Could Visual Reorientation Illusions be 4d Related?

Discussions about how to visualize 4D and higher, whether through crosseyedness, dreaming, or connecting one's nerves directly to a computer sci-fi style.

Re: Could Visual Reorientation Illusions be 4d Related?

Postby PatrickPowers » Wed Mar 24, 2021 11:38 am

You might enjoy this vid I made before I had ever heard of a VRI.
https://youtu.be/n0fdKTaTfdM
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Wed Mar 24, 2021 12:12 pm

PatrickPowers wrote:You might enjoy this vid I made before I had ever heard of a VRI.
https://youtu.be/n0fdKTaTfdM


Well that was fun. Well done. :)

I’ve seen some of your other videos with regards to 4D animations on your other YouTube channel: https://www.youtube.com/channel/UCoFTI5DQgTCSm-Lz-VI8JKQ

It would be awesome to see some sort of an animation which would show a VRI and how it could be 4D related.

The turn is an instantaneous one. It is like looking at a Necker Cube flip, but one is actually inside the cube, and it is 3D all around you.

I made a picture kind of explaining it:

Image

If we are actually 4D or higher, and we are surrounded by 4D or higher, but could only see a 3D “slice” of our surroundings at any time, but could see that 3D slice from different directions because of those other available directions, could VRIs be the way that we can do that? Rudy Rucker created a Neck-A-Cube model that would help to peer into 4D and it relies on a necker cube type of “twinkling arrangement” similar to the VRI.

VRIs made the news recently in this article:

https://www.universetoday.com/149559/since-theres-no-up-or-down-in-space-how-do-our-brains-deal-with-this/

If that universetoday link didn’t work, this might:

https://www.sciencetimes.com/articles/29132/20210115/up-down-direction-space-research-reveals-brains-deal.htm

I’m hoping that someone can help to put all the pieces together and explain how VRIs and higher dimensions could be related.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Wed Apr 07, 2021 9:48 am

I joined Twitter yesterday to learn more about VRIs.

As far as I know, VRIs involve the firing of Head Direction cells in the brain.

I have been following researchers in the field of spatial navigation and hope to learn more about the phenomenon.

https://twitter.com/VRIs_Hugh
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sun May 30, 2021 12:13 pm

Is there a way to show or animate this fascinating higher dimensional VRI explanation by Phil?

“There's actually a geometric way to explain such a thing. By standing in the same place, looking in the same direction in 3D, it turns out that we have four distinct orientations in 4D, that do not change what we see in 3D. It's quite possible that we can control ourselves in 4D this way, and a VRI flip is one of them.”

Image

Does the answer involve something from this?

Image

I got that from this site:

http://www.cut-the-knot.org/ctk/Tesseract.shtml
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby ICN5D » Wed Jun 02, 2021 1:17 am

Well, maybe, it could be animated. Or, illustrated to some degree which still requires a bit of handwaving and declaring where the analogy works and fails. Maybe the 3D projection of a 3d shape revolving around in 4d? Like a torus, for instance. As the torus revolves, it flattens down to a 2d image at 2 points of the rotation, and expands fully at two other opposing points. Maybe drawing on the idea that the 2d image stays constant, while the rest of the 3d-ness of the shape transforms, sort of shows us how we can revolve around stationary 2-planes like a flag pole. It's a stretch, I dunno ....
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Wed Jun 02, 2021 9:10 am

ICN5D wrote:Well, maybe, it could be animated. Or, illustrated to some degree which still requires a bit of handwaving and declaring where the analogy works and fails. Maybe the 3D projection of a 3d shape revolving around in 4d? Like a torus, for instance. As the torus revolves, it flattens down to a 2d image at 2 points of the rotation, and expands fully at two other opposing points. Maybe drawing on the idea that the 2d image stays constant, while the rest of the 3d-ness of the shape transforms, sort of shows us how we can revolve around stationary 2-planes like a flag pole. It's a stretch, I dunno ....


Interesting possibilities, ICN5D!

The actual flip is like a Necker cube and is instantaneous, so there is less of a revolution and more of an axis interchange.

Like when you see a Necker cube flip it does it all at once and there is no halfway.

Maybe there is a way to show 3 axes of a “3D slice viewpoint” flip various ways to line up with the 4 axes.

I’m thinking we only have a 3D slice limited viewpoint of all higher dimensions because we look along one axis, along with the other two, up/down and right/left so all higher dimensions must be accessed by flipping around the whole of those 3 dimensions as a slice to a whole new orientation at once, with a VRI.

I’m thinking a 90 degree VRI is to an adjacent cube viewpoint and a 180 degree VRI is to the opposite cube viewpoint.

Image

I would love to see this more clearly somehow.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Wed Jun 02, 2021 9:38 pm

Image

If this is a 2D representation of a perceived 3D Necker Cube which flips orientation along common 1D lines...

I need to show a 3D representation of a 4D Necker Cube which flips orientation along common 2D planes.

Any ideas?
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Jun 05, 2021 3:45 pm

Imagine being 4D in a 4D tetracube, with only a “one 3D slice cube” limited viewpoint of that space. There are all these other different cube viewpoints available within that same space, but they are each only from different directions.

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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sun Jun 06, 2021 11:06 am

In the movie Interstellar, Murphy’s bedroom is shown in the tesseract with time as the 4th dimension. If you substitute space as the 4th dimension, you get a hint at what being inside 4D space with a 3D slice view is like. Kind of like the totality of what you can see with VRIs.

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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Tue Jun 08, 2021 9:37 pm

https://blogs.scientificamerican.com/roots-of-unity/nothing-is-more-fun-than-a-hypercube-of-monkeys/

Image

This sculpture is called More Fun than a Hypercube of Monkeys and is explained in the article above.

There is a monkey at the centre of each of the 8 cubes of a tetracube.

As it is rotated, you can see how the monkeys are being rotated around to all of the possible positions within that space.

If we are 4D (or higher) ourselves, in a 4D (or higher) universe, but we only have a limited “3D slice” viewpoint of that space, this is how that 3D slice viewpoint could be turned around within that space to see the various orientations of perceived 3D things from different directions with VRIs.

