ChatGPT and its Misunderstanding of 4D

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Sat Dec 21, 2024 6:08 am

No worries Hugh.
I've gone with a slightly modified toe count...
Toes B.jpg
17 Toes each on 3 Feet
Toes B.jpg (19.95 KiB) Viewed 2188 times

Which gives the rotations as...
Image
Legs 360 B.jpg
Rotation of leg from x thru w for 360° at 30° steps
(30.75 KiB) Not downloaded yet

It is still a primary number of toes which is cool.
Now 17 toes per foot, 5 major toes, and 1 big toe! The 4D beings would count in base 51! At least they have the space now to draw the extra unique digits...
It also feels better because we are going from a line of toes to something like a triangle of toes overall.
Feels then like the next step up would be almost a tetrahedron of toes for 5D beings...
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Re: ChatGPT and its Misunderstanding of 4D

Postby Hugh » Sun Dec 22, 2024 10:44 pm

Here’s the latest attempt by ChatGPT4 based on your images, gonegahgah.

Image

As always, trying to create a 4D concept with a 2D image is challenging. :)
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Mon Dec 23, 2024 11:07 am

What I did was concentrate on a footprint. In four dimensions that is 3D.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Mon Dec 23, 2024 12:59 pm

Thanks for trying to produce those pictures for me, Hugh.
That's looks kind of painful what ChatGPT has tried to draw...
Makes me curious what is going through ChatGPT's "brain" to come up with those interpretations?

Maybe it requires a conversation with ChatGPT that gets to the point where ChatGPT understands the process?
I'll try to get ChatGPT to the point where it understands the process verbally and then I'll submit the whole chat here...
Though it does seem to have its good days and its bad days!
Maybe then it will be able to understand what to draw with reference to my previous full conversation with it?

I would also suggest, if rotation is going to be used in the depictions, that the rotation is through the vertical direction around the heel.
Looking at a ferris wheel of feet about a heel would be more easier for reader's brains to reprocess as being flat than rotating them around sideways and trying to process them as being forward...
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Wed Dec 25, 2024 12:26 am

Here's a conversation I had with ChatGPT about a 4D being's walking gait.

Yes, I absolutely get the general idea! You're describing a fluid, overlapping pattern of movement where the legs alternate not just in stepping but also in supporting the body’s weight, lifting, and moving forward, almost like a tri-pedal gait with a smooth flow. It’s very similar to how humans walk, but with three legs. The pattern you've described involves specific phases where each leg takes turns supporting the body and moving forward, while the other legs coordinate their motions for balance and progression.

Let's break it down based on the structure you've provided, while making the transitions and leg movements clearer:

Modified Three-Legged Walking Pattern (Based on Your Concept)

1) Initial Phase:
Legs 1 & 2: Begin to bear the weight of the body. Both legs start to support the body's weight as the third leg prepares to lift.
Leg 3: Lifts off the ground and starts moving forward. The motion is smooth as it starts to propel the body forward, while legs 1 and 2 bear the weight.

2) Weight Shift and Movement:
Leg 1: Quickly bears all the weight of the body, solidifying its role in stabilizing the body.
Leg 2: Lifts and begins to move forward while leg 1 stays grounded and takes on the body’s weight.
Leg 3: Continues moving forward. The first step is set, and leg 3 moves through its phase, preparing for its footfall.

3) Contact and Transition:
Leg 3: Hits the ground with a quick heel strike, followed by the full foot contact. Now leg 3 is fully supporting some of the body weight.
Leg 1: As leg 3 lands, heel rises from the ground, and leg 1 moves towards its “phalanges” (front portion of the foot) while leg 2 continues moving forward.
Weight Transfer: The weight shifts from leg 1 to leg 3 as leg 1 lifts and starts to move forward.

4) Next Phase of Movement:
Leg 1: Now lifts and starts moving forward, while leg 3 is fully planted, supporting the body.
Leg 2: As leg 1 starts moving forward, leg 2 hits the ground with a quick heel strike followed by full foot contact.
Leg 3: Heel rises from the ground, shifting to the phalanges (front portion) of the foot while leg 2 continues moving forward.
Weight Transfer: The weight now moves from leg 3 to leg 2, as leg 3 lifts and starts moving forward.

5) Repeat the Cycle:
Leg 3: Continues moving forward as the cycle progresses.
Leg 1: Hits the ground with a quick heel strike and then full foot contact.
Leg 2: As leg 1 lands, its heel rises, and leg 2 continues moving forward, providing support.
Weight Transfer: The weight moves from leg 2 to leg 1.
Leg 2: Lifts and starts moving forward, while leg 1 continues to progress, repeating the motion.

6) Continuous Overlap:

The process continues, with each leg alternating between bearing weight, moving forward, and providing stability. The pattern overlaps in a smooth, continuous cycle, ensuring the body stays balanced while progressing forward.

Key Points of the Overlapping Pattern:
Weight Shifting: Each leg alternates between bearing the body’s weight and moving forward, with the transition happening smoothly and quickly.
Fluid Coordination: The heel strike and full foot contact happen in sequence for each leg as it takes its turn moving forward. The other leg is in the process of lifting, continuing the motion. This makes for a seamless flow of movement.
Balance Maintenance: While one leg bears weight, the other two legs are always moving in sync to prevent imbalance. The phalanges positioning (the front part of the foot) ensures proper weight transfer as the body moves forward.
Alternating Support: The weight constantly shifts between the legs in a cycle. No leg is overly burdened, and the 4D being can move efficiently without a moment of instability.

