beings life?

[moonlord]

View quickfur's introduction. That's how I understood it. The main idea is to consider rotations as taking place IN A PLANE, and not AROUND something.

[thigle]

hugh, did you read this thread on rotations ?
glomar (glomic ?) rotation & spins of spins in general

[Hugh]

hugh, did you read this thread on rotations ?
glomar (glomic ?) rotation & spins of spins in general

Yes I did thigle, but there are many concepts within it that are far beyond my comprehension.

What do you think with regards to if a 4d being could line its body sections up with four perpendicular axes?

View quickfur's introduction. That's how I understood it. The main idea is to consider rotations as taking place IN A PLANE, and not AROUND something.

Quickfur's 4d visualization document is a very informative piece of work. I've read through it several times. The step-by-step process he takes is really helpful. The only problem for me though, is taking that last step from seeing shifting, distorted 3d cubes to seeing what it's like to be in 4d space as a whole. To put my consciousness into 4 spatial dimensions and explore around with a 4d body.

Moonlord, when you say "this means you can never actually know where you're heading to", are you saying that a 4d being could not have a directional sense due to the geometry of the space? (I'd ask how a 4d being with only a 3d directional sense would sense 4d space but that's another thread. )

[moonlord]

They can have a sense of orientation but not a relative one. My left is not necessary your left, even if we're looking in the same direction. It's somewhat like orienting in space. You can both look in the same direction, but your left is not necessary his left. It can be his up, his right or his down aswell.

[Hugh]

Okay, I understand, maybe this is what Wendy was saying too, that directions can be relative to one's own body. Two people standing beside each other may only share the same "up" direction. Two people standing on opposite sides of the earth don't even share the same "up". But relatively speaking, one could say to the other "turn to your right", or "look forward" or "look down" and the other would know which direction is meant relative to their own body. In the same way, wouldn't a tetronian be able to say to another "turn to your wint" and be understood? (assuming that tetronians have a 4d sense of direction :wink: ).

Wint is "90 degrees clockwise from left" in 4d. So if a tetronian knows where its left is, he turns 90 degrees clockwise from there to get to wint. For our sense of direction, that position is already occupied by our "forward". For a tetronian the order of 90 degree clockwise turns are "left, wint, right, zant". For us its left, forward, right, back. Complicating this is the fact that 4d rotations take place through a plane, not around an axis. So I'm trying to picture being in 4d space, and turning through the planes to rotate around...

Seems like you'd have a lot of common planes in that space, that could be seen from many different directions as you rotated through them.

[wendy]

Note that wint/left/zant/right is not the same as our left/forward/right/backwards. This same circle exists in 4d as well, too.

What happens, is that we see perpendicular to up/down front/back, a line. We call one end, left, the other right.

In 4d, the perpendicular to up/down front/back is a circle.

There is no way you can call a point on this circle 'left' or 'zant' like you can have up/down and forward/backwards.

The sole effect you can do is to give an angle between things, eg 30 clockwise (right) from x. There is no intrinsic left/right or zant/wint.

W

[moonlord]

In 4d, the perpendicular to up/down front/back is a circle.

Really? I could swear it's a plane...

[bo198214]

In 4d, the perpendicular to up/down front/back is a circle.

Really? I could swear it's a plane...

The vectors of length 1 from the same origin (i.e. the directions (of the points of the plane)) form a circle.

[moonlord]

Oh, I see. So, in fact, in 3D, what is perpendicular to front/back and up/down is represented by two points (or the segment between them). Now I get it.

By the way, how DO you handle rotations in 4D? Octonions? Tensors?

[bo198214]

Yes, two points is the zero-dimensional Sphere.
A circle is the one-dimensional Sphere.

You have rotation plane, say (x,y). Then the (left) rotation (around 0) is the same as in a plane and z,w remains unchanged. I.e.
x' = cos(a) x - sin(a) y
y' = sin(a) x + cos(a) y

If you want to have an arbitrary rotation in a by two orthonormal vectors a,b given plane then you simply transform this plane into the x,y-plane make the rotation and transform it back. Or in other words for each point p you (parallely) project p onto the plane (a,b) resulting in q = p_aa + p_bb. Then you know that p-q is perpendicular to (a,b) and hence remains unchanged by the rotation in a,b. Then you do the rotation with the above formula (where x=p_a and y=p_b) yielding q'=p_a'a+p_b'b and then p-q+q' is the rotated point p.

[moonlord]

Nice. Wouldn't have thought of that.

[wendy]

Four dimensional rotation is done by simply watching the things spin. Is there any other way?

W

[bo198214]

There is no way you can call a point on this circle 'left' or 'zant' like you can have up/down and forward/backwards.

Oh, I see ways.
Usually we distinguish the direction of a body by its asymmetries. up/down is a bit of exception to it, because it is not relative to the body, but a direction in the world (determined by gravity) like north/south (determined by magnetism).
Forward/backward has simply to do with the asymmetry of moving, or the looking direction. If our back-front were mirrored (for example we had one eye at the front and one eye on the back, and equally often going forward and backward) we had the same difficulties discerning forward/backward as we now have with left and right.

So if I put up/down in the same scheme I would define (the personal) up/down by the location of head and feet (because they are distinguishable). Left and right can be distinguished by slighter asymmetries as with the inner organs. (Though I know that there are also mirrored people).

