ChatGPT and its Misunderstanding of 4D

[gonegahgah]

I've noticed that ChatGPT is clever at putting things together but is also imaginative as well at times.

Case in point is that I asked it:
If you had 4D beings in a 4D world that had 3 legs, 3 eyes, 3 arms, 3 ears, 1 nose, 1 mouth, 3 feet (one per foot), and maybe 16 toes per foot (giving a total of about 48 toes), then which direction would their toes point?
Originally a few weeks ago it mentioned with this that "the toes of a 3D being generally point along the 'forward' direction of it body, ...".
Today it mentions that, "In our 3D world, beings typically have two legs that extend in a primarily vertical direction (in relation to the ground) with toes pointing outward or downward, depending on the stance."

That's interesting that it can change... though it also confessed that it forgets everything discussed between sessions...

What was more interesting is that it initially concluded during the first discussion that for 4D beings that, "Some toes would point forward (y), sideways (x), up (z), or even in the w-direction, which is a unique 4D "sideways" direction."
I became better, with each fresh subsequent discussion, at explaining to the new session that none of the toes will ever point up (except if your sleeping or something)!
But, this seems to be a repeated misconception by ChatGPT for each new session...

[gonegahgah]

Part of the trick seems like it may be to get ChatGPT to move from thinking X, Y, Z (and W for a 4D being) to forward-backward, up-down, sideways.
This seems to bring ChatGPT to a better conclusion. I am curious to find what minimal initial phrase is needed to get it to a useful 4D interpretation of the direction of the 4D toes?
It does seem to me to be evolving a sense of my question being within a 4D biome but I'm not absolutely sure of that. It's claim was that it won't remember anything from session to session.

When questioned ChatGPT responded:
"You're right that my reasoning about the toes pointing "up" was more of an extrapolation from mathematical structures rather than a natural, intuitive process rooted in how a living creature’s body might behave in 4D space."

Originally it felt like it was trying to convince me that the toes would point in all directions. Maybe that was a graph misinterpration more than a body relation interpretation.
However, it did still seem to be telling me, in the way that I phrased my questions that some of the toes would have to be pointing in the 4D sideways directions for the purpose of sideways rotation while facing forward.
It did take me a few days for me to get to the point of understanding what would actually be the correct answer to this question...

When questioned ChatGPT responded:
"This could mean the toes are articulated in a way that allows them to align with the direction of movement (forward, sideways, or rotational)."

But, is that so...

[PatrickPowers]

Suppose the N dimensional beings have two legs side by side. 2D beings don't need feet at all -- the left-right and up-down dimensions are already taken care of. 3D beings need feet that go in the new forward-back dimension. 4D beings need feet that are angled so that they extend in both the forward-back and ana-kata dimensions. By the Pythagorean Theorem, to have the same extent that they have in 3D the feet need to be longer by a factor of the square root of two. 6D beings need feet that are angled so that they extend in the forward-back dimension and in the extra three sideways dimensions. The feet need to be twice as long as those of 3D people. And so forth.

Two legs are all you need no matter how many dimensions you may have. All you need is longer feet and a different angle to the feet.

I found this riding on the Metro. Standing with legs at an angle of 45 degrees from the motion of the train gives stability in both the forward-back and left-right dimensions.

[gonegahgah]

I've been promoting your book Patrick on some youtube videos though youtube doesn't allow links to be included sadly.

I must admit that I was surprised that your comment here was that two legs were sufficient so I did a recheck of your book and, sure, that's what you have written:
"While 3D bodies cannot exist in a 4D Universe, the basic design of one head, two eyes, and two legs remains quite useful in four dimensions."
(from "Elsewhere: Everyday Life on a Hypergeometric Earth" by Patrick Powers).

So, forget about toes for the moment I guess...

Obviously in 3D we balance quite well on our legs with just the two.
They work effectively to stop us falling sideways just by defending the sideways directions, one left, one right.
Having feet that project frontwards help us to not fall forward or backwards, as well as serving some cushioning and springing effect at times.
On that we agree. I certainly agree too that an extra leg takes more energy than extended feet.

So your suggestion seems to be to have a semi-circular like foot that extends around to the ana-kata directions from the front. (Or is it to have T-feet?)
Why ana-kata particularly I'm not sure but maybe that's just arbitrary and it just has to be any pairing of opposite sideways?
This is remembering that all sideways are equally sideways in 4D...

Another alternative in your 4D world could be to have semi-sphericular feet (taking it even further) where the foot extends from the front around to all sideways directions.
The closest equivalent in our world have been to have semi-circular feet. We do have some birds whose feet spread out, like ducks, and also those ones that walk on lilies.
Us having two side-by-side legs on land means that of course is not necessary for us and our feet are largely pointing forward.

So what is the trade off?
The problem is the muscular support for the feet.
Our legs and feet are driven by muscles that are largely front muscles and back muscles with some smaller muscles in the feet and some movement control by tendons on top of the foot and in the Achilles.
These together gives us extension and retraction which is a necessary pairing. This is definitely a number 2 type article of inclusion (type 1 - mouth, type 2 - muscles, type 3 - ?).
For something of our bulk there is not really enough space to additionally provide side muscles (bulk is even more of an issue in 4D).
I mean, we do have twisting muscles, but the majority action is forward-backward.

But, first off, my question to you is how much of your 4D sideways needs to be covered with your 4D feet extending around?
Is it strictly just forward and both ana-kata? (Which can equally be considered to be left-right at other times or any other sideways angle).
Or would you wrap the feet around even more into the 4D perhaps even to fully semi-sphericular foot print?

[PatrickPowers]

I like to go for minimal weight and materials so I figured the feet would be just as narrow as what we have here in 3D, no matter how many dimensions you've got.

"This is remembering that all sideways are equally sideways in 4D..."