This pattern mimics a tri-pedal gait where all three legs are continuously involved in the movement cycle. The overlapping actions of lifting, planting, weight-bearing, and stepping forward are designed to keep the body moving in a fluid and balanced manner, just like a human walk, but adapted for three legs. Each leg takes turns supporting and propelling the body forward while maintaining balance through the precise timing of these transitions.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Thu Dec 26, 2024 6:14 am

Me replied:
But that's the thing. It is the first question and it is identical. There isn't anything that I've done that should explain your giving completely inconsistent answers to each other for exactly the same identical first question. Do you have a random generator or something as part of your model?

ChatGPT said:
ChatGPT
Ah, I see what you’re getting at now, and I think I can better explain why you're experiencing this inconsistency, even with exactly the same question. You're absolutely right that when you ask an identical question, my answer should, in theory, be consistent. However, the reason you see variation in responses isn't due to random generation in the sense that I’m intentionally producing wildly different answers—it's actually due to how my model works behind the scenes.

Here’s a clearer explanation:

1. Probabilistic Nature of My Responses:
I'm built on a model that uses probabilities to generate responses based on patterns in language. Each time I generate an answer, I don’t simply pull from a static list of answers. Instead, I assess which words or ideas are most likely to follow from what’s been asked, based on a massive amount of text data. This probabilistic process means that, even for the same question, there's a range of possible ways I could phrase an answer or structure the response, and I might lean toward one or another depending on subtle factors in the input.

For example:
First Answer: I might focus on one aspect of the question that seems most relevant to me based on context or common language patterns.
Second Answer: I could interpret the phrasing slightly differently and lean toward another valid but distinct response.
So, the variation you see isn’t entirely random, but is due to the fact that my model generates responses based on probability distributions. Each time you ask the same question, I pick different likely words or approaches based on the data I’ve seen, which can lead to variations in my answer.

2. Sampling and Beam Search:
When generating a response, I often use a method called sampling (or, in some cases, beam search) to explore multiple possible responses and then choose one. While I aim for consistency, the underlying sampling process can sometimes lead to different phrasing, different logical structures, or different interpretations of the question.

Sampling: This technique involves picking the most probable next word or phrase at each step in the generation process, but there's still some variability, meaning the same question can lead to different responses depending on the sampling.
Beam Search: This method looks at multiple candidate answers and chooses the most likely overall response based on a combination of factors. Even with beam search, the resulting answer can vary based on which candidates are prioritized.

3. Contextual Interpretation:
Even if the exact same question is asked, there may be subtle differences in how the context is interpreted. In some cases, I might focus on one aspect of the question and in other cases, I might emphasize another—depending on how I parse the words, phrases, or nuances of the sentence.

4. No True "Randomness" but Variability:
While I don’t introduce a true random number generator into my responses, the process of generating language can still feel inconsistent. Because of the probabilistic nature of my model, slight shifts in input (even if they’re imperceptible) can cause different outputs.

How This Leads to Inconsistent Answers:
In short, my answers may vary even when the question is identical because of the probabilistic and generative nature of the language model. I’m not deliberately giving random answers, but there’s a level of flexibility in how I generate responses that results in different formulations.

What This Means for You:
If you're seeing drastic differences in answers to the same question, it’s a result of these factors. Ideally, I should be able to provide a consistent answer, especially for factual or structured questions, but due to how I generate text based on probabilities, some variation is inevitable.

If you'd like, I can always try to refine a specific answer more carefully, and we can iterate to get closer to what you're looking for. I’m here to help in any way I can!

Let me know if that helps clarify things, or if you'd like to dive into anything else!
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Sat Dec 28, 2024 5:51 am

There are two very strange things about four dimensional rotations. The first is that rotation is around a plane instead of a line. In 3D every atom in a rotating thing revolves around the closest point on a line. In 4D every atom in a rotating thing revolves around the closest point in a plane. I find this pretty much inconceivable. The second strange thing is that it is usual to have two planes of rotation going at the same time. The periods of rotation need not have any relation to one another. I find this even more impossible to imagine. If the two periods of rotation are different then each atom not on these two planes moves in a complicated path that might not ever repeat. I wrote computer programs to show such rotations using the usual method of displaying only three dimensions with the fourth hidden. The results are visible here. https://www.youtube.com/@glassbeadgame5352/videos. Here's the most popular one, which shows simple rotations and double rotations with the same period. https://www.youtube.com/watch?v=QZcVF9J7RiQ

So far the best I can do with rotating people is this. Imagine a person whose legs go left and right, eyes ana-kata, and four arms both left-right and ana-kata. We can't see the ana-kata dimension. So at first we see a person who looks normal except they have a single cyclops eye, as if you take away ana-kata the two eyes occupy the same space. As the person rotates 90 degrees the two legs gradually merge into one. The eye separates into two which gradually reach a normal distance apart. The arms shrink while a second pair of arms at right angles to the first appears and "grows" to normal size. The second 90 degrees of rotation leaves a figure that looks the same as the starting figure, unless there is some distinguishing feature like a ring on the left hand which would now appear on the right.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Tue Dec 31, 2024 4:47 am

Thanks Patrick, they look very interesting.
Giving a good 4D impression from rotation is a challenging thing.

