Dimensional Baby Steps

[quickfur]

[...]Step 28. Immersing our 3Der in a 4D image via direct brain stimulation:



This shows how 3Der hooked up to a box that is receiving multiple pictures from a great number of rooms.

Hmm. I'm not sure I understand why we would need so many rooms. I thought it should be sufficient to use something like Patrick Stein's nD ray-tracer and deliver slices of the 3D projection volume into whatever is stimulating the brain cells?

[...]The biggest problem with this is that there is no feedback to help the 3Der grow and interact with their environment such as hands.

Yeah, that would be the second big stumbling block once we figure out how to deliver 3D data to the brain directly. Clearly, contemporary 3D VR motion sensors will be inadequate, since they can only capture 3D motion of our 3D limbs. Perhaps this can somewhat be mitigated by designing a set of controls manipulable by our 3D hands that control a virtual 4D arm (or arms) that manipulate the virtual 4D environment -- then it'd be a matter of learning how to operate a "4D robot" that serves as our surrogate body in the 4D world.

[...] Although this would allow us to see inside the cubes that form the faces of the tesseract across the rooms; this approach would fail to allow us to see the compound faces of the end cubes; which a 4Der can. The 4Der can see 4 of the cube-'faces/volumes' at once of a tesseract (like we can 3 of the square faces at once of a cube).

I think if our hypothesis is correct, and the brain is able to rationalize a set of 3D nerve impulses into coherent 3D image, then we shouldn't have a problem with this at all. The 4D simulator will deliver the 3D array of pixels including lighting and shade, parallax, and what-not, and, assuming our brain is able to make any sense of this data, we should be able to perceive the cube's volume directly, judge its angle with the light source(s) based on the 3D shading within it, and infer the square boundaries where it touches the other cells in the tesseract.

[...] I think we need an extra eye and we need to place it in a series of parallel rooms that are looking at the cube from a different aspect. I'll have to think some more about that though...

I'm not sure if that relates to the debate on the need for three eyes yet - though I had already been pondering about where the 4Der's third eye would lay if they had three eyes. [...]

On my way home today, I thought a bit more about the whole number-of-eyes issue. I've come to the conclusion that only two eyes are needed for stereoscopic vision in any dimension. Except for some exceptional cases which are unlikely to occur (or aren't very important) in real-life, it doesn't really matter which direction the binocular disparity happens in. I believe that any correspondence between the number of eyes and the dimension of space only depends on the advantage conferred by the widening of one's lateral peripheral vision.

First, let's talk about stereopsis. Since we have two eyes, what each eye sees isn't exactly the same as the other; the difference is called binocular disparity. Our brain instinctively tries to reconcile these two different images by inventing a 3rd dimension, by means of which the difference between the two images can be rationalized. This produces a 3D model out of the 2D images that we see.

Now, our eyes are spread out horizontally, so the disparity in the two images lie in the horizontal direction. For example, if we were looking at a vertical pole in front of us, its position relative to the background will appear shifted differently when seen from each eye; when our brain tries to reconcile this disparity, it constructs a 3D model where the pole "protrudes" from the background; i.e., we see 3D depth. What about a horizontal line? If the line is infinite (or at least, extends beyond our field of vision), and completely featureless, then there will be no binocular disparity, and we wouldn't be able to tell if it was part of the background, or something in front. If our eyes were laid out vertically instead, then we would see the vertical disparity of the line, and we'd be able to tell where it is in relation to the background. So here we see a difference between horizontal binocular vision (i.e., "normal" vision) and vertical binocular vision (if our eyes were laid out one on top of the other).

However, such cases are contrived; your average general object isn't a featureless infinite line or any featureless infinite construct that is vertically or horizontally homogenous. Our world is filled with all manner of shapes, most of which are irregular; even smooth objects like very long tree trunks or pipes aren't completely homogenous; they usually have enough features that binocular disparity will still happen. In other words, if your two eyes were lined up vertically on your face instead of horizontally, you'd still be able to perceive depth without any problems.

Where would this make a difference, though? It makes a difference when we consider peripheral vision. If your eyes were lined up vertically, you'd have a wider range of vision vertically; you'd be able to detect peripheral movements above or below your field of vision. Since we're mostly confined to the ground by gravity, though, this expanded vertical range of vision is largely useless: it's usually just the floor or ground below you, and a largely empty sky above. The interesting things lie mainly on the across dimension, to use wendy's terminology. Having two eyes horizontally laid out expands our horizontal range of vision -- very useful because that's where most of the interesting things are. The fact that two eyes also give binocular vision is a happy coincidence.

In 4D, the same applies: having just two eyes is good enough to have binocular disparity -- save for exceptional, contrived cases which aren't really that important in real-life situations. However, having 3 or 4 eyes laid out in a planar fashion will enhance lateral peripheral vision -- remember that, as gonegahgah repeatedly pointed out, in 4D there aren't just left and right, but a whole 360° of sideways. Having only 2 eyes will limit peripheral vision to mainly two opposite directions; having 3 or more eyes will have a better coverage of the 360° of periphery.

So I think one can get by with just 2 eyes in 4D, though for better peripheral vision, 3 or 4 would be more advantageous.

[gonegahgah]

Hmm. I'm not sure I understand why we would need so many rooms. I thought it should be sufficient to use something like Patrick Stein's nD ray-tracer and deliver slices of the 3D projection volume into whatever is stimulating the brain cells?