In this case we already have the two legs to the left and right (by definition). So ana-kata is ninety degrees from that. Forward-back is distinct because legs are locomotion. Once we've gotten this far then these three directions are all different. There then are two basic schemes. You can have one foot angle off in the 45 degrees between forward and ana, the other angles off in the the 45 degrees between forward and kata. More slick is if both feet angle off 45 degrees between forward and ana. At first this seems screwy, but consider that our mundane 3D feet stabilize us not only in the forward direction but also back. This is because the center of gravity is over the center of the foot, not above the ankle. If we start to fall backward then we can reduce pressure on the toes. Taking it a step further, we can even balance on one foot here in 3D if we feel like it. Our foot gives us some stability that way. So maybe we don't even need that 45 degree angle. The forward direction is more important, so it could be that 20 degrees between forward and ana is enough. Or maybe even zero degrees. Who knows? Maybe we don't actually need longer feet.

Bird feet -- the are designed for minimal weight and grabbing twigs, with walking secondary. I figure they need N+1 toes.

"For something of our bulk there is not really enough space to additionally provide side muscles (bulk is even more of an issue in 4D)." Something I haven't considered. Muscles flex in one dimension only. So it seems my usual economy doesn't apply in this case. You've got to have more muscles, small ones for balance. Now something that I didn't know circa the 0.3 or whatever version of the book is that muscles in 4D are far stronger. The strength of a muscles depends on the cross sectional area. The cross section of a 4D muscle is 3D, so it contains a lot more mass. Consider that a big strong 120kg 4D man is 3 millimeters high. A leg muscle is only 1 mm long but has the same mass we do. So the cross section is much larger than here in 3D. (It also means that those Flatland 2D creatures would have tiny 1D muscular cross sections and would be very weak, but never mind that.) So you can make the muscle a lot smaller in radius and still have plenty of strength.

We can see this sort of thing here in our 3D world. Consider the ant. It's common to see an ant dragging something much heavier than itself. It's because the muscle-cross-section-to-weight ratio of an ant is much higher than ours. So they are proportionally much stronger than we. All and all the physics that 4D people would have to deal with would be a lot like that of our 3D insects. The surface-to-mass ratio is a thousand times higher for a 4D person than a 3D person. I figured that terminal velocity when falling is about seven miles per hour. 4D people would have no need of parachutes when jumping out of airplanes, just like our insects. There's so much friction that rock climbing would be a lot easier. Maybe they don't even bother with space-consuming stairs and just climb walls. I

[gonegahgah]

"I like to go for minimal weight and materials so I figured the feet would be just as narrow as what we have here in 3D, no matter how many dimensions you've got."
.. I would agree with this, just remembering of course that bulking up from 3D to 4D is as an exponent factor.

"In this case we already have the two legs to the left and right (by definition). So ana-kata is ninety degrees from that."
.. Fair enough.

"Forward-back is distinct because legs are locomotion. Once we've gotten this far then these three directions are all different."
.. No worries.

"There then are two basic schemes.
You can have one foot angle off in the 45 degrees between forward and ana, the other angles off in the the 45 degrees between forward and kata.
More slick is if both feet angle off 45 degrees between forward and ana."
.. Instead you are trading off foward-backwards with ana-kata.
It doesn't in this case matter which angle they are pointing to between ana-forwards-kata-backwards as these are all the same to each other in this case.
You are just making the balance direction be whichever angle the feet are facing. This still leaves the remaining perpendicular unbalanced with nothing holding it in place.

Also, just because your face and torso are facing a different direction doesn't mean that they will contribute anything to your balance.
Whether your feet were exactly like ours pointing forward, or they're pointing fully ana, or fully kata, or one ana and one kata, or any angle in between, they will offer you no more support than what ours do, for your example.
Maybe you could have one pointing forward and one pointing kata but that is going to make running extremely interesting... Sort of a hop step basically like missing one foot.

Your feet really need to be pointed in the direction that you want to run which should really be the direction that you are facing. Not ana, or kata, or something in between.
You really don't want them pointing towards some other direction... It stops being just an issue of balance then.

"At first this seems screwy, but consider that our mundane 3D feet stabilize us not only in the forward direction but also back. This is because the center of gravity is over the center of the foot, not above the ankle. If we start to fall backward then we can reduce pressure on the toes. Taking it a step further, we can even balance on one foot here in 3D if we feel like it. Our foot gives us some stability that way. So maybe we don't even need that 45 degree angle. The forward direction is more important, so it could be that 20 degrees between forward and ana is enough. Or maybe even zero degrees. Who knows? Maybe we don't actually need longer feet."
.. This feels like your improving sense of the matter is growing even further with these thoughts so excellent...

"Now something that I didn't know circa the 0.3 or whatever version of the book is that muscles in 4D are far stronger."
.. Absolutely, gravitational considerations aside (always the 4D elephant in the room), they may have to strain across a broader output area.
They are also shorter as 4D beings are going to tend to be a lot more squatter than we are just as 2D beings would most likely be much taller proportionally than us...

"Consider that a big strong 120kg 4D man is 3 millimeters high. A leg muscle is only 1 mm long but has the same mass we do."
.. Cool, sorry I wrote first than read on... Though it is always difficult to compare bulk between dimensions because it is always infinity by the next exponent greater than the last lower dimension.

"The surface-to-mass ratio is a thousand times higher for a 4D person than a 3D person."
.. Surface to mass ratio is certainly an interesting beast and is, I believe, the main setting factor for the constants of nature and size limitations in our 3D universe too.

"Maybe they don't even bother with space-consuming stairs and just climb walls."
.. There's not really much point to upper floors in 4D for "space" purposes as there is just so much of that. Maybe for observation purposes?

Thank you for your counter arguments. There are a lot of interesting things in there to consider.

[PatrickPowers]

"Though it is always difficult to compare bulk between dimensions because it is always infinity by the next exponent greater than the last lower dimension."

I assume that atoms have the same radius and mass no matter how many dimensions you have, that chemistry is the same. There's no reason to believe this. Indeed there is no reason to believe that protons or electrons or photons exist in 4D, but if you don't assume such things then one is limited to imaginary geometric shapes. That's not my game.