If we assume a 4d being's body having enough assymetries we can assign those directions. (Has onybody btw already made a model of the hypothetical tetronian?!) I would suggest:

• up/down via head/foot which usually should be aligned with gravity
• forward is the looking direction, backward opposite
• left/right is along the line between the two eyes (which a tetronian needs to see 4d). I would suggest our tetronian should have a nearly mirror-symmetry between left and right side. And the left side is where the heart is (i.e. determined by the left/right-assymetry of the inner organs).
• wint/zant is the remaining line perpendicular to the previous. We define one end as wint by orientation. (though I am not sure if it complies with the original definition, but the other directions are quite intuitive, so wint/zant must be the remaining line.)

[wendy]

Suppose you look at a dance-room. In 3d, you can draw its floor-plan as a rectangle, and along one wall, put the "wallflowers", young hopefuls hoping that Mr Right will step out and ask them for a danse.

For this, you can represent a person by a line, L-R. You can't see the height, because it's a plan. There is a definite front, too. Our young hopefuls face the floor, and so you have a wall, with lots of L-R along it, eg L-R, L-R L-R.

In four dimensions, you have the whole room as the "plan". The walls, roof and floor are the "walls" of the 4-room. Let's represent our people by clocks. The clock faces forward, and so if the circle is anticlockwise, you are looking at the person from behind.

We now arrange clocks on the floor (literally, a wall).

We now ask, what makes us put all of the 12's pointing the same direction. Neither gravity nor facing forward does this. Even if you had these directions, different people could have different numbers facing the north.

While one does not habitually walk upside down, or back to front, there is no reason that all of the 12's have to face north, or east, or even to align to the same direction. Therefore clock1's 12 might be clock2's 3. That is, my "left" could be read as your "zant", and the distinction can not be made clear unless you look at the orientation of the clock-faces.

On the other hand, the closest you can come, is that a clockwise turn is always clear. But there is no overriding sense of a uniquely identifyable left or right or zant or wint.

And this, is why i do not hold much faith in these directions.

[bo198214]

I dont know whether you read my post, but simply repeat what you have already told.
Defining these directions is for me simply assigning a coordinate system to a body. And this is uniquely possible if all bodies are similar and enough asymetric (i.e. if the symmetry group is only the identity element).

We now ask, what makes us put all of the 12's pointing the same direction.

We simply put a lead dancer on the stage and ask the crowd to imitate her/him. Then all 12s pointing into the same direction.

I can not see your problem, it seems at least different from uniquely assigning a coordinate system to a body. And it looks as if it is also unclear to yourself if you still mix up the local coordinate system (looking forward) with the global coordinate system (up and north).

[PWrong]

I don't see the problem either. A "left" turn is a turn in the direction of the left arm. A "wint" turn is in the direction of the wint arm. We can easily define the "up" vector, and the "forward" vector. In 3D we can define "left" as the cross product of up and forward. In 4D, there is no cross product, so all we get is a plane. If our 4D creature had no arms or legs, it wouldn't have a sense of left/right and zant/wint.

Now suppose our 4D creature had a single arm, perpendicular to up and forward, but otherwise in a random direction. Then the arm would define the "left" vector, and the "wint" vector would be (up x forward x left). Now that we have two perpendicular vectors, we can put three more arms in their proper places.

(Has onybody btw already made a model of the hypothetical tetronian?!)

We've tried a few times. The main problem is that we have no universal model of a trionian. How many legs does a 3D animal have? We do know that a stool or table in 4D needs four legs to stand up (generally, an nD table needs n legs). But that rule doesn't stop us from walking on two legs.

I once worked out the number of fingers required for a 4D monkey to swing on a spherindrical tree branch. It was 2k fingers and a thumb, for some integer k. http://tetraspace.alkaline.org/forum/viewtopic.php?t=266

[wendy]

Please remember that the room i described is a floor-plan of a 4d room. You can put clocks in anywhere, facing any direction. You are not restricted to the floor.

What happens, then if the lead dancer's 12 is facing the same direction as your front? This is like saying my left can be the same direction as your front.

The thing is, that in such a room, the nature of the danse could have people swirling around the up/front axis, ie the clock itself rotating!

The thing is, that unless there is some kind of interaction between two people, there is no way of saying this is left, this is zant. You can't for example, say "turn left/zant at the service station".

The thing is, that even if the clock was spinning crazy, you can be up and walking, because the clock is perpendicular to up and forward.

[bo198214]

What happens, then if the lead dancer's 12 is facing the same direction as your front? This is like saying my left can be the same direction as your front.

This is nothing 4d-special. If, for example, we stand parallel and I then turn to the right, my left is your front (and my front is your right). And this doesnt hinder people, to call a certain direction left and right.

The thing is, that in such a room, the nature of the danse could have people swirling around the up/front axis, ie the clock itself rotating!

Then lets dance!

You can't for example, say "turn left/zant at the service station".

Yes thats right, and the possibility to say so in 3d comes from the incidently imposed gravity. So we simply impose an other global (for example electrical) field in 4D and supply the tetronians with the corresponding sense. The second field direction we call wint and zant. And readily a 4d-pedestrian can describe the way to an 4d motorbiker by "turn left at the service station and then zant at the next crossing then you already can see the dance club on the wint side."