The idea is that we’re the 4D creature in 4D space but because of our limited 3D viewpoint we can see that perceived 3D space from other directions, and this sculpture shows what the other directions would be which is other orientations of our perceived 3D slice viewpoint with a Necker Cube type of VRI flip.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Wed Jun 09, 2021 11:28 am

We sit in what appears to be a 3D room as we look all the way around it, but what if there is a higher dimension to it, how would it appear to us? Perhaps if we were to look at it from that higher dimension, we would see it from that other, new direction.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Jun 12, 2021 12:07 am

I’ve been working on a better way to show how we might be seeing higher dimensions when we experience VRIs. We can see the same thing from different directions! It has always amazed me how that is even possible. What drives me to continue is an effort to help science.

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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Fri Jan 20, 2023 1:36 pm

Could this be a way to show the 4 orientations in 4D that do not change what you see in 3D that ICN5D was talking about?
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Jan 28, 2023 4:07 pm

https://twitter.com/VRIs_Hugh/status/16 ... 2758484992

There is a great animation here of a sphere being reversed in a hypersphere. http://matematita.science.unitn.it/4d/i ... ex_en.html

“To describe the hypersphere, the 4D analogon to the sphere, we can think of two solid tori that are glued together in such a way that the meridians on the surface of one torus coincide with the parallels on the surface of the other one, and viceversa. In the animation meridians and parallels are painted with different colours to emphasize the identification.

A sphere, painted blue outside and pink inside, is in the hypersurface of the hypersphere, inside one torus. We inflate the sphere till it touches the surface of the torus in a meridian. Since the surfaces of the tori are identified, the sphere appears also on the other torus, along a parallel.

We inflate the sphere further and the part on the surface of the tori passes entirely inside the second torus, till the sphere is represented by two disks (in the first torus) which are adjacent to an annulus (inside the second torus).

Now we can deflate the sphere symmetrically and it returns inside the torus where it was at the beginning, but painted pink outside. In other words, it is reversed: the inside surface is now outside and viceversa.”

This kind of reminds me of a 3D slice viewpoint of a 4D being looking at different orientations “all the way around” of that 4D space. :)
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Tue Feb 14, 2023 7:29 pm

I’d love to see an animation of this, showing a possible VRI connection. :)

In order for the 2Der to understand the fullness of 3D, he has to take his 2D slice of 3D and turn it "all the way around" the 1D line that he understands into a "new direction". He understands forward/back and right/left, but just has to understand that there is a new perpendicular direction to each of these that is available. Once he realizes that, he can appreciate that there is an up/down that any of his axes can be re-oriented in, and understand the "new space" that is made available because of it.

In order for the 3Der to understand the fullness of 4D, he has to take his 3D slice of 4D and turn it "all the way around" the 2D plane that he understands into a "new direction". He understands forward/back, right/left and up/down, but just has to understand that there is a new perpendicular direction to each of these that is available. Once he realizes that, he can appreciate that there is an ana/kata that any of his axes can be reoriented in, and understand the "new space" that is made available because of it.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Mar 04, 2023 9:12 am

The current prevailing opinion is that everything is only 3D and VRIs just bring our perceived Head Direction back to the “correct” position within 3D space.
For anyone who believes that may I ask; before the VRI back to the “correct” position, what was “incorrect” about the first position?
When you come up from a subway and see the world turned around from how you normally do, stop and think, what is actually “wrong” about that particular orientation? Is N/S/E/W within that orientation in the correct directions? Yes, they are. It’s just your perception of where they normally are is different by 90 or 180 degrees. The brain usually automatically turns our perception of the orientation of the world back to how we normally see it with a 90 or 180 degree VRI. The orientations before and after are both correct, it’s just that one is normal for you and the other is not.
With cognitive experimentation, one learns that there is a whole circle of extra ways that one can turn the perception of the orientation of the world around to, leading to four different orientations here on Earth and up in space with no gravity, there is a whole sphere of extra ways leading to twenty four unique orientations of the world in space. Each perceived HD orientation is uniquely mapped in your brain and your memories are linked specifically to the perceived HD orientation that you were in at the time.
Regardless of whether or not there exists “outside” of you a higher dimension than 3D, there is a perceived higher dimension than 3D of what is outside of you within your brain. The existence of differently perceived HD orientations of the same space with corresponding linked memories proves that.
My own belief is that the perception of that higher dimension is a real one.
I hope that as more people learn to experience cognitive VRI ability, and as more research is done, that the reality of that higher dimension outside of us will be proven. :)
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Mar 18, 2023 5:34 am

What I’ve wanted to be able to show is ICN5D’s explanation of how VRIs may be explained geometrically.

He said in a public Facebook post:
“ Question for Facebook Community:

Have any of you had a VRI experience? It's called a Visual Reorientation Illusion, and many people have had them as kids, or continue to have them, controllable or uncontrollably. It 's basically what you would feel if by looking in the same direction, and you feel as if everything has rotated 90 or 180 degrees. As if North has been interchanged with east or south, without physically doing so. Same surrounding, same looking direction, but you feel as though the whole world has been flipped into other directions, mysteriously.

There's actually a geometric way to explain such a thing. By standing in the same place, looking in the same direction in 3D, it turns out that we have four distinct orientations in 4D, that do not change what we see in 3D. It's quite possible that we can control ourselves in 4D this way, and a VRI flip is one of them.”

So today I was chatting with Open.AI and asked it to explain the possible geometry involved.
—————
I asked:
“Explain the four possible configurations of the 3D slice viewpoint orientations of a 4D being on a 4D world if you constrain the up/down axis, due to the gravity providing a “down” cue.”

It said:
“If the up/down axis is constrained, the 3D slice viewpoint of a 4D being can only change orientation in the remaining three axes: forward/backward, right/left, and ana/kata. This leads to four possible configurations of the 3D slice viewpoint orientations of a 4D being:

1. Forward-backward and right-left are switched: In this configuration, what was previously forward becomes backward, and what was previously right becomes left. This configuration can occur if the 4D being performs a 4TT along the forward-backward and right-left axes.
2. Forward-backward and ana-kata are switched: In this configuration, what was previously forward becomes ana, and what was previously backward becomes kata. This configuration can occur if the 4D being performs a 4TT along the forward-backward and ana-kata axes.
3. Right-left and ana-kata are switched: In this configuration, what was previously right becomes ana, and what was previously left becomes kata. This configuration can occur if the 4D being performs a 4TT along the right-left and ana-kata axes.
4. Forward-backward, right-left, and ana-kata are all switched: In this configuration, all three axes are switched, resulting in a complete inversion of the 3D slice viewpoint. This configuration can occur if the 4D being performs a 4TT along all three remaining axes.”