[gonegahgah]

I'm not sure what you are saying with that? Perhaps it is the same with what I said?
I assumed I was working under 3D gravitational rules and I assume you are too and I was ignoring the difficulties of things like spirals in a 4D world, etc.
But I was still working on different step ups between dimensional additions.
Like, a cube is made of infinite squares and a tesseract is made of infinite cubes.
But, the proportion of infinite cubes to the single cube (another infinite) is immeasurably bigger than the proportion of infinite squares to the single square (another immeasurably smaller infinite).
Proportions are interesting because they affect things like cell sizes in real life. It was good to see you interest in those things.
But, it's a side issue.

Sorry, did I say something wrong?
I'm more interested in the interesting thing I realised about feet and toes in relation to the 4D being rotating while remaining upright and facing forward?
I was hoping the conversation would circle back to that...

I'm certainly way more into the possible biologies than geometric shapes.

[PatrickPowers]

I'm not sure what you are saying with that? Perhaps it is the same with what I said?
I assumed I was working under 3D gravitational rules and I assume you are too and I was ignoring the difficulties of things like spirals in a 4D world, etc.
But I was still working on different step ups between dimensional additions.
Like, a cube is made of infinite squares and a tesseract is made of infinite cubes.
But, the proportion of infinite cubes to the single cube (another infinite) is immeasurably bigger than the proportion of infinite squares to the single square (another immeasurably smaller infinite).
Proportions are interesting because they affect things like cell sizes in real life. It was good to see you interest in those things.
But, it's a side issue.

Sorry, did I say something wrong?
I'm more interested in the interesting thing I realised about feet and toes in relation to the 4D being rotating while remaining upright and facing forward?
I was hoping the conversation would circle back to that...

I'm certainly way more into the possible biologies than geometric shapes.

This is all reasonable. I'm going to think about feet for a while before responding.

[Hugh]

Hi gonegahgah, nice to see you exploring ChatGPT!

I asked ChatGPTo1 to summarize this thread and attempt an answer to your original question.

ChatGPTo1:

Below is a summary and explanation of the conversation, followed by an attempt to answer the original question posed by gonegahgah:

Summary of the Conversation:

The discussion begins with gonegahgah describing an interaction they had with ChatGPT about a hypothetical 4D being. This being supposedly exists in a 4D world and has a body structure that includes 3 legs, 3 feet, and multiple toes on each foot (48 toes total). The original question posed to ChatGPT was: “In which direction would such a 4D being’s toes point?”

Key Points Raised:
1. ChatGPT’s Changing Explanations:
Initially, ChatGPT tried to draw analogies from 3D beings, suggesting toes point “forward” in the creature’s body frame. In 4D, it momentarily suggested toes could point in the additional w-direction (the fourth dimension), or even “up,” before corrections were made. Over time, the user got ChatGPT to shift from pure geometric extrapolations (simply adding a “w-axis”) toward using more naturalistic reasoning aligned with how limbs and feet might function in a 4D biome.
2. Difficulties in Reasoning About 4D Directions:
The conversation shows that interpreting directions in 4D is tricky. In 3D, we have forward-backward, left-right, and up-down. In 4D, there’s an extra set of perpendicular directions (often called ana-kata) that are neither simply “up” nor “sideways” as we understand them. Early attempts by ChatGPT conflated mathematical axes with biological orientation, leading to confusion.
3. Evolving Understanding of Stability and Locomotion in 4D:
Participants, including PatrickPowers, discussed how beings in higher dimensions might still only need two legs for balance and how feet and toes might be arranged to handle additional directions. In 4D, legs and feet could be angled or shaped to provide stability not just forward-backward and left-right, but also in the ana-kata directions. However, several key points arose:
• Just as in 3D we mostly have forward-facing feet, 4D beings would likely want toes oriented along their primary direction of locomotion (the “forward” direction they face).
• To handle balancing in extra dimensions, feet might be somewhat more complex, or angled to provide support in additional “sideways” directions.
• There were suggestions that the complexity of muscle arrangement and biological design could differ substantially in 4D, possibly allowing more compact yet stronger muscles due to differences in how volume and cross-sectional areas scale with dimensionality.
4. Minimal Foot Modifications vs. Complex Multi-Directional Feet:
On one hand, larger or differently angled feet might help a 4D being maintain stability. On the other, it might remain minimalistic—just as our feet do the job in 3D. With stronger muscles and potentially different gravity or frictional properties, 4D beings might not need drastically different feet. They might simply rely on slightly angled or more flexible feet to maintain equilibrium and locomotion in 4D space.
5. Biological vs. Geometrical Considerations:
The conversation emphasized the difference between a purely mathematical 4D space and a hypothetical biological ecosystem adapted to that space. While geometric reasoning might suggest toes pointing in every available direction, biological reasoning suggests that the being’s structure would evolve in a way that is efficient, stable, and functional, not just a simple geometric extrapolation of axes.

Answering the Original Question:

Original Question by gonegahgah:
If you had 4D beings in a 4D world that had 3 legs, 3 feet, and numerous toes (3 feet × 16 toes/foot = 48 toes), which direction would their toes point?

Proposed Answer:

In a 4D biome, directions are more complex than in our familiar 3D world. There is an additional perpendicular axis (often referred to as ana-kata) to consider. However, much like in 3D, a 4D being’s toes would likely be adapted for efficient locomotion, stability, and balance in its everyday environment, rather than pointing in arbitrary “mathematical” directions.

If the being primarily moves and faces “forward” in a chosen direction, it is natural that its toes would predominantly point forward as well, aligning with its direction of travel and the orientation of its legs. The toes might have a range of motion or a slight spread into the additional sideways dimensions (ana-kata), helping the creature maintain balance and adjust its stance. They need not point “up” (in the sense of a new vertical dimension), nor would they aim in every possible 4D direction simultaneously.

Instead, the toes would serve as stabilizers and support structures. Much as our feet and toes help us balance and move forward while providing minor lateral support, a 4D being’s toes would do similarly in a richer dimensional context. They would likely aim forward (the primary locomotion direction) with subtle angling or flexibility that enables stability and maneuverability within the extra dimension. The end result might look like toes arranged around the “front” of the foot, with some capacity to pivot or shift slightly into the additional sideways dimension for balance, but still predominantly oriented toward the being’s main direction of travel and function—just as our own toes point in the direction we face and move.