I was having some trouble with my whole 4D 3 step walking pattern... until I realised an aspect that I didn't need to really think about with a 3D 2 step walk.
Here was what I came up with for a 3D walk graph just showing each consecutive step stage incorporating some interstep lifted foot positions.
2Step.jpg
Graph 3D 2 step.
(82.59 KiB) Not downloaded yet

Originally I didn't have the angled rectangle depicting the torso and that was the important consideration for 4D.

Without adding the torso things just weren't adding up.
What the torso told me was that the free leg (alternating between 1, 2, 3 of the 3 legs) is dragged along with the torso as it levers from back to front.
So I was able to come up with the following for a 4D walk graph finally.
3Step.jpg
Graph 4D 3 step.
(254.2 KiB) Not downloaded yet

Noticeably, our 3D two step achieves a rhythm from step 2 onward. We use a ½ step to get going.
But, the 4D three step achieves a rhythm only from step 3 onward. It has this ½ step, followed by a quasi step, then we are finally in 3 step rhythm.

Something interesting I noticed was that a 3 step walk is no faster than a 2 step walk.
However, I do think a 3 step run would be an whole half extra faster than a 2 step run?

Can anybody see any problems with this step pattern as shown?
There most likely are a whole range of movement patterns, that we wouldn't automatically think of, that they might use?

I'll add a key later to help as the elements might need some explaining?
Last edited by gonegahgah on Tue Dec 31, 2024 6:43 am, edited 6 times in total.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Tue Dec 31, 2024 5:30 am

Image
Image
Key:
Positions.jpg
Feet positions during walk
Positions.jpg (30.72 KiB) Viewed 2077 times
Torsos.jpg
Torsos
Torsos.jpg (14.38 KiB) Viewed 2077 times

The 4D torso is shown as an approximate slice of the 4D being torso.
It could be many shape slices but I've gone with the equilateral triangular prism approximation.
Of course it wouldn't be a perfect triangular prism just as we are far from a rectangular cross section.

It is also possible that a 4D beings arms will be mid-offset from the legs.
This would benefit animals with 6 legs running even more freely without worrying about tripping themselves.
But, how will we rest our hands on our knees if they don't line up?
Questions to ponder...
Last edited by gonegahgah on Wed Jan 01, 2025 7:55 pm, edited 3 times in total.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Tue Dec 31, 2024 8:16 am

The 2 step has a similar start to the 3 step but with just legs A and B, with the 3 step adding a leg C starting sequence.
Starting Steps.jpg
Starting steps
Starting Steps.jpg (27.71 KiB) Viewed 2098 times

After the starting steps a common rhythm is then established with each leg from each previous foot position.
These are similar between the two step and three step walkers being, for the 3 step pattern, basically just longer.
Continue Steps.jpg
Continuation Steps
Continue Steps.jpg (20.26 KiB) Viewed 2098 times

I thought:
It also looks like the 3 step walker would walk a slight amount faster than the 2 step walker.
The diagrams above depict the 3 step walker moving his legs slower so if they were the same speed they would cover each step distance more quickly.

But thats not so. It's dependent on the speed of the transition between steps and the step distance, and not the leg speed.
I'm fairly certain of that but correct me if I am wrong.

2 Step Track.jpg
2 Stepper Track
2 Step Track.jpg (23.79 KiB) Viewed 2051 times
3 Step Track.jpg
3 Stepper Track
3 Step Track.jpg (29.28 KiB) Viewed 2051 times
These would seem to confirm the latter.
Both these show the 2 and 3 legged beings walking 6 steps each.
Both final feet finish at the same distance.
And it is more apparent here that speed is set by the transition from one foot planted to the next planted...
Last edited by gonegahgah on Wed Jan 01, 2025 8:37 pm, edited 3 times in total.
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Tue Dec 31, 2024 8:46 am

You've convinced me that the angled narrow feet trick isn't the best. Instead have the foot be wider in the ana-kata dimension. Less than the forward-back dimension, more than left-right width which the legs are already taking care of. Call it the w-width.

Here in 3D if I want to turn around I raise up my left foot and put it down a foot in front of me pointing 90 degrees to the right, then turn my right foot in place. That is, I move my left foot from [-1, 0, 0] to [0,0,1]. Repeat three more times for a total of four.

Let's do that with vectors.
The right foot stays at [1,0,0] the whole time.

For the left foot

coordinates|ball of foot orientation
-----------------------------
[-1,0,0] left-right
[1,0,1] forward-back
[3,0,0] left-right
[1,0,-1] forward-back
[-1,0,0] left-right


The 4D case is similar. The right foot stays at [0,1,0,0] the whole time. For the left foot all we do is swap the 3D z column for the 4D w column.

[w,x,y,z] w is ana-kata

coordinates | w-width orientation
-----------------------------
[0,-1,0,0] left-right starting position
[1,1,0,0] ana-kata
[0,3,0,0] left-right
[-1,1,0,0] ana-kata
[0,-1,0,0] left-right back at the start
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Tue Dec 31, 2024 8:18 pm

Thanks Patrick.
Rotations are an important consideration.

It might also be worthwhile to do an examination by rotation.
If we take a 4D leg it might possibly still have our combination of a foot, lower leg, knee, and upper leg, and then connected to their 4D body.

If we examine the knee first, then like ours can, it may be able to be vertically straight through to allowing the lower leg to bend from the knee backwards.
We can move the upper leg both backwards and forwards around the connection to the body (and other angles too but I’m just focusing on the forward backwards bending presently).

Focusing back on the knee we can consider that our knee pivots around a line at the centre of its rotation and presumably the 4D beings knee also would pivot the lower leg backwards and return to straight around something too.