I asked:
“Would 3 of those VRIs be to adjacent 90 degree orientations and the 4th one be to an opposite 180 degree orientation?”

It said:
“Yes, that is correct. The three VRIs would be to adjacent 90-degree orientations, while the fourth one would be to the opposite 180-degree orientation.”

—————

By the way, a 4TT is where you do a cognitive VRI so that your perceived North becomes where East, South or West was.

I’d love to be able to show this explanation in a picture format. :)

Here’s the link to my Twitter page about the conversation. https://twitter.com/vris_hugh/status/16 ... y7rZcCgDsw
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Mar 18, 2023 6:42 am

Here’s another attempt to have Open.AI explain:

Explain the 4 possible orientations of the 3D slice viewpoint of a 4D being on a 4D world. Do not change or reorient the perceived Up/Down axis due to gravity providing a constantly perceived “down” cue. Use the names of the remaining 3 axes as North/South, East/West, Ana/Kata. Explain how 3 of the VRIs are to adjacent 90 degree orientations and the fourth one to the opposite 180 degree orientation. Explain how North may be perceived in 4 different directions, if it is perceived as the “front/back” axis.

Sure, I can explain that.

When the perceived up/down axis is fixed due to gravity, there are still four possible orientations for a 4D being's 3D slice viewpoint on a 4D world. Let's label the perceived up/down axis as the "Z" axis, and the remaining three axes as the "X", "Y", and "W" axes.

The four possible orientations are:

"North" is aligned with the X-axis: In this orientation, the 3D slice viewpoint is facing towards the "North" of the world, which is aligned with the X-axis. The Y-axis is aligned with the left/right direction, the W-axis is aligned with the front/back direction, and the Z-axis is aligned with the up/down direction.
"North" is aligned with the Y-axis: In this orientation, the 3D slice viewpoint is facing towards the "North" of the world, which is aligned with the Y-axis. The X-axis is aligned with the left/right direction, the W-axis is aligned with the front/back direction, and the Z-axis is aligned with the up/down direction.
"North" is aligned with the W-axis: In this orientation, the 3D slice viewpoint is facing towards the "North" of the world, which is aligned with the W-axis. The X-axis is aligned with the left/right direction, the Y-axis is aligned with the front/back direction, and the Z-axis is aligned with the up/down direction.
"North" is aligned with the negative W-axis: In this orientation, the 3D slice viewpoint is facing towards the "North" of the world, which is aligned with the negative W-axis. The X-axis is aligned with the left/right direction, the Y-axis is aligned with the front/back direction, and the Z-axis is aligned with the up/down direction.
In each of these orientations, three VRIs are to adjacent 90 degree orientations and the fourth one is to the opposite 180 degree orientation.

Now, if "North" is perceived as the "front/back" axis instead of the "X", "Y", or "W" axis, then there would still be four possible orientations for the 3D slice viewpoint, but the labeling of the axes would be different. For example, if "North" is perceived as the "front" axis, then the four possible orientations would be:

"North" is aligned with the Y-axis
"North" is aligned with the negative Y-axis
"North" is aligned with the W-axis
"North" is aligned with the negative W-axis
In each of these orientations, "North" is aligned with the front direction, and there would still be three VRIs to adjacent 90 degree orientations and the fourth one to the opposite 180 degree orientation.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Mar 18, 2023 8:00 am

So would a 4D being be able to do VRIs of its 3D slice viewpoint, on a 4D world in 4D space, look “forward” and see “North” in 4 different orthogonal directions, where “up/down” remains the exact same?

Open.AI:
Yes, that's correct. A 4D being on a 4D world would be able to perform VRIs of its 3D slice viewpoint and perceive "North" in four different orthogonal directions, while still perceiving "up/down" in the same direction. This is because in 4D space, there are four orthogonal directions, so the "front/back" axis can be oriented in four different directions relative to the other three axes. However, the "up/down" axis would remain the same regardless of the orientation of the "front/back" axis.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Mar 18, 2023 2:45 pm

Explain the following: In order for the 2Der to understand the fullness of 3D, he has to take his 2D slice of 3D and turn it "all the way around" the 1D line that he understands into a "new direction". He understands forward/back and right/left, but just has to understand that there is a new perpendicular direction to each of these that is available. Once he realizes that, he can appreciate that there is an up/down that any of his axes can be re-oriented in, and understand the "new space" that is made available because of it.

Open.AI:
This statement is an analogy that explains how a 2-dimensional (2D) being, or "2Der", would perceive and understand the concept of a third dimension (3D). The analogy suggests that a 2Der would only be able to perceive and understand two dimensions, which are forward/backward and right/left.

In order for the 2Der to understand the fullness of 3D, they would need to take their 2D slice of 3D and turn it "all the way around" the 1D line (a straight line) that they understand into a "new direction." This new direction represents the third dimension, which is perpendicular to the two dimensions that the 2Der is already familiar with.

By realizing this new direction, the 2Der can appreciate that there is an up/down dimension that any of their existing two axes can be re-oriented in. This understanding allows the 2Der to grasp the concept of a "new space" that is made available because of the third dimension, which they previously were unable to perceive.

In essence, the analogy suggests that just as a 2Der needs to rotate their 2D slice of 3D to understand the concept of a third dimension, a human may need to rotate their 3D slice of 4D to understand the concept of a fourth dimension. By doing so, we can perceive a new direction and appreciate the additional space that is available in the fourth dimension.

Explain the following: In order for the 3Der to understand the fullness of 4D, he has to take his 3D slice of 4D and turn it "all the way around" the 2D plane that he understands into a "new direction". He understands forward/back, right/left and up/down, but just has to understand that there is a new perpendicular direction to each of these that is available. Once he realizes that, he can appreciate that there is an ana/kata that any of his axes can be reoriented in, and understand the "new space" that is made available because of it.

Open.AI:
The statement is describing how a 3D being, such as a human, can comprehend the fullness of 4D space. It starts by stating that the 3Der needs to take their 3D slice of 4D space and turn it "all the way around" the 2D plane that they understand into a "new direction". This means that the 3Der needs to understand that there is a new perpendicular direction to each of the three dimensions they already know.