[gonegahgah]

Thanks Patrick.

Thanks Hugh too. It was most probably your adventures into ChatGPT that eventually prompted me to put various questions to it too.
I do find that sometimes talking to ChatGPT feels a little like chatting with the old Eliza program in some respects.
The conversations can feel sometimes a little like rewording what you say, better of course, and feeding it back to you.
There does need to be caution in building our bias into ChatGPT I think. Perhaps this is even one of the reasons it does not feed user sessions back into the overall data.
But, that does not mean that it doesn't provide some interesting new, to us, insights from what exists out there, and can incorporate new parameters very nicely too.
At least that is my assessment from using it, and from pusing back on its reponses, both for and against my own arguments, when chatting with ChatGPT.

But, that was quite a useful response that you have posted I believe.
My main current thread focus, aside from ChatGPT itself, are the motions a 4D being would engage with when moving.
In 3D our locomotions mainly concern moving forward, turning (changing the forward direction), and dodging (changing our approximately parallel line of travel).
We know how we achieve these things in 3D.

In 4D you also have these as well. These include:
- walking forward
- turning (but towards any sideways angle for a new forward direction)
- dodging (to any sideways keeping the same forward direction)
But 4D also adds:
- rotating sideways while continuing to face the same forward.

The question is, in relation to the evolution of the 4D legs and feet, how are each of these above actions achieved for them?

[Hugh]

Thanks, gonegahgah, here’s what ChatGPTo1 responded with:

Below is a reasoned analysis and response to the question as posed by gonegahgah: how would the evolution of 4D legs and feet handle the various modes of movement available in a four-dimensional environment—namely walking forward, turning to a new forward direction, dodging sideways without reorienting forward, and uniquely, rotating sideways while continuing to face the same forward direction.

Contextualizing 4D Locomotion:
In 3D, our locomotion relies on a stable stance provided by two legs and feet oriented primarily forward, with some lateral stability derived from the foot’s width and the slight flaring of toes. We handle forward motion by stepping ahead, turning by pivoting our bodies around a vertical axis, and dodging to the side by sidestepping. Each of these actions involves shifting our center of gravity and leveraging friction to push in new directions.

In 4D, there is an extra dimension of “sideways” space beyond the usual left-right plane. Instead of having a single perpendicular direction to forward (sideways), a 4D being would have a continuum of perpendicular directions forming a sort of “sphere” of sideways possibilities. Terms like ana and kata often represent these new, orthogonal directions. This means that simple concepts such as “forward” and “turning” become richer and more complex, and the anatomy of the feet and legs would evolve to handle these complexities.

1. Walking Forward:
Walking forward in 4D would not differ drastically from 3D forward motion. The being would still orient itself so that one dimension is designated as “forward,” and it would place one foot after the other along this direction. The major difference is that while the foot in 3D only needs to counter forward-back and left-right imbalances, a 4D foot must also provide stability against tipping into the ana-kata directions.

Evolution might handle this by:
• Foot Structure: A foot that is somewhat broader in a “base plane” to provide traction not only forward and backward, but also in more than one perpendicular direction. The foot might have a slightly more complex, possibly radially symmetric or fan-like distribution of toes, each able to push against the ground in a variety of subtly different directions.
• Musculature and Joints: The ankle and toe joints might be more flexible, allowing fine control of balance corrections into these extra dimensions. Muscles, now effectively “3D cross-sectional” structures (in a 4D body), could be stronger and more compact, providing ample support without overly increasing bulk.

2. Turning (Changing the Forward Direction):
In 3D, turning involves pivoting around a vertical axis, rotating our orientation on the ground plane. In 4D, turning is more complex: you must choose a new direction to face from a 2D “sphere” of sideways options rather than a simple line. You are not just rotating on one axis; you can rotate along multiple axes simultaneously.

Evolution might yield:
• Multi-Axial Ankles: The joints at the ankles and hips could allow the leg to pivot smoothly in any chosen “side” direction, not just left or right.
• Dynamic Toe Engagement: By slightly shifting weight onto different sets of toes arranged at various angles, the being could generate pivot forces that rotate the body smoothly toward a new forward direction. The toes, through specialized tendon layouts, could press into the ground to create torque in these extra dimensions.

3. Dodging (Side-Stepping) Without Changing Forward Orientation:
Dodging in 3D is basically a lateral shift—stepping left or right while still facing forward. In 4D, you could dodge not just in one lateral line, but anywhere along a “sideways circle” around your forward axis.

Evolution might produce:
• Radially Distributed Toes: Instead of toes all pointing in nearly the same direction, toes could be arranged around a broader arc. By pushing off using toes oriented slightly into the ana or kata directions, the being can smoothly shift sideways without turning to face that direction.
• Stable, Low-Profile Body Form: Given the complexity of balancing in extra dimensions, natural selection might favor a somewhat more compact body form (in relative 4D terms), making quick lateral shifts more stable.
• Enhanced Proprioception: Neurological adaptations would allow the being to sense and coordinate subtle weight shifts into multiple sideways directions, making dodging a fluid, natural movement.

4. Rotating Sideways While Continuing to Face Forward:
This is the unique and most challenging new motion: remaining oriented forward while “twisting” one’s body into the ana-kata dimension. Imagine being able to rotate your upper body around an axis that does not force your gaze to shift left or right, but rather moves you into or out of the “screen,” so to speak.

For this:
• Multi-Degrees-of-Freedom in the Leg/Foot Complex: The hip, knee, and ankle joints might be arranged so that rotation can occur in a plane orthogonal to both forward-back and left-right. In other words, these joints would not be limited to a single hinge or ball-and-socket arrangement but might be something more complex—an evolved joint capable of controlled rotation in multiple perpendicular planes simultaneously.
• Foot Traction in All Directions: The toes and foot pads might be slightly domed or segmented, providing grip and torque generation in any direction required. Certain clusters of toes might specialize in providing traction for these “twisting” motions.
• Dynamic Postural Control: The being’s nervous system and muscle coordination would be sophisticated, enabling it to maintain orientation forward while effectively “rolling” its body into or out of the ana-kata axes. Think of it as having the innate ability to swivel part of its body dimensionally, like pivoting a wheel along an invisible axis that we can’t visualize easily in 3D.