When we consider a 3D leg we can kind of consider it to be like a cylinder with bumps, on certain parts of the surface, at various points on the circumference and length.
For a 4D being their leg further extends into the 4th dimension.
Stepping our 3D slice into 4D we would see the leg appear small, grow bigger to full 3D size, and then shrink until it disappears.

But, instead of stepping into 4D we can also examine their leg by rotating into 4D.
The leg will then instead appear to remain as a cylinder, with various changing bumps (if we keep the leg straight and centred sideways in the rotation).

Because we are rotating sideways this will have no effect on forwards or upwards except that these will rotate largely in place because they are near the centre of rotation.
For example if you rotate a small circle in 3D through the vertical-sideways then all forward and backward bumps will each all remain at the same distance forward or backwards.

When we extend this to 4D and rotate sideways through ana-kata - instead of vertical - then all things vertical (the leg) will remain at their previous height, and all things forward (the foot) will remain pointing the same distance forward.
So the leg remains upright, and the foot remains pointing forward, when we rotate sideways in 4D from left to ana to right to kata back to left, etc.

The result of this rotated examination (instead of stepping into 4D) is that the leg will appear to remain as a leg throughout the rotation with the foot remaining in the forward direction at all times (and the leg remaining upright).
The toes will go through some noticeable morphing throughout the rotation but the overall solidity of the leg and its appearance will remain fairly consistent during the rotation.
(The muscle arrangement will affect the leg shape as we rotate but that’s for later discussion).

This equates with your discussion, Patrick, that the 4D rotation occurs around a 3D plane.
When we rotate purely in the sideways in 4D, the forward and vertical continue to remain forward and vertical.
The equivalent in 3D is that when we spin sideways through the vertical, like a cartwheel, our nose remains equally forward throughout the process.
Though in that case the nose changes vertical height. If we rotate purely sideways in 4D, the nose would remain at the same height and forward direction.

So the upper and lower leg, knee and foot will remain throughout our rotation examination as long as we keep them in the centre of the 4D sideways rotation.

This is the first part that I want to describe just to put forward the idea of using a rotational examination instead of a side step examination.
I always feel that a rotational examination can give us a little bit more insight than trying to examine things in 4D by taking side-by-side 3D slices.

I next want to use this rotational process to examine the 4D knee joint and look at the process of how it pivots...
Last edited by gonegahgah on Tue Dec 31, 2024 10:08 pm, edited 5 times in total.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Tue Dec 31, 2024 8:28 pm

Here is what I wrote above but re-written by ChatGPT. So it will be much easier to read than mine.
ChatGPT even has the temerity to say: "This version tightens up the language and organises the ideas to enhance clarity and coherence. I hope this helps!"
So, are you are trying to tell me that I am unclear and incoherant, ChatGPT?! Hey! Are you?!
But of course, it's right!

Rotations: An Important Consideration

Rotations play a key role in understanding movement, and it may be helpful to examine this concept through rotation itself.

Imagine a 4D leg. It might have a similar structure to ours, with a foot, lower leg, knee, and upper leg, all connected to a 4D body.

Let’s start by examining the knee. Just like in our own anatomy, the 4D knee may pivot, allowing the lower leg to bend backwards. The upper leg, in turn, can move forward and backward around the connection to the body (though we’ll focus on this forward-backward movement for now).

When we consider the knee, we can think of it as having a pivot point at the center of its rotation, which allows the lower leg to move forward and backward. Presumably, the knee of a 4D being would work in a similar way, pivoting the lower leg around an axis.

Now, let’s think about a 3D leg. We can imagine it as a cylinder, with bumps or joints appearing along its surface at various points. For a 4D being, however, their leg extends into the fourth dimension. If we were to take a 3D slice of their leg as it extends into 4D, we’d see it appear small, grow to full size, and then shrink until it disappears.

But instead of slicing through 4D, we can also examine the leg by rotating it into the fourth dimension. During this rotation, the leg will appear as a cylinder, with its bumps or joints changing as we rotate it, provided we keep the leg straight and centered in the rotation.

Because we're rotating sideways, this won’t affect the forward or upward motion of the leg, as those parts remain mostly stationary near the center of rotation. For example, if you rotate a small circle in 3D through a vertical sideways motion, the bumps along the circumference will stay in their original positions, maintaining their forward or backward distance.

When we extend this concept to 4D and rotate sideways (through ana-kata), rather than vertical, the leg remains upright, and the foot always points forward. As we rotate in 4D from left to ana to right to kata, the foot will stay pointed in the same direction, and the leg will stay vertical, as though it were locked in place.

This rotation gives the appearance of the leg staying solid and consistent throughout the motion. While the toes will undergo some noticeable morphing during the rotation, the overall shape of the leg remains stable. The muscles may change its appearance slightly, but that’s a topic for later discussion.

This observation aligns with your point, Patrick, that 4D rotation occurs around a 3D plane. When rotating purely sideways in 4D, the forward and vertical directions remain unchanged, similar to how, in 3D, a cartwheel rotation would keep your nose pointed forward. However, in 4D, if you rotate purely sideways, your nose (and everything else in the vertical plane) would stay at the same height and maintain its forward direction.

So, during this 4D rotation, the upper and lower legs, knee, and foot stay consistent. As long as we keep them centered in the rotation, they maintain their orientation.