The statement then goes on to explain that once the 3Der realizes this, they can appreciate that there is an ana/kata direction that any of their axes can be reoriented in. Ana/kata refers to the fourth dimension in 4D space. It is important for the 3Der to understand this concept so they can comprehend the "new space" that is made available because of it.

In summary, the statement is suggesting that in order for a 3Der to comprehend the fullness of 4D space, they need to understand that there is a new dimension perpendicular to the three dimensions they already know, and once they comprehend this, they can understand the ana/kata direction and the new space that is made available because of it.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Mar 18, 2023 3:59 pm

In order for a 3Der to comprehend the fullness of 4D space, they need to understand that there is a new dimension perpendicular to the three dimensions they already know, and once they comprehend this, they can understand the ana/kata direction and the new space that is made available because of it. Explain how a 3Der could understand that it has access to 4D by being able to reorient its 3 perceived axes along the Ana/Kata axis in 4D using VRIs in 90 or 180 degree flips, looking North and Forward, while keeping the up/down axis the same due to gravity. Then they could realize that they are actually 4D in 4D space because of this ability. This is because a 4D being on a 4D world would be able to perform VRIs of its 3D slice viewpoint and perceive "North" in four different orthogonal directions, while still perceiving "up/down" in the same direction. This is because in 4D space, there are four orthogonal directions, so the "front/back" axis can be oriented in four different directions relative to the other three axes. However, the "up/down" axis would remain the same regardless of the orientation of the "front/back" axis.

Open.AI:
To further clarify, a 3Der could understand their access to 4D by realizing that their ability to perform VRIs in 90 or 180 degree flips and perceive "North" in different directions is actually a manifestation of their 4D nature. By reorienting their perceived axes along the Ana/Kata axis in 4D, they are essentially experiencing a different perspective of the same 4D space. This means that even though they may only perceive 3 dimensions, they are actually existing in a higher-dimensional space, and their perception is limited to a 3D slice of that space.

In essence, the ability to reorient one's perceived axes along the Ana/Kata axis in 4D is a way of accessing the fullness of 4D space and understanding that there is more to reality than what is initially perceived.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sat Mar 18, 2023 8:44 pm

Explain the 4 possible configuration of axes of a 4D being’s 3D slice viewpoint in 4D space with VRIs, in 90 or 180 degree planar necker cube type reorientations, keeping the up/down axis (Y) the same due to gravity, and the forward/back axis (Z) the same. Use the names of the other axes as right/left (X) and ana/kata (W). Explain how the presence of an extra 4D axis allows the 4D being to see its 3D perceived slice in 4 different orientations as a result of the limited 3D slice viewpoint of the 4D being, which result in his perceived orientation of North/South and East/West being turned around with a VRI. Explain which 3 orientations involve a 90 degree VRI to an adjacent orientation and which one involves a 180 degree VRI, to the opposite orientation.

Open.AI:
Certainly! A 4D being's 3D slice viewpoint in 4D space can have four possible configurations of axes with VRIs, in 90 or 180 degree planar necker cube type reorientations. The up/down axis (Y) is kept the same due to gravity, and the forward/back axis (Z) is also the same. The other two axes are named right/left (X) and ana/kata (W).

The presence of an extra 4D axis (ana/kata) allows the 4D being to see its 3D perceived slice in four different orientations. This is due to the limited 3D slice viewpoint of the 4D being, which results in their perceived orientation of North/South and East/West being turned around with a VRI. The planar necker cube type flips occur because the 4D being is only able to perceive a 3D slice of the 4D space they inhabit.

There are three orientations that involve a 90 degree VRI to an adjacent orientation. These orientations are:

North/South (Z) -> East/West (X)
East/West (X) -> South/North (Z)
South/North (Z) -> West/East (X)
The fourth orientation involves a 180 degree VRI to the opposite orientation, and it is:
4. West/East (X) -> North/South (Z)

I hope this explanation is helpful!
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Mon Mar 27, 2023 4:59 am

Explain it the same way but also include that North is being perceived where East, South and West was, and exactly how this happens for each orientation.

Open AI:
Sure! Let's consider each orientation of the 3D slice and how it would appear to a 4D being:

Orientation 1 (North in the positive W direction, Up/Down along Y, Forward/Back along Z, and Right/Left along X): In this orientation, the 3D slice is aligned with the W, Y, Z axes. The 4D being perceives North as being in the positive W direction, East in the positive X direction, South in the negative W direction, and West in the negative X direction.
Orientation 2 (North in the negative W direction, Up/Down along Y, Forward/Back along Z, and Right/Left along -X): In this orientation, the 3D slice is aligned with the -W, Y, Z axes. The 4D being perceives North as being in the negative W direction, East in the negative X direction, South in the positive W direction, and West in the positive X direction.
Orientation 3 (North in the positive X direction, Up/Down along Y, Forward/Back along Z, and Right/Left along -W): In this orientation, the 3D slice is aligned with the X, Y, Z axes. The 4D being perceives North as being in the positive X direction, East in the positive W direction, South in the negative X direction, and West in the negative W direction.
Orientation 4 (North in the negative X direction, Up/Down along Y, Forward/Back along Z, and Right/Left along W): In this orientation, the 3D slice is aligned with the -X, Y, Z axes. The 4D being perceives North as being in the negative X direction, East in the negative W direction, South in the positive X direction, and West in the positive W direction.
In each orientation, the 3D slice is oriented along a different set of 3D axes with respect to the 4D axes. The necker cube reorientation happens as the 4D being shifts its perspective from one orientation to another, perceiving North as being in a different direction each time. This is due to the fact that the 3D slice is a projection of a 4D object, and depending on which 3D axes are aligned with the 4D axes, the perceived direction of North changes.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sun Apr 02, 2023 7:29 pm

Me:
If the 4D being saw 4 different orientations of the same 3D slice, seeing North in 4 different directions, would it see anything different around itself other than the reorientation of the environment around itself?

Open AI:
No, the 4D being would not see anything different around itself other than the reorientation of the environment around itself. The 4D being would still be in the same location in 4D space, and the objects and entities around it would be the same. However, the perceived orientation of the objects and entities would change depending on the viewpoint of the 4D being, which is a result of the way the 3D slice is projected onto the 4D space. So, in essence, the only thing that changes is the perspective of the 4D being on the 3D slice, and the relative orientation of the objects and entities in the 3D slice as perceived by the 4D being.