Evolutionary Trajectory:
Over countless generations, minor structural changes that allowed better exploitation of the 4D space would accumulate. Early 4D ancestors might have simple limbs not optimized for complex lateral rotations. As they adapted to predators, obstacles, and the need to move efficiently, limbs would become more flexible, feet would gain more versatile toe arrangements, and joints would evolve to allow stable movement in all available dimensions. Muscles and tendons would strengthen and diversify, providing finely tuned control.

In short, evolution would favor feet and legs that:
• Are minimally more complex and versatile than ours, not extravagantly so.
• Use subtle joint, muscle, and toe arrangement differences to handle extra directions.
• Maintain a primarily “forward” orientation for locomotion, with the ability to pivot, dodge, and even rotate sideways all built into the foot-leg-hip structure through more flexible and multidimensional joint designs.

Conclusion:
Each mode of movement in 4D—forward walking, turning, dodging, and sideways rotation—would be achieved by incremental adaptations that allow a 4D being’s feet and legs to exert force and maintain balance in multiple perpendicular axes simultaneously. Rather than drastically different anatomy, expect subtle but significant joint and toe arrangement modifications, stronger and more compact muscles, and a neuromuscular system tuned for the complexity of controlling and exploiting extra-dimensional freedom of movement.

[gonegahgah]

Thanks Hugh.
It makes some interesting observations but gets just a little too overawed by the additional rotation movement I believe...
I'm guessing it is trying to struggle with a new concept that is not really described well enough elsewhere?
The main thing I would say its argument seems to be missing is where and when do all the feet go to achieve the various movements.
That is the critical distinguisher.

But, nice work on its part to describe things in 4D sideways terms though just at one point it seemed to be tripping up with the 4-axis thinking that all (including up) were somehow available?
However, it didn't seem to get too stuck on that...
It also spoke about a "sphere of sideways possibilities". I'm not exactly sure what it meant by that? Maybe it means "almost a sphere of possible new directions when turning"?
However, it did seem to be trying to connect a combination of movements, major and minor, which was kind of cool.

Maybe suggesting that "the placement of the feet during each type is a critical factor" might get it further along?

[PatrickPowers]

- rotating sideways while continuing to face the same forward.

The question is, in relation to the evolution of the 4D legs and feet, how are each of these above actions achieved for them?

Let's say your feet are to the x and -x of center. You move one leg-foot in the w direction, the other in the -w direction.

For one of the feet it could be w = sin(theta) x = cos(theta)
For the other foot then it's w'= -sin(theta) x'= -cos(theta)

Theta goes from zero to 2pi. This doesn't seem terribly useful though. There are various tricks to visualize this -- presumably this is what you would like to have -- but for me they don't provide much insight. I can't imagine doing such a thing. But there is nothing to hinder it so it will happen.

In 4D baseball what would be the shape of the batter's box? There's no answer -- it's whatever 4D people prefer. There could be one batters' box shaped like an annulus with home plate in the center. Or there could be one much smaller cube so that everyone has to stand the same place and can then rotate to their favorite ana-right orientation. Or there could be four -- for ana-right, ana-left, kata-right, and kata-left handers.

[gonegahgah]

I think I get what you are saying Patrick. Are you questioning the need to rotate? If not then please help me.

And, yes, that is correct about the placement of the feet. Whether 2 feet in your example or 3 feet the answer would be similar.
That is that they move towards the other feet in a circular pattern in the desired direction of rotation one at a time as you are moving.
It took me a few days of twisting my mind around ChatGPT's conversation and thinking about it to come to that direct approach.
I think ChatGPT's answer to me at the time, and even in its answer to Hugh, described more complicated body abilities than was necessary.

In regards to rotating. Turning and sidestepping are obviously useful, but why rotate at all?
If we take your 2 footed 4D beings and have each facing the other and one has their arms left and right and the other has their arms ana and kata.
But, if they want to hold both each other's hands, like people in 3D do, then it will be a little awkward if the arms are not on the same sides.
It's possible but they will be much more comfortable if they line up their arms, and the way to do that would be for one or both to rotate sideways while facing each other until they are aligned.
Also, it may be useful for fitting through things at times, or you would make some things align with you (like dressing) with the same motion to those things.

Or did I misread your answer? Apologies if so.

[gonegahgah]

When I asked the opening question better:

In a 4D world where 4D beings are likely to have 3 legs, and if we think of our directions as forward-backwards, up-down, and we don't limit ourselves to left-right/ana-kata and instead think of these and all the angles between those as sideways, in fairly simple terms how would all three legs and feet animate while walking 1) forward, 2) turning towards a sideways angle, 3) dodging something ahead, and 4) rotating sideways while facing forward to align with something ahead?

ChatGPT gave me the following response:

In a 4D world where beings have three legs and can move in a more complex space than our three-dimensional world, their movement would follow certain principles similar to how we walk, but with additional dimensions and directions to account for. To simplify this, I'll break down how these 4D beings might walk in the scenarios you described:
1) Walking Forward
In this case, imagine the 4D world as having four directions (which we can map to our understanding of 3D space):
• Forward-Backward (like our "forward-backward")
• Up-Down (like our "up-down")
• Sideways (combining left-right/ana-kata and other angles in the sideways plane)
To animate walking forward, a 4D being with three legs would likely use the three legs in a pattern that mirrors how we walk but in 4D. One possible motion sequence:
• One leg (e.g., the front leg) lifts and extends in the forward direction, while the other two legs are lifted to support the movement.
• As the front leg extends, the second leg (on the opposite side) would lift and move forward, while the third leg adjusts to maintain balance and stability in the remaining dimensions.
• The three legs would cycle in a way that allows forward motion along the "forward-backward" axis, while the other two legs provide stability in the sideways and vertical dimensions.
2) Turning Towards a Sideways Angle
Turning would involve a rotation in the 4D space. When turning sideways (in any direction between up, down, and sideways angles), a 4D being's legs would need to adjust dynamically:
• One leg might pivot or rotate around its base (like turning a wheel) to reorient the body in the desired sideways direction.
• The other two legs would shift to maintain balance. This would involve adjusting their positioning in the sideways plane and possibly lifting and planting at angles that correspond to the direction of the turn.
• The overall effect would look like a smooth reorientation of the body along the new "sideways" axis, while the legs continuously move to keep the creature balanced and moving efficiently.
3) Dodging Something Ahead
Dodging something ahead involves rapid, non-linear movement that likely involves shifting quickly to the side, while still maintaining some forward momentum:
• One or two of the legs would push the body sideways (perpendicular to the forward direction) to dodge the object, while the third leg would either anchor the body or help push further in the new direction.
• The body would lean sideways and rotate in that direction to avoid the object.
• The foot placement might mimic a zig-zag pattern, where each leg is lifted and extended rapidly in new angles to sidestep while keeping the rest of the body balanced and facing forward or slightly sideways, depending on the trajectory of the dodge.
4) Rotating Sideways While Facing Forward to Align with Something Ahead
In this situation, the 4D being would rotate its body along a sideways axis (like spinning but staying oriented toward a target ahead):
• The legs would adjust like in a dance, where one or two legs are planted at an angle while the other legs move to help rotate the body.
• To rotate, the front leg might pivot on its axis, helping the body rotate around it, while the other legs support the movement, rotating in place or stepping in circular motions.
• This allows the 4D being to rotate sideways while keeping its body mostly aligned with whatever it’s facing, like a dancer spinning while maintaining focus on an object or target ahead.
Key Points of 4D Walking:
• Leg Movements: The three legs work together to create a balance in the 4D space, which involves more complex rotations and placements than in our 3D world.
• Sideways Rotation: Because sideways angles in 4D involve multiple axes, rotating or turning would feel more fluid and multidimensional, requiring the legs to adjust and shift in many directions at once.
• Adaptation to Dimensions: The extra dimensions in 4D allow for a wider range of movement, so walking and turning would feel less constrained than in 3D, with the legs moving more freely through a variety of possible directions.
This movement might look almost like a dance, with each leg constantly adjusting to keep the being balanced and moving smoothly in the higher-dimensional space.

Which was pretty dam good actually! Is ChatGPT learning? Or did I just manage to frame the question better this time?

[PatrickPowers]

I think I get what you are saying Patrick. Are you questioning the need to rotate?

No.

And, yes, that is correct about the placement of the feet. Whether 2 feet in your example or 3 feet the answer would be similar.
That is that they move towards the other feet in a circular pattern in the desired direction of rotation one at a time as you are moving.
It took me a few days of twisting my mind around ChatGPT's conversation and thinking about it to come to that direct approach.
I think ChatGPT's answer to me at the time, and even in its answer to Hugh, described more complicated body abilities than was necessary.

In regards to rotating. Turning and sidestepping are obviously useful, but why rotate at all?

Well, why do we rotate here in 3D? It's a very basic thing to do. All sorts of reasons.

If we take your 2 footed 4D beings and have each facing the other and one has their arms left and right and the other has their arms ana and kata.
But, if they want to hold both each other's hands, like people in 3D do, then it will be a little awkward if the arms are not on the same sides.
It's possible but they will be much more comfortable if they line up their arms, and the way to do that would be for one or both to rotate sideways while facing each other until they are aligned.
Also, it may be useful for fitting through things at times, or you would make some things align with you (like dressing) with the same motion to those things.

Or did I misread your answer? Apologies if so.

[PatrickPowers]

A little puzzle : here in 3D, how can we make a full 360 degree rotation while facing forward the whole time?

[gonegahgah]

The Wheel of Death... Or, just a cartwheel. An aeroplane doing a rotation. Out in space lining up the ISS.
It's cool in 4D that you don't have to upend yourself.

[gonegahgah]

Here's a possible toe arrangement for a 4D being with three legs:

Toe Arrangement for Three Feet

Toes.jpg (25.47 KiB) Viewed 522 times

1) There is one big toe per leg as these (like our big toes) are all side on with the other big toes, and also facing the middle between all the feet and their toes.
2) Next there are wide toes (ones we don't have) which are all side on with a matching wide toe on another foot.
3) Lastly come the little toes (like out 8 little toes) that are separated from the other feet's toes by their big and wide toes.
This is an approximation of what it might look like if you were look at the feet from the front at foot height.

These are showing the toes from font on only, and not the feet themselves behind this.
Also, you can only see a horizontal cross section of the toes here, as the usual height of the toes are off in 4th dimension in this depiction.

[gonegahgah]

Exactly the same posit as last time and the following reply today...
Super weird! Super wrong! When it was fairly correct the previous time.
What drugs is ChatGPT taking today?

My question (exactly the same as last time):
In a 4D world where 4D beings are likely to have 3 legs, and if we think of our directions as forward-backwards, up-down, and we don't limit ourselves to left-right/ana-kata and instead think of these and all the angles between those as sideways, in fairly simple terms how would all three legs and feet animate while walking 1) forward, 2) turning towards a sideways angle, 3) dodging something ahead, and 4) rotating sideways while facing forward to align with something ahead?

ChatGPT said:
ChatGPT
In a 4D world with 4D beings, where they likely have three legs, we would need to imagine how their movements unfold along three spatial dimensions and how those dimensions interact. Here's a breakdown of the four scenarios you mentioned:

1) Walking Forward
In a 4D world, forward and backward are distinct, but so is the concept of sideways in the 4D sense. The three legs of the 4D being would likely move in coordination with each other, akin to how we move in a three-dimensional world. For simplicity, let’s assign the three legs to the following directions:

Leg 1: Moves forward in the "forward-backward" direction.
Leg 2: Moves upward/downward in the "up-down" direction.
Leg 3: Moves sideways in the "sideways" direction, which includes all angles between the primary axes, given that we're imagining a world without a strict "left-right" definition.
When walking forward, the 4D being would alternate between lifting each of the three legs and placing them forward, maintaining balance. This would look like a smooth motion, with legs moving sequentially in a coordinated manner. All three legs would likely move in the general direction of forward motion, but each would contribute by moving along different axes (forward-backward, up-down, sideways).