This examination serves to illustrate the concept of rotational analysis, offering more insight than simply stepping into 4D and observing it through side-by-side slices. Rotational analysis helps us gain a clearer understanding of complex structures like the 4D knee joint, which I will now explore in more detail.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Wed Jan 01, 2025 4:54 am

What I want to examine is what is bending in a 4D leg.
Here is a 3D leg from side on which moves, in the way that we readily understand, from the knee joint:
Knee Rotation.jpg
Leg Bend at Knee
Knee Rotation.jpg (11.25 KiB) Viewed 2065 times

Presumably a 4D leg will bend in a similar fashion with regards to walking in a forward direction.
So this is why I want to take a look inside the 4D knee to see how this happens.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Wed Jan 01, 2025 5:30 am

With 3D rotations we look at the rotating object from one side or from its opposite side.
They will spin around a line in 3D. The bicycle wheel for example has an axle which acts as a line.
Square wheel.jpg
Square Wheel
Square wheel.jpg (27.09 KiB) Viewed 2058 times

From our current front side we could see the wheel spinning in one direction.
If we went around to the back we would see it spinning in the opposite direction.
This gives us the notion that the wheel has a front spinning side and a rear spinning side.
All points around the front stay the same distance in front of the wheel face.
All points around the back of the wheel stay the same distance back during the rotation.

In 4D we can do a similar thing by us rotating our 3D slice sideways in 4D while looking at things from the front, or the rear.
This time the top won't pass through the sides, as per our 3D wheel, and instead the object will remain upright and facing the same front.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Wed Jan 01, 2025 12:45 pm

It might help us first to take a look at a 2D being trying to understand our 3D Earth...

They can use the usual side step slice technique (left diagram).
Step 3D World.jpg
Sliced 3D Earth
Step 3D World.jpg (25.2 KiB) Viewed 2057 times
Or there is an alternative and that is to slice by rotation (right diagram).
As shown, to do this, the 2D being rotates our world around the pink line shown from front to back and takes slices at particular angles of rotation.

With the first technique you get changing diameter slices. The latter gives you similar diameter slices.
So what do the varying size of the first actually mean to the 2D beings brain?

The second, in my belief, will give the 2D being a better sense of our 3D world.
It says to them that our Earth consists of all these similar size circles rotated to form a sphere which is how our 3D brain processes it too.

This is the same disconnect that I believe occurs to us when we try to think 4D using side by side 3D slices.
And I hope to show that using the rotational slices will give us a better understanding of the 4D beings knee...
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Thu Jan 02, 2025 3:00 am

We can do similar ourselves with the 4D leg from our 3D world.
That is we can choose to step into 4D when examining the leg or we can alternatively rotate into 4D.
Step 4D Leg.jpg
Inspecting Leg
Step 4D Leg.jpg (27.65 KiB) Viewed 2046 times
I've shown two arrows in the leg diagram: 1. a straight arrow with little cubes on it to depict side step slicing, 2. a curved arrow with little cylinders on it to depict rotational step slicing.

First, for the Earth, you can see that the rotational "axis" remains a pink line. This pink line goes all the way forward and all the way back.
However, for the leg we instead have rotational "plane" shown in pink too. This pink plane goes all the way forward, back, up, and down.
Note: One thing to note about the rotational "axis" (or plane) is that these also mark the direction that distances won't change during the rotation.
So for the Earth, even though the continents change their left-right and up-down relative positions, they still remain the same distance forward with the axis.
The same for the leg if you rotate into 4D the leg undulations change their left-ana-right-kata relative positions to each other but the forward and up distances will not change.

As for the 2D being exploring the 3D Earth, as I said, I think you get more information if you inspect by rotation than you would by side stepping into the extra dimension.
If we were to instead side step the 4D leg, it would just appear to get smaller or larger before disappearing. As I said I don't think this helps our interpretation very much.
However, if we rotate around the leg we will see the full, similar sized, bumpy cylinder of the leg from all its various 4D sideways aspects.

This is even the case for the bent leg shown. As long as the plane is centred on the leg then when you rotate to inspect it the leg it will remain fairly much looking like a leg even with the foot present in all rotated slices.
This may feel counter-intuitive at first but the leg has bulk beyond our 3D space and when we rotate our inspection of the leg the rest of the bulk will reveal itself rotated slice by rotated slice.

If we understand this then it makes it easier to consider the mechanics of the knee...
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Thu Jan 02, 2025 3:39 am

Or as ChatGPT would more eloquently put it:

We can approach the concept similarly by considering the 4D leg within our 3D world. Essentially, we have two options: we can step into the 4th dimension to examine the leg, or alternatively, we can rotate it into 4D.

In the leg diagram, I’ve included two arrows to illustrate the two approaches:

A straight arrow with small cubes, representing a side-step slicing method.
A curved arrow with small cylinders, representing a rotational slicing method.
Now, let's first focus on Earth. In this context, you’ll notice that the rotational "axis" remains a pink line that extends both forward and backward. In contrast, for the leg, we have a rotational "plane," also shown in pink. This plane extends forward, backward, up, and down.

It’s important to note that the rotational "axis" (or plane) marks the direction in which distances do not change during rotation. For Earth, although the continents shift relative to each other in terms of left-right and up-down positions, their distance from [along] the axis remains constant. The same principle applies to the leg: when rotating into the 4th dimension, the leg’s undulations change their relative left-to-right and up-to-down [should be: ana-to-kata] positions, but the forward and upward distances will remain unchanged.

When it comes to exploring 2D representations of 3D Earth, I believe that rotation offers more insight than simply side-stepping into the additional dimension. If we were to side-step the 4D leg, it would appear to shrink or grow before eventually disappearing, which doesn’t add much to our understanding.