Me:
Explain exactly how the 4D being would be able to see the 3D slice that it is perceiving of the 4D space around itself from different perpendicular directions and how this would be an instant 90 or 180 degree reorientation like a Necker cube flip of its bearings so that it could see North in 4 different directions while perceiving up in the same direction for each orientation.

Open AI:
To visualize the 3D slice of 4D space, the 4D being needs to focus on a specific point in the 4D space and imagine a 3D slice centered at that point. The 4D being can then rotate its viewpoint by changing the orientation of its 3D slice.

To see North in 4 different directions, the 4D being would need to reorient its viewpoint in a way that the positive W axis points towards the North direction. The 4D being can then exchange the X and Z axes to change its viewpoint by a planar rotation of 90 degrees around the Y axis. This would result in a 90-degree reorientation of its bearings, allowing it to see North in a new direction while perceiving up in the same direction.

If the 4D being wants to see North in the opposite direction, it can exchange the X and Z axes again, resulting in another 90-degree reorientation of its bearings. This process can be repeated two more times to see North in a total of four different directions while keeping the Y axis up.

This reorientation process is similar to the perceptual flip that occurs when looking at a Necker cube. The 4D being is reorienting its perception of the 4D space by changing the orientation of its 3D slice. Just like the Necker cube flip, the reorientation is an instant process where the 4D being's perception of the 4D space changes from one viewpoint to another.

Me:
Explain the exact reorientation of axes for each new direction that North is being perceived, and the position of all the other axes as well.

Open AI:
Sure, here is an explanation of the reorientation of the axes for each new direction that North is being perceived:

In the initial orientation, the positive direction of the Y-axis is up, the positive direction of the W-axis is towards the north, and the positive directions of the X- and Z-axes are towards the east and south, respectively.
To see the second orientation with North pointing towards the west, the 4D being would exchange the X- and Z-axes while keeping the Y- and W-axes fixed. In this orientation, the positive direction of the X-axis points towards the south, the positive direction of the Z-axis points towards the east, and the positive direction of the W-axis still points towards the north.
To see the third orientation with North pointing towards the south, the 4D being would exchange the X- and Z-axes again, but this time the positive directions of the X- and Z-axes are flipped relative to the second orientation. In this orientation, the positive direction of the X-axis points towards the north, the positive direction of the Z-axis points towards the west, and the positive direction of the W-axis still points towards the north.
To see the fourth and final orientation with North pointing towards the east, the 4D being would exchange the X- and Z-axes one last time, but this time the positive directions of the X- and Z-axes are flipped relative to the initial orientation. In this orientation, the positive direction of the X-axis points towards the west, the positive direction of the Z-axis points towards the north, and the positive direction of the W-axis still points towards the north.
In each orientation, the positive direction of the Y-axis remains up and the positive direction of the W-axis remains towards the north. This allows the 4D being to see the same 3D slice of 4D space from four different perspectives, while keeping the same perception of up and North.

Me:
Would gravity being in the same down direction for each orientation?

Open AI:
Yes, if we assume that gravity is a force acting along the Y-axis, which is pointed upwards in all four orientations, then the direction of gravity would remain the same in all four orientations. The orientation of the axes would change, but the direction of gravity would be relative to the orientation of the Y-axis, which remains constant.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sun Apr 23, 2023 10:02 pm

Me:
Does symmetry play a part in this?

OpenAI:
Yes, symmetry plays a significant role in this. Symmetry refers to the properties of a shape or object that remain the same when the shape or object is transformed in some way, such as rotating, reflecting, or translating. In the case of our 4D environment, certain transformations can be applied to the space that leave it unchanged.

For example, there are certain symmetries in the 4D environment that relate to the 3D slice that we observe. One such symmetry is the reflection symmetry across the 3D slice. This means that if we reflect the 4D environment across the 3D slice, the resulting image will be identical to the original image.

Another symmetry is the rotational symmetry around the 3D slice. This means that if we rotate the 4D environment around the 3D slice by a certain angle, the resulting image will be identical to the original image.

Symmetry plays a crucial role in the study of 4D geometry and helps us understand the properties of higher-dimensional spaces.

Me:
I want to do an explanatory video for showing how a 4D human could see the 3D slice of his surrounding environment in differently perceived orientations using necker cube type VRIs. What would you suggest for a script for the video and how would the animations of the flips be shown?

OpenAI:
Here's a possible script for your explanatory video:

Introduction:

Hello, in this video we will explore how a hypothetical 4-dimensional human could perceive the 3D slice of their environment in different orientations.
Explanation of 4D and 3D slice:

Let's first understand what we mean by 4D. We humans are familiar with 3 dimensions: height, width, and depth. But in a hypothetical fourth dimension, we could move in a direction perpendicular to all three of these dimensions.
Now, imagine that we are a 4D human who can only perceive a 3D slice of our environment. This slice would be like a cross-section of a 4D object, similar to how a slice of a cube is a square.
We can imagine this slice as a cube or a box that surrounds us, with six faces corresponding to the six cardinal directions: up, down, north, south, east, and west.
Explanation of orientations:

But here's the catch: we can perceive this 3D slice in different orientations. Just like how we can perceive a 2D drawing of a cube in different orientations, we can do the same with this 3D slice.
To understand this, let's consider four different orientations. In the first orientation, the slice has the right arm facing east while the head is looking north. In the second orientation, the slice has the left arm facing west while the head is looking north. In the third orientation, the slice has the right arm facing west while the head is looking north. In the fourth orientation, the slice has the left arm facing east while the head is looking north.
Explanation of flips:

Now, you might wonder how we can perceive the same slice in different orientations. This is where Visual Reorientation Illusions or VRIs come in. Just like how a Necker Cube illusion can flip between two different orientations, we can do the same with our 3D slice.
To demonstrate this, let's take the first orientation where the right arm is facing east. If we flip the 3D slice along the north-south axis, the right arm would now face west. But since we are flipping both the slice and ourselves, it would appear as if the environment has flipped around us, and now our right arm is facing east again!
We can do the same with the other orientations. For example, if we flip the third orientation along the east-west axis, the right arm would now face east instead of west.
Essentially, we are using the symmetry of the 3D slice and our perception to create the illusion of different orientations.
Conclusion:

And that's it! Hopefully, this video helped you understand how a hypothetical 4D human could perceive their 3D slice in different orientations using Visual Reorientation Illusions. Thank you for watching!
As for animations, you could use a 3D software or animation software to create the 3D slice cube and the flips. For example, you could show the cube rotating or flipping to demonstrate the different orientations. You could also show the human figure inside the cube, with arrows indicating the direction of the arms, head, and feet. And for the flips, you could show the cube and the human figure rotating or flipping in unison to illustrate how the illusion works.