2) Turning Towards a Sideways Angle
When the being turns towards a sideways angle (which we can assume is not just left-right, but any angle in a 3D plane perpendicular to "forward"), its movement becomes more complex. In this case:

The leg that was aligned in the original "sideways" direction would rotate or slide to accommodate the change in direction.
The other two legs might either lift to reposition or extend along their respective axes (forward-backward, up-down).
The movement could resemble a twist, where one leg swings or rotates in a perpendicular plane to the being’s current orientation, and the other two legs adjust in unison. This could be a smooth, coordinated motion where the entire being’s posture shifts to reflect the new direction, and the legs move in angles corresponding to the new sideways alignment.

3) Dodging Something Ahead
Dodging something ahead would require a rapid shift in the 4D being’s position. Since dodging is a lateral (side-to-side) movement in our world, in the 4D world, dodging would likely involve moving the legs in all three axes:

One leg might extend or retract quickly along the sideways direction, helping the being to sidestep.
The second leg could push forward or backward to aid in the shift.
The third leg could be lifted and repositioned to stabilize the body, especially if the dodging involves a quick change in position.
This would result in a rapid, dynamic repositioning of the legs, similar to how we sidestep or leap to avoid an obstacle, but involving a more complex array of leg motions due to the additional dimension of movement.

4) Rotating Sideways While Facing Forward to Align With Something Ahead
When the 4D being rotates sideways while facing forward, it is essentially aligning with something along a new angle of orientation. This would require:

One leg (likely the one most aligned with the sideways axis) would rotate or move to a new position, helping to support the being’s weight in the new direction.
The second leg might move along its forward-backward axis to maintain balance while helping to adjust the direction.
The third leg could be used to stabilize the rotation, pivoting or shifting along a direction that helps balance the body during the realignment.
This would involve a smooth, rotational movement, where all three legs might contribute to a coordinated shift in the 4D space. The being would adjust the angles between its legs as it rotates, with each leg performing a task that stabilizes and supports the realignment.

In Summary:
Walking forward: Legs move in a coordinated sequence in the forward-backward, up-down, and sideways axes.
Turning sideways: One leg moves along the new direction, while the other two adjust to the new angle.
Dodging: Rapid movement in multiple axes to avoid an obstacle.
Rotating sideways: Legs adjust to allow a smooth pivot or realignment, maintaining balance while facing forward.
In all these movements, the 4D legs would exhibit motions across three spatial axes (forward-backward, up-down, sideways), with the coordination between legs being key to maintaining balance and direction in the complex 4D space.

[Hugh]

It’s always interesting to see the images that ChatGPT4 comes up with about 4D concepts but we always have to remember it’s trying to show a 4D concept in a 2D image.



Here was the prompt for the image it came up with:

A surreal depiction of a tripod-like creature in a highly abstract 4D world, illustrating complex multidimensional movements. This creature has three legs evenly spaced around its lower torso, each capable of moving in unfathomable directions due to the extra dimension. The actions include: 1. Walking forward with a stable tripod motion, yet appearing to traverse through unseen dimensions. 2. Turning by leaning deeply into a broad ‘sideways plane’ that spans multiple directions, showing a dynamic shift in spatial alignment. 3. Dodging by ‘sidestepping’ into a dimension unseen to the human eye, creating an effect of partial invisibility or phasing out. 4. Rotating sideways while facing forward, with the legs moving through complex curves and arcs in the extra dimensional space, resembling a cosmic dance. The background should be filled with abstract geometric shapes and ethereal textures, enhancing the perception of a four-dimensional space.

[Hugh]

Here's a possible toe arrangement for a 4D being with three legs:

Toes.jpg

1) There is one big toe per leg as these (like our big toes) are all side on with the other big toes, and also facing the middle between all the feet and their toes.
2) Next there are wide toes (ones we don't have) which are all side on with a matching wide toe on another foot.
3) Lastly come the little toes (like out 8 little toes) that are separated from the other feet's toes by their big and wide toes.
This is an approximation of what it might look like if you were look at the feet from the front at foot height.

These are showing the toes from font on only, and not the feet themselves behind this.
Also, you can only see a horizontal cross section of the toes here, as the usual height of the toes are off in 4th dimension in this depiction.

I asked Chat GPT4 to comment on the image and explanation:

The illustration depicts a conceptual arrangement of toes in a multi-legged creature, oriented in a way that aligns with our understanding of spatial dimensions while incorporating an aspect of a fourth spatial dimension. Here’s the breakdown of what each part represents:
1. Big Toes (B) - These are centrally located in relation to each corresponding cluster of toes. Similar to human anatomy, the big toes are analogous to a key structural toe but are represented here as being face-on, implying a direct line of sight across from each big toe to its corresponding opposites on other legs, and centrally oriented towards the middle point among all feet.
2. Wide Toes (W) - These toes, which are unique to this creature and not present in humans, are also face-on and side-aligned with their counterparts across other feet. This configuration suggests they play a role in balancing or might have a unique function in the movement or stability of the creature, particularly in a 4-dimensional context.
3. Little Toes (L) - The arrangement of little toes here mirrors the less dominant toes in human feet, positioned around the big and wide toes. These toes are shown separated from the toes of other feet by the intervening bigger toes, potentially indicating their standard role in grip or fine adjustment during movement.

The diagram also hints at a four-dimensional aspect by noting that what we see is only a horizontal cross-section of the toes. The usual vertical dimension of these toes extends into the fourth dimension, which is not visible in this representation. This would imply that these toes have an additional degree of freedom or movement that we cannot perceive, playing a crucial role in how this creature interacts with its environment in four-dimensional space.