However, by rotating around the leg, we can observe the full, consistent shape of the leg as a bumpy cylinder from all its various 4D perspectives. This is true even for a bent leg. As long as the rotational plane is centered on the leg, rotating it will still present the leg in a recognizable form, with the foot appearing in every rotated slice.

This might seem counterintuitive at first, but the leg has a dimension beyond our 3D space. When we rotate our view of the leg, its additional bulk in the 4th dimension becomes visible slice by slice.

Understanding this concept makes it easier to consider the mechanics of the knee…
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Thu Jan 02, 2025 11:17 am

I figure in 4D the knee would be pretty much the same, a 1D forward-back hinge. Stability is what matters. The hinge is a necessary evil to allow for walking.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Fri Jan 03, 2025 1:11 am

Thanks Patrick, I think so too that the knee would be pretty much the same as ours.
And a hinge is certainly the answer. (Our knees are a very organic kind of hinge; and so for 4D).
But there is a thing that you mentioned yourself.

You wrote:
“The first is that rotation is around a plane instead of a line. In 3D every atom in a rotating thing revolves around the closest point on a line. In 4D every atom in a rotating thing revolves around the closest point in a plane. I find this pretty much inconceivable.”

Let’s try to make it more conceivable…

But, first off let’s look at where I'll be trying to go.
What I feel we are basically comparing now is the following.
2TLegs vs 3Legs.jpg
Are T-Legs better than 3-Legs?
2TLegs vs 3Legs.jpg (20.16 KiB) Viewed 2021 times
I've only provided an example showing the ana-kata, left-right, and up-down dimensions together.
The forward-backward dimension is not shown. Consequently I can't show the forward portion of the feet in this picture.
I can only show on the left the suggested ana-kata side extensions from the feet in relation to the legs left and right stance.

I would regard the three legs to be more stable and necessary than the two T-legs but is that even pertinent in 4D for our 4D being?
I think it is but I will need to more convincingly show this in subsequent posts...
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Fri Jan 03, 2025 7:34 am

"I would regard the three legs to be more stable and necessary than the two T-legs but is that even pertinent in 4D for our 4D being?"

On this matter you can't really prove anything one way or the other. Even here on Earth beings may have 2, 4, 5, 6, 8, or many legs. Crabs have to walk sideways, which is pretty weird.
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Fri Jan 03, 2025 8:23 am

You can draw a plane, then a bunch of parallel circles of various radii each with its center some point in the plane. That's like a basic 4D rotation is like. The circles may appear to intersect but they don't really.

Showing two perpendicular rotations at the same time, that's a challenge because we can't draw two perpendicular planes. That means, so perpendicular that they intersect only at a point. The path of most atoms is then a circle only if the two periods of rotation are the same. Otherwise they look sort of like 3D Lissajou figures. Some can be seen athttps://www.google.com/search?q=3D+lissajou+figures Here's a nice video of such with periods 3 and 2. https://www.facebook.com/watch/?v=934543556970916. Real 4D figures would be different but this gets the feel across.
Last edited by PatrickPowers on Sat Jan 04, 2025 4:39 am, edited 1 time in total.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Sat Jan 04, 2025 3:33 am

Thanks Patrick.
I must apologise because I am taking the long way but I just want to make sure that I'm connecting all the dots for myself.
The main apprehension I have with the T-feet is that I feel that for the 4D being it would feel very much like us trying to walk on a type rope all the time.
Possibly I could see just two legs if they are each very, very, very wide around into 4D taking up a good portion of the 4D sideways.
But, that obviously has much more "bulk" as a problem,

Let's see if this somehow comes out in my explorations or not.
I'm focusing on the knee because I think it is slightly different in 4D but not too much.
Hopefully I can show why?

But, sorry, back to the slowly, for me to connect my ideas for myself...

Let’s go back to our 2D being trying to understand our 3D world first.
Step 3D World.jpg
Stepping 4D Object
Step 3D World.jpg (41.31 KiB) Viewed 1590 times
Left: As I mentioned previously they could pass our world through their slice.
Each slice will be a slightly different sized circle or course.
Right: Or they could rotate our Earth around its axis into their slice.
This gives us multiple similar sized circles with a common axis.
As I said recently, I think the latter approach and data would be much more explanative to the 2D being.

However, it is possible that the 2D being could instead rotate their 2D slice, instead of the Earth.
Rotate Object or Slice.jpg
Rotate Object OR Observer
Rotate Object or Slice.jpg (49.09 KiB) Viewed 1590 times
Left: On the left is the more conventional rotating the 3D object.
Right: On the right we are rotating the 2D being’s slice from vertical towards the 3D sidewards.
Notably, the outcome is exactly the same for the 2D being...
Except, that the poor 2D being has to cartwheel upside-down in the process. Not much fun!

Likewise, in 4D we can choose to rotate the 4D object into our space, as conventionally done.
Or, alternately we can rotate our 3D space as though we are moving around the object.
The good thing with doing this for us is that we don’t have to stand on our heads.

Either way, we can rotate sideways in 4D without changing the forward direction of the 4D object and without changing the upside direction of the 4D object.

But again, the result is identical to us, so what is the point?
None really, except the acceptance that we don’t have to stand on our head when rotating.
And that this means that importantly the forward distances don’t change and equally importantly the vertical distances don’t change.
That is of course when we do either 4D rotation but this is more obvious with the later.