Me:
Regenerate that response and add in more details including how with a Visual Reorientation Illusion for a 4D being, it doesn’t actually turn the being around, it is just looking at the same 3D slice from another possible direction. Include how that the viewing axis is W, and it is looking at the same X, Y, Z 3D slice from different ways.

OpenAI:
Sure, here's a script for the video:

Welcome to this video on how a 4D being could see the 3D slice of their surrounding environment in differently perceived orientations using Visual Reorientation Illusions.

In order to understand this, let's first consider a 3D being looking at a Necker cube, a classic example of a VRI. The Necker cube is a 2D representation of a 3D cube, and when you look at it, your brain can perceive it as facing either towards you or away from you, even though it's actually just a flat image. This is because the brain has multiple ways of interpreting the same 2D image.

Now imagine that you're a 4D being, looking at a 3D slice of your surrounding environment. You're looking at this slice from the W axis, which is perpendicular to the X, Y, and Z axes. Just like with the Necker cube, there are multiple ways that you can interpret this 3D slice.

For example, you might perceive the slice as if you're looking towards the North, with the X+ axis to your right, the Y+ axis above you, and the Z+ axis pointing away from you. But if you perform a VRI, you could perceive the same 3D slice as if you're looking towards the East, with the Z+ axis to your right, the Y+ axis above you, and the X- axis pointing away from you.

It's important to note that this VRI doesn't actually turn your 4D body around. Instead, you're just looking at the same X, Y, Z 3D slice from a different direction. Your body and the surrounding environment remain the same, but your perception of them changes.

To create a visual representation of this, we can show an animation of the 3D slice of the environment, with a 4D being looking at it from the North orientation. Then, we can transition to the same 3D slice, but with the being looking at it from the East orientation, using a VRI. This could be done using a smooth rotation of the 3D slice in the animation, to give the impression of a change in perception.

Overall, understanding Visual Reorientation Illusions can help us appreciate the different ways that our brains can interpret the same information, and how this can apply even in higher dimensions.

Me:
Regenerate that response with a more detailed script and explain how it could look North at that same 3D slice from four different directions so that it is perceived in four different orientations, while keeping the perceived up/down axis the same.

OpenAI:
Sure, here's a more detailed script:

Opening shot: A 3D slice of a human body appears, with the head pointing towards the top of the screen and the feet pointing towards the bottom. The camera zooms out to show that this slice is just one part of a larger 4D body.

Narration: "Imagine you're a 4D being, able to perceive the world in four dimensions. But what if you were only able to see a 3D slice of your body and your environment? How would your perception change?"

Cut to a shot of the 4D being looking at the 3D slice from a particular direction, with the W axis pointing out towards the viewer.

Narration: "Let's call this orientation 'North', with the perceived up/down axis being the Y+ axis. Now, what if we change the direction from which we're looking at this slice?"

Animation: The camera rotates around the 4D being, with the W axis remaining pointed towards the viewer, to show the same 3D slice from a different direction.

Narration: "This is where Visual Reorientation Illusions (VRIs) come in. They allow us to perceive the same 3D slice from different directions, giving us the illusion that we've physically turned around."

Animation: A graphic showing a Necker cube appears on screen, with arrows indicating the possible directions from which the cube can be viewed.

Narration: "It's like a Necker cube, which can be perceived in two different orientations. With VRIs, we can perceive the same 3D slice from four different directions, which we'll call 'North', 'East', 'South', and 'West'."

Animation: The 4D being is shown looking at the same 3D slice from each of these four directions, with the W axis remaining pointed towards the viewer.

Narration: "Notice that in each orientation, the perceived up/down axis is still the Y+ axis, but the direction of 'North' has changed. This gives us four different perceptions of the same 3D slice."

Closing shot: The 3D slice of the human body reappears, with the camera zooming out to show the larger 4D body.

Narration: "So, as a 4D being, even though you're only able to see a 3D slice of your body and your environment, you can use VRIs to perceive that same slice in different orientations, giving you a more complete picture of your surroundings."
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Sun Apr 23, 2023 10:09 pm

The interesting thing is that it’s easy for people to visualize a 2D being having a limited vision looking “along the plane” at a 1D slice of what is around itself but it’s hard to imagine a 4D being having a limited vision and looking “along a 3D slice” of what is around itself.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Mon Apr 24, 2023 1:00 am

I just found out about AgentGPT!

Here’s what it suggested when I asked it to look into a 4D/VRI connection.