In essence, this is an attempt to visualize how a multi-legged creature with complex toe arrangements might function in a world with an additional spatial dimension, exploring how conventional movement and balance could be radically different in such a context.

I asked it to create an image based on this description.



The exact prompt that was used to create the image was: “A highly detailed diagram illustrating the conceptual arrangement of toes on a multi-legged, four-dimensional creature. The image should include clusters of toes organized in three types: 1. Big Toes (B) - Central, face-on position, suggesting a pivotal role in balance and orientation, directly aligned across from each other and towards the middle of all feet. 2. Wide Toes (W) - Unique toes, side-aligned with matching wide toes on another foot, indicating specialized functionality in a 4D context. 3. Little Toes (L) - Surrounding the big and wide toes, separated by them, demonstrating their role in finer movements. This arrangement implies a fourth dimension by showing only a horizontal cross-section of the toes, with their true vertical extent stretching into an unseen dimension, suggesting additional movement capabilities beyond our three-dimensional perception. The diagram should be clear, highly detailed, and scientific in appearance, capturing the intricate relationships and orientations of the toes in a four-dimensional space.”

[gonegahgah]

Thanks Hugh, cool drawings!!!
But, also looks a little creepy! Makes me think a little of Thing in the Addams Family.
Not sure why it shows toes coming out of the top of the feet?

A better depiction would be to draw the foot with leg from different angles between x and w. I would just put each angle side by side to demonstrate.
I was going to try drawing something like that soon but if we can get ChatGPT to understand then maybe it can do a much better job than me?
This may have been what ChatGPT was trying to do, however doing so with the feet in a circle (which they wouldn't be), but its result is a little wrong.

Here's a new diagram:

Angles of Toe Rows

Toe Angles.jpg (28.43 KiB) Viewed 475 times

What we need it to do is draw each angle of the 4D leg at counter clockwise 60°, 30°, then 0°, the clockwise 30°, 60° & 90° which is 6 angle views (of 12).
If we do it like ChatGPT's picture which is dimensional skewing we could do at these angles too.
If we did it as a circle we would more accurately depict this by showing all 30° steps from -180 to 180 which is 13 angle views (of 12). But that depiction may or may not confuse people too?

Some of the important things are that at 60° counterclockwise, 0°, and 60° clockwise, we get legs and feet that look much like ours.
But in this case the big toe should be the largest. ChatGPT has made the wide toes the largest which is wrong.
Further, at 30° counterclockwise, and 30° clockwise, and 90° we get big gaps between the toes with only 3 toes on the foot at that angle.
This is just because of the toe arrangement (and is not a bug).
At 90° you have 3 spaced little toes. At both 30° you have spaced toes too but one of them is a wide toe...

Do you think you could convince ChatGPT to draw that in the way I describe?
If you need further clarification just let me know?

[Hugh]

Here’s what it came up with based on your explanation and diagram.





Unfortunately the DALLE model that it uses still has a long way to go in understanding precise orientations. I found when I was trying to create some pictures for my 4D/VRI idea I had to ask it to create Python code to get a more precise diagram, but even that had its limitations. I also tried having it create p5js code which came with some interesting results. ChatGPT also has Canvas mode now where you can have it create code right in the same window which is handy. I still have so much to learn and it has so much more to grow in its abilities. I wish I could jump ahead to 5 years from now and A.I. could totally understand, explain and create exactly what I want it to now!

[gonegahgah]

Thanks for trying Hugh.
I'll have to draw up my simple diagrams soon and hopefully that can help us work out how to get ChatGPT to draw better illustrations.

[gonegahgah]

Here are my simple diagrams...

Rotating the view between w and x with the leg in the centre of view in 30° increments from -180° to 180°.

A closer up view showing just the unique angles.

Legs unique angled rotations between w-x

Legs -60..90.jpg (31.7 KiB) Viewed 409 times

Single Feet's Toes

Foot Toes.jpg (20.72 KiB) Viewed 399 times

If you imagine that you are rotating your 3D space sideways into the 4th dimensions between w and x then this is approximately what you would see for a single leg.
Only 30° increments are shown and these would all morph together including all infinite between angles to give you that leg.
The pattern of the toes follows the approximate example given previously...

You can see how at particular angles the feet look fairly similar to ours!
Hopefully we between us can somehow work out how to get ChatGPT to do up some nicer depictions of this?

[Hugh]

Here are the latest attempts by ChatGPT4, using your text and image inputs.



It’s interesting as I look at these, and think of seeing the different 3D slice viewpoints of the 4D legs from different directions, it reminds me of different VRI orientations.

[gonegahgah]

ChatGPT just doesn't want to leave bigger gaps between the toes at 30° and 90°, does it?
At 29°, 30°, 89° and 91° there is not a gap but exactly between it may be considered to be the case for these angles.

I was thinking about why ChatGPT defaulted the feet in a circle, earlier on.
I imagined a 2D being trying to draw a 3D foot. They might draw the foot from a common heel and then rotate it up one foot rotation at a time for each toe.
So, I think this may be the same thinking that makes ChatGPT default to drawing the feet outwards from a common heel with toes off at different angles.

For the 2D being they would be better off drawing the five feet with the flat 3D toes one above the other.
Also, for cylinders, ChatGPT can't really understand how biology might develop using a cylinder plus height for our feet.
A 2D being might draw a cylinder as 5 rectangles above each other and imagine that these can some be rotations 3D-wards (when they know only forward-backwards rotations normally).
The same goes for us. I'm not sure that ChatGPT can understand that a 4D foot would be a cylinder as the extra dimension allows unseen cylinders to form.
This is why the 4D being can lie down in such a way that we see some of there toes approximately in the arrangement I describe.

In VRI you wouldn't ever see missing toes like I've drawn at anytime; would you Hugh?

[Hugh]

I’m sorry, gonegahgah, I started writing a lengthy reply regarding 2D beings, what they could see, VRIs for 4D beings etc but I realized that this was going way off topic.

As far as illustrating the 4D being foot idea you might try some of the software programs that others have used on here to illustrate 4D concepts.

I’d love to see any results with these.