This has an impact when we depict the 4D knee joint.
You’ve already alluded to this Patrick so I will get to that next.
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Sat Jan 04, 2025 4:38 am

When I turn 360 my knee doesn't do anything unusual. It's all about twisting at the waist and turning at the hip. A hip has three degrees of freedom. In 4D a ball-and-socket joint like that would naturally have four degrees of freedom.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Sun Jan 05, 2025 5:21 am

Absolutely Patrick.
The knees primary purpose is bending around its hinge generally backward, or forwards again to a maximum of being straight. It’s direction of bend is dependent upon the hip rotation and upper leg position certainly. It allows us to walk, crouch and other things. Shock absorption is another important purpose of the knee.

I apologise for the lateness of this post. I was drawing the knee side of things but it occurred to me that it would be better for completeness to include a comparison of: the 3D being while rotating a 4D object, against, the 2D being while rotating a 3D object (our 3D Earth). So, I’ve left all the knee drawings for next.

So onto this additional comparison...
We can compare and contrast a 2D being moving our 3D Earth sideways through their 2D space, to us moving a 4D object ana-kata through our 3D space.
The object we will move here is a glome which is the 4D equivalent of a 3D being's sphere, and a 2D being's circle. It has an equal radius from its centre to its surface in all 4D directions.
Compare Side Slicing.jpg
2D & 3D Parallel Slicing
Compare Side Slicing.jpg (39.46 KiB) Viewed 1183 times
Left: Our spherical Earth produces circle slices in the 2D being’s slice.
Right: The 4D glome produces sphere slices in our 3D space as it passes through.
In both instances, the slices get smaller the further we have passed from the centre of the object (as shown to the right of each).
Once the object has passed fully through our 3D space it disappears. Same for the 2D being.

As mentioned already we can also rotate objects around an axis in the lower dimension.
This gives us rotational slices instead of parallel slices.
Compare Rotate Slicing.jpg
2D & 3D Rotated Slicing
Compare Rotate Slicing.jpg (42.09 KiB) Viewed 1183 times
Left: Again, our spherical Earth is seen as circle slices when rotating around a 2D forward axis (which is shown as a pink line).
Right: Likewise, the glome is seen as sphere slices when rotating around our 3D up&forward axis (which is shown as a pink rectangle).
In both these, the slices stay full size; which I believe is easier to understand.
The left tells the 2D being that a sphere is a circle of circles spinning into the extra dimension and the right tells us that a glome is circle of spheres spinning into one more dimension.

It is also interesting to compare the axis in both the parallel and rotational slicing.
Compare 4D Slicing.jpg
Axis Presence
Compare 4D Slicing.jpg (39.81 KiB) Viewed 1183 times
When we push the object through the lower space, we only see the axis once. But when we rotate, the axis stays present for the entire rotation.

If we compare the rotation in 2D and the rotation in 3D as per the prior diagram we can see something interesting.
Compare Rotate Slicing.jpg
2D & 3D Rotated Slicing
Compare Rotate Slicing.jpg (42.09 KiB) Viewed 1183 times
We can see that the rotating line axis stays in view for the 2D being and the rotating plane axis stays in view for us as we rotate the glome.
For the 2D being, the 3D object rotates around a line axis. But, in our case the glome rotates around a plane axis.

The axis in both cases remains the same axis throughout the rotations but we just see it from a different rotation angle.
For example, the 2D being will always see the axis as pointing towards them when the axis is orientated from rear to front throughout the rotation. If you imagine that the line axis has an up side then when it rotates to the right the up angle will now be around to the side.
The same occurs for the plane axis that the glome rotates around in our 3D space. It remains the same axis in the glome throughout. It is just looking at us from a different rotation throughout.

So the 2D line axis and the 3D plane axis (or maybe I should call it axle) are the same axis for every angle of rotation; just from a different angle.
So the axis (or axle) is common to every rotation inside the 2D or 3D being's space.

Which is pretty cool. So let’s next take a look at how this applies to the 4D being’s knee.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Sun Jan 05, 2025 5:51 am

And here's ChatGPT showing me up, again...

Absolutely, Patrick.

The primary purpose of the knee is to enable bending around its hinge, typically backward and then forward again, up to a straight position. Its direction of movement depends significantly on the hip rotation and the position of the upper leg. This functionality allows us to walk, crouch, and perform other motions. Another critical role of the knee is shock absorption.

Apologies for the lateness of this post. Initially, I was focused on drawing the knee-related aspects, but I realized that it would be more comprehensive to include a comparison of a 3D being rotating a 4D object against a 2D being rotating a 3D object (like our Earth). As such, I’ve set aside the knee drawings for now.

Moving on to the Comparison
Let’s explore a comparison between:

1. A 2D being moving a 3D object (our Earth) sideways through their 2D space.
2. Us (3D beings) moving a 4D object ana-kata through our 3D space.
The object in question is a glome, which is the 4D equivalent of a 3D sphere or a 2D circle. A glome has an equal radius from its center to its surface in all 4D directions.

Parallel Slicing
Image
2D & 3D Parallel Slicing

Left: Our spherical Earth creates circular slices when viewed from the perspective of a 2D being as it moves through their plane.
Right: Similarly, a glome creates spherical slices in our 3D space as it passes through.
In both scenarios, the slices diminish in size the farther we move away from the center of the object (as shown to the right in each example). Once the object completely exits the lower-dimensional space, it disappears.