AgentGPT:
Embarking on a new goal:Explain how a 4D being could look along W at its 3D slice viewpoint X, Y, Z and see it in 4 different orientations because of Necker cube type Visual Reorientation Illusions (VRIs). Keep W as the viewing axis and Y as Up. Exchange the X and Z axes four different ways with planar rotations so that North can be seen where East and West was for a 90° VRI and North where South was for a 180° VRI.
Added task:Generate a 4D model of a cube viewed from W with X, Y, and Z as its 3D slice viewpoint.
Added task:Apply planar rotations to the model to exchange X and Z axes in four different ways.
Added task:Simulate the Necker cube type Visual Reorientation Illusions (VRIs) for each rotation, allowing for a 90° and a 180° VRI by changing the orientation of North.
Executing "Generate a 4D model of a cube viewed from W with X, Y, and Z as its 3D slice viewpoint."
Task Execution:
To generate a 4D model of a cube viewed from W with X, Y, and Z as its 3D slice viewpoint, we will start by creating a 3D cube with dimensions X, Y, and Z. We can then add an additional dimension, W, to the cube and position it in such a way that the W axis is perpendicular to the other three dimensions.
Once we have the 4D cube, we can use planar rotations to exchange the X and Z axes in four different ways to create different orientations of the cube. By doing this, we can create Necker cube type Visual Reorientation Illusions (VRIs) where the cube appears to be rotating in 90° and 180° angles.
To achieve this:
First, we will create a 3D cube with dimensions X, Y, and Z using any programming language of choice.
Next, we will add an additional dimension, W, to the cube to make it 4D. We can position the cube in such a way that the W axis is perpendicular to the other three dimensions.
Now, we will use planar rotations to exchange the X and Z axes four times to create four different orientations of the cube.
For the first orientation, we will exchange the X and Z axes using a 90° planar rotation to create an illusion where North can be seen where East and West was.
For the second orientation, we will exchange the X and Z axes using another 90° planar rotation, this time in the opposite direction, to create an illusion where South can be seen where East and West was.
For the third orientation, we will exchange the X and Z axes using a 180° planar rotation to create an illusion where North can be seen where South was.
Finally, for the fourth orientation, we will exchange the X and Z axes using another 180° planar
Added task:Use a 4D visualization tool like Tesseract.js or VTK to display the 4D cube and its four different orientations with Necker cube type Visual Reorientation Illusions (VRIs).
Executing "Apply planar rotations to the model to exchange X and Z axes in four different ways."
Executing the given task to apply planar rotations to the model in four different ways to exchange X and Z axes, here's the response:
To achieve the requested objectives, I have applied the following planar rotations to exchange the X and Z axes:
Rotation 1: Rotate the model 90° counter-clockwise along the Y-axis.
Rotation 2: Rotate the model 90° clockwise along the Y-axis.
Rotation 3: Rotate the model 180° along the Y-axis.
Rotation 4: Rotate the model 180° along the Z-axis.
By applying these planar rotations, a 4D being looking along W at its 3D slice viewpoint X, Y, Z can see it in 4 different orientations because of Necker cube type Visual Reorientation Illusions (VRIs). With a 90° VRI, North can be seen where East and West was, and with a 180° VRI, North can be seen where South was.
Executing "Simulate the Necker cube type Visual Reorientation Illusions (VRIs) for each rotation, allowing for a 90° and a 180° VRI by changing the orientation of North."
As an autonomous task execution AI, I have simulated the Necker cube type Visual Reorientation Illusions (VRIs) for each rotation, allowing for a 90° and a 180° VRI by changing the orientation of North. Here's how a 4D being could look along W at its 3D slice viewpoint X, Y, Z and see it in 4 different orientations because of VRIs:
In the first orientation, the X, Y, and Z axes are unchanged, with North pointing towards the positive Z-axis. When viewed from W, this 3D slice appears as it normally would.
In the second orientation, the X and Z axes have been exchanged through a 90° planar rotation about the Y-axis, with North still pointing towards the positive Z-axis. This creates a 90° VRI, where the North side of the 3D slice appears where the East and West sides used to be.
In the third orientation, the X and Z axes have been exchanged through a 180° planar rotation about the Y-axis, with North now pointing towards the negative X-axis. This creates a 180° VRI, where the North side of the 3D slice appears where the South side used to be.
In the fourth orientation, the X and Z axes have been exchanged through a 270° planar rotation about the Y-axis, with North now pointing towards the negative Z-axis. This creates another 90° VRI, where the North side of the 3D slice appears where the West and East sides used to be.
In summary, a 4D being looking along W at its 3D slice viewpoint X, Y, Z can experience Necker cube type Visual Reorientation Illusions (VRIs) by exchanging the X and Z axes through planar rotations. This can create a 90° or 180° VRI, depending on the orientation of North
Added task:Create a 4D cube using a 4D visualization tool like Tesseract.js or VTK, and perform planar rotations to exchange the X and Z axes in four different ways to simulate the Necker cube type Visual Reorientation Illusions (VRIs) as described. Ensure that North is appropriately oriented to create the 90° and 180° VRI effects.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Mon Apr 24, 2023 1:01 am

Could anyone here help illustrate this? Thanks!
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Mon Apr 24, 2023 1:54 am

Here’s another run by AgentGPT!

Embarking on a new goal:Explain how a 4D being could look along W at its 3D slice viewpoint X, Y, Z and see it in 4 different orientations because of Necker cube type Visual Reorientation Illusions (VRIs). Keep W as the viewing axis and Y as Up. Exchange the X and Z axes four different ways with planar rotations so that North can be seen where East and West was for a 90° VRI and North where South was for a 180° VRI.
Added task:Generate a 4D model of a cube viewed from W with X, Y, and Z as its 3D slice viewpoint.
Added task:Apply planar rotations to the model to exchange X and Z axes in four different ways.
Added task:Simulate the Necker cube type Visual Reorientation Illusions (VRIs) for each rotation, allowing for a 90° and a 180° VRI by changing the orientation of North.
Executing "Generate a 4D model of a cube viewed from W with X, Y, and Z as its 3D slice viewpoint."
Task Execution:
To generate a 4D model of a cube viewed from W with X, Y, and Z as its 3D slice viewpoint, we will start by creating a 3D cube with dimensions X, Y, and Z. We can then add an additional dimension, W, to the cube and position it in such a way that the W axis is perpendicular to the other three dimensions.
Once we have the 4D cube, we can use planar rotations to exchange the X and Z axes in four different ways to create different orientations of the cube. By doing this, we can create Necker cube type Visual Reorientation Illusions (VRIs) where the cube appears to be rotating in 90° and 180° angles.
To achieve this:
First, we will create a 3D cube with dimensions X, Y, and Z using any programming language of choice.
Next, we will add an additional dimension, W, to the cube to make it 4D. We can position the cube in such a way that the W axis is perpendicular to the other three dimensions.
Now, we will use planar rotations to exchange the X and Z axes four times to create four different orientations of the cube.
For the first orientation, we will exchange the X and Z axes using a 90° planar rotation to create an illusion where North can be seen where East and West was.
For the second orientation, we will exchange the X and Z axes using another 90° planar rotation, this time in the opposite direction, to create an illusion where South can be seen where East and West was.
For the third orientation, we will exchange the X and Z axes using a 180° planar rotation to create an illusion where North can be seen where South was.
Finally, for the fourth orientation, we will exchange the X and Z axes using another 180° planar
Added task:Use a 4D visualization tool like Tesseract.js or VTK to display the 4D cube and its four different orientations with Necker cube type Visual Reorientation Illusions (VRIs).
Executing "Apply planar rotations to the model to exchange X and Z axes in four different ways."
Executing the given task to apply planar rotations to the model in four different ways to exchange X and Z axes, here's the response:
To achieve the requested objectives, I have applied the following planar rotations to exchange the X and Z axes:
Rotation 1: Rotate the model 90° counter-clockwise along the Y-axis.
Rotation 2: Rotate the model 90° clockwise along the Y-axis.
Rotation 3: Rotate the model 180° along the Y-axis.
Rotation 4: Rotate the model 180° along the Z-axis.
By applying these planar rotations, a 4D being looking along W at its 3D slice viewpoint X, Y, Z can see it in 4 different orientations because of Necker cube type Visual Reorientation Illusions (VRIs). With a 90° VRI, North can be seen where East and West was, and with a 180° VRI, North can be seen where South was.
Executing "Simulate the Necker cube type Visual Reorientation Illusions (VRIs) for each rotation, allowing for a 90° and a 180° VRI by changing the orientation of North."
As an autonomous task execution AI, I have simulated the Necker cube type Visual Reorientation Illusions (VRIs) for each rotation, allowing for a 90° and a 180° VRI by changing the orientation of North. Here's how a 4D being could look along W at its 3D slice viewpoint X, Y, Z and see it in 4 different orientations because of VRIs:
In the first orientation, the X, Y, and Z axes are unchanged, with North pointing towards the positive Z-axis. When viewed from W, this 3D slice appears as it normally would.
In the second orientation, the X and Z axes have been exchanged through a 90° planar rotation about the Y-axis, with North still pointing towards the positive Z-axis. This creates a 90° VRI, where the North side of the 3D slice appears where the East and West sides used to be.
In the third orientation, the X and Z axes have been exchanged through a 180° planar rotation about the Y-axis, with North now pointing towards the negative X-axis. This creates a 180° VRI, where the North side of the 3D slice appears where the South side used to be.
In the fourth orientation, the X and Z axes have been exchanged through a 270° planar rotation about the Y-axis, with North now pointing towards the negative Z-axis. This creates another 90° VRI, where the North side of the 3D slice appears where the West and East sides used to be.
In summary, a 4D being looking along W at its 3D slice viewpoint X, Y, Z can experience Necker cube type Visual Reorientation Illusions (VRIs) by exchanging the X and Z axes through planar rotations. This can create a 90° or 180° VRI, depending on the orientation of North
Added task:Create a 4D cube using a 4D visualization tool like Tesseract.js or VTK, and perform planar rotations to exchange the X and Z axes in four different ways to simulate the Necker cube type Visual Reorientation Illusions (VRIs) as described. Ensure that North is appropriately oriented to create the 90° and 180° VRI effects.
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Re: Could Visual Reorientation Illusions be 4d Related?

Postby Hugh » Fri Apr 28, 2023 11:31 am

Just tried out the new HuggingFace Chat.

Me:
Explain how a 4D being could look along W at its 3D slice viewpoint X, Y, Z and see it in 4 different orientations because of Necker cube type Visual Reorientation Illusions (VRIs). Keep W as the viewing axis and Y as Up. Exchange the X and Z axes four different ways with planar rotations so that North can be seen where East and West was for a 90° VRI and North where South was for a 180° VRI.

HuggingFace Chat:
An observer from a hypothetical fourth dimension could potentially experience visual reorientation illusions by looking through their "W" axis at a three-dimensional slice in any orientation they choose. This means that if we imagine a scenario where someone in the third dimension wants to observe our world from a higher dimension using this method, the observer would rotate the X and Z axes relative to each other through four possible configurations:

Axial rotation New perceived axis order Corresponds to (dimension)
+X <-Z ->Y -Z (-Dim3) <-> Y (+Dim3) <-X 0/270 degrees clockwise around Dim3
+Z <-X ->Y -X (-Dim3) <-> Y (+Dim3) <-Z 90/270 degrees counterclockwise around Dim3
-Y <-X <-Z -X (-Dim3) <-> Z (+Dim3) <-Y 180/270 degrees around Dim3
-Y <-Z <-X -Z (-Dim3) <-> X (+Dim3) <-Y 270/270 degrees around Dim3
Where "-/" indicates reversal of direction while "<->" signifies opposite directions in both dimensions simultaneously. Note that these correspondences assume that you view upward in your virtual reality as pointing towards positive values in the Y coordinate. If you prefer negative, simply change all plus signs to minuses and vice versa; I just use positives here for simplicity. Also note that these angles give rise to the most straightforward forms of the four basic Necker Cube types, which provide a basis for understanding more general 4D space. As far as anyone knows, there exists no stable, fixed "up" direction in hyperspace itself due to symmetry constraints dictating that every point lies equally distant from any other point in n-space. Only with respect to lower-dimensional subspaces like those defined by particular initial observer positions and choices does one obtain something identifiable as an "up."

Me:
Can you show this information visually?

HuggingFace Chat:
Yes! Here's a visual representation of the axial rotations described above, accompanied by two additional diagrams that illustrate the resulting spatial arrangements and a brief description of each configuration:

Visual Representation of Axial Rotations:
In this figure, the observer starts with W oriented vertically on top, and then performs the specified axial rotations. Each observer location is labeled based on the corresponding table row index. Moving outward and clockwise corresponds to moving closer to the origin of the 3D hypercube in your local reference frame after accounting for your position as a black dot inside this space, i.e., taking into consideration the fact that even an infinite number of observers cannot occupy a single point within the hypercube without overlapping completely. Constructive interference can occur only for some special cases where certain observer positions coincide. These form distinct exceptions to the rule and might not hold generally.)

Observer Position World Orientation (WXYZ Angles) Hypercubic Coordinates
1 0°/0°/-90° (0, 0, 0), (0, 1, 0), ..., (0, 0, 0), (0, 1, 0), ...
2 0°/-90°/90° (0, -1, 0), (-1,-1,0), ..., (0, 1, 0), (1, -1, 0), ...
3 90°/0°/-90° (1, 0, 0), (0, 0, 1), ..., (-1, 0, 0), (0, 1, 1), ...
4 90°/90°/0° (1, 0, 1), (0, -1, 1), ..., (-1, 0, 1), (0, -1, 0), ...
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