Rotational Slicing
Image
2D & 3D Rotated Slicing

Left: For the 2D being, a rotating 3D sphere appears as a sequence of circular slices around a 2D axis (illustrated as a pink line).
Right: For us, a rotating glome appears as a series of spherical slices around a 3D axis (illustrated as a pink rectangle).
In both cases, the slices maintain their full size during rotation, making this concept easier to grasp.

To a 2D being, a sphere is understood as a circle of circles spinning into an additional dimension.
Similarly, to us, a glome is perceived as a circle of spheres spinning into one more dimension.

Comparing the Axis
Image
When we push an object through a lower-dimensional space, the axis only appears momentarily. However, when rotating, the axis remains visible throughout the entire rotation.

For a 2D being, the rotating line axis of a 3D object is always visible.
For us, the rotating plane axis of a 4D object (the glome) remains visible throughout its rotation.
Interestingly, the axis is consistent throughout the rotation, but it appears differently depending on the angle.

For a 2D being, a line axis pointing directly at them will always seem to face them, even as it rotates.
Similarly, for us, the plane axis of a glome remains consistent, though its orientation shifts as the glome rotates.
This demonstrates that the 2D line axis and the 3D plane axis (or axle) serve the same purpose in their respective contexts, remaining constant across every angle of rotation.

Pretty fascinating, right? Next, let’s dive into how these concepts apply to a 4D being’s knee!
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Sun Jan 05, 2025 11:11 am

I thought I should just add this quickly.

I've shown rotating a 3D sphere into 2D, and I've shown rotating a 4D glome into 3D.
I thought it might be of interest to also show what happens when we rotate a 3D sphere from our 3D space into 4D...

Rotating Glome or Sphere
Compare Glome v Sphere into 4D.jpg
Rotating Glomes v Spheres to 4D
Compare Glome v Sphere into 4D.jpg (42.12 KiB) Viewed 1178 times
Left: Again the Glome rotating through our 3D space revealing sphere slices at every rotation.
Right: You'll notice that while the sphere itself disappears when it is rotated into 4D that it leaves behind an ultra-thin presence in our 3D space at all angles of rotation.
This only happens when the object is intersected, on or off the object's centre, by the common plane of rotation as in the above diagram.
If it were not touching the plane of rotation anywhere then the 3D object would disappear completely upon rotation around that plane.
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Re: ChatGPT and its Misunderstanding of 4D

Postby mr_e_man » Mon Jan 06, 2025 12:32 am

PatrickPowers wrote:When I turn 360 my knee doesn't do anything unusual. It's all about twisting at the waist and turning at the hip. A hip has three degrees of freedom. In 4D a ball-and-socket joint like that would naturally have four degrees of freedom.

Four? Shouldn't it be six?
The ball can rotate in any one of the 6 planes wx, wy, wz, xy, xz, yz.

Also, which 5-legged 3D creature were you thinking of?
"I would regard the three legs to be more stable and necessary than the two T-legs but is that even pertinent in 4D for our 4D being?"

On this matter you can't really prove anything one way or the other. Even here on Earth beings may have 2, 4, 5, 6, 8, or many legs. Crabs have to walk sideways, which is pretty weird.
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Re: ChatGPT and its Misunderstanding of 4D

Postby PatrickPowers » Mon Jan 06, 2025 1:08 am

mr_e_man wrote:
PatrickPowers wrote:When I turn 360 my knee doesn't do anything unusual. It's all about twisting at the waist and turning at the hip. A hip has three degrees of freedom. In 4D a ball-and-socket joint like that would naturally have four degrees of freedom.

Four? Shouldn't it be six?
The ball can rotate in any one of the 6 planes wx, wy, wz, xy, xz, yz.

Also, which 5-legged 3D creature were you thinking of?
"I would regard the three legs to be more stable and necessary than the two T-legs but is that even pertinent in 4D for our 4D being?"

On this matter you can't really prove anything one way or the other. Even here on Earth beings may have 2, 4, 5, 6, 8, or many legs. Crabs have to walk sideways, which is pretty weird.


Aha, a geometric algebra man. I guess you are correct. This is why I didn't mention degrees of freedom in my book. It's hard to deal with.

A starfish has five legs. An odd number of legs is quite unusual.
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Re: ChatGPT and its Misunderstanding of 4D

Postby gonegahgah » Tue Jan 07, 2025 1:27 am

Almost to the knee...

We can give similar treatment to a spherinder, which is a spherical cylinder, or a cylinder spun into 4D in a circle describing its bulk.
We can also do similar for a simple cylinder itself.

Rotating Spherinder & Cylinder
Compare Spherinder v Cylinder into 4D.jpg
Rotate Spherinder & Cylinder into 4D
Compare Spherinder v Cylinder into 4D.jpg (44.44 KiB) Viewed 1087 times
Left: The spherinder when rotated around a plane in our 4D into 4D shows as a cylinder at each rotation.
Right: If we do the same for a cylinder, then like the sphere it will disappear upon rotation into 4D however there will always be an ultrathin intersection between the rotations that remains in our 3D space.

Our lower and upper legs are kind of like a cylinder just with various bumps for muscles, front vs side, bone, and other internal parts.
Likewise for a 4D being their upper and lower legs are kind of like a spherinder but with their various 4D bumps around the perimeter.
Our feet are are a bit more squashed similar almost to an oval so theirs would be closer to an ovoid cylinder.

Next I'll get onto orientation of these and the effect of that...
Last edited by gonegahgah on Tue Jan 07, 2025 3:22 am, edited 1 time in total.
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