Yes, that would be a whole lot better. We would need to send the brain input as points of stimuli, from imaginary seeing cells equivalent to the number of seeing cells required for 4D eyes, into the brain; I guess? (Again, probably our number of seeing cells multiplied by the squareroot of that to create a cube). Then the brain would have to sort it all out...
The rooms are mainly to help our readers to visualise; though I did, in my haste this morning, over-exaggerate the number of needed rooms. The rooms themselves provide the 3D resolution already. So really you would only need to make up to the 4D resolution by having the 'squareroot of our existing eye cell count' number of rooms. Plus multiply this by enough rooms to allow us to rotate into the fourth dimension (we can rotate in 3 dimensions in the rooms but can't into the fourth dimension) and to move through the fourth dimension (which is done by turning off the right cameras in sequence) and to zoom in on objects as well, subtracting rooms where redundancy can be utilised. ie. if two rooms would look identical you need only one room and can send its 3D slice in both instances.

[...]The biggest problem with this is that there is no feedback to help the 3Der grow and interact with their environment such as hands.

Yeah, that would be the second big stumbling block once we figure out how to deliver 3D data to the brain directly. Clearly, contemporary 3D VR motion sensors will be inadequate, since they can only capture 3D motion of our 3D limbs. Perhaps this can somewhat be mitigated by designing a set of controls manipulable by our 3D hands that control a virtual 4D arm (or arms) that manipulate the virtual 4D environment -- then it'd be a matter of learning how to operate a "4D robot" that serves as our surrogate body in the 4D world.

Cool, that's a good idea. It also occurred to me, just now, that if we can deliver data direct to the brain we might also be able to simulate touch and resistance as well.

[...] Although this would allow us to see inside the cubes that form the faces of the tesseract across the rooms; this approach would fail to allow us to see the compound faces of the end cubes; which a 4Der can. The 4Der can see 4 of the cube-'faces/volumes' at once of a tesseract (like we can 3 of the square faces at once of a cube).

I think if our hypothesis is correct, and the brain is able to rationalize a set of 3D nerve impulses into coherent 3D image, then we shouldn't have a problem with this at all. The 4D simulator will deliver the 3D array of pixels including lighting and shade, parallax, and what-not, and, assuming our brain is able to make any sense of this data, we should be able to perceive the cube's volume directly, judge its angle with the light source(s) based on the 3D shading within it, and infer the square boundaries where it touches the other cells in the tesseract.

Using the software to generate the eye cell stimuli; absolutely; that's the best way to do it. I'm a little concerned with the real room approach because the end cube would not appear as the other three tesseract cube/faces would. But, as you say, this is not a concern with software. One thing I have to remind myself is that the collection of squares across the rooms to form a cube are not the actual faces the 4Der sees; as we are seeing them in alignment with our world. They would tend to look across the rooms stacked side by side in the 4th dimension and not from the front of the rooms as we do. Although this technique would help us to see what a 4Der sees, and our brains may actually process seeing the whole cube at once via this, it is still from a different direction then the 4Der. The same goes for us when we see a 2Der's lines forming a square. We don't see the lines from their direction. We see the lines via the non-connection side of those lines. If a 2Der were to have a series of 2D rooms projected into their head to make 3D it would be from their direction too. But it would give them a picture of a square side on. And we would see a cube as a 4Der sees it. Our brain would (if it is true) meld all the squares together to form a total cube space in our mind. I'll draw pictures later to illustrate.

Where would this make a difference, though? It makes a difference when we consider peripheral vision. If your eyes were lined up vertically, you'd have a wider range of vision vertically; you'd be able to detect peripheral movements above or below your field of vision. Since we're mostly confined to the ground by gravity, though, this expanded vertical range of vision is largely useless: it's usually just the floor or ground below you, and a largely empty sky above. The interesting things lie mainly on the across dimension, to use wendy's terminology. Having two eyes horizontally laid out expands our horizontal range of vision -- very useful because that's where most of the interesting things are. The fact that two eyes also give binocular vision is a happy coincidence.

I like both your terms: 'across dimension' and 'peripheral vision'. They are both very effective for this discussion.

In 4D, the same applies: having just two eyes is good enough to have binocular disparity -- save for exceptional, contrived cases which aren't really that important in real-life situations. However, having 3 or 4 eyes laid out in a planar fashion will enhance lateral peripheral vision -- remember that, as gonegahgah repeatedly pointed out, in 4D there aren't just left and right, but a whole 360° of sideways. Having only 2 eyes will limit peripheral vision to mainly two opposite directions; having 3 or more eyes will have a better coverage of the 360° of periphery.

And, 'lateral peripheral vision'; I like that term too.

So I think one can get by with just 2 eyes in 4D, though for better peripheral vision, 3 or 4 would be more advantageous.

Thanks QuickFur, what you wrote was very interesting to read. I'm going to try and do up a picture of a 4D archery target next and see if that helps this discussion too...

[gonegahgah]

Here's just some deliberations on this archer's target that I'm drawing. It's a challenge to draw real world objects in 4D but fun as well.

It's taken me a while to work out what to do with the target's feet. The target itself is pretty easy but the legs don't have the same uniformity.

What I have gone for is to have 3 front feet and one back foot for our 4D target.
A 2Der just needs one front foot and one back foot for their 2D target. We need a least to have two front feet and one back foot for our 3D targets:

(picture not done by me)

As I said, for our 4Der I am going with three front feet and one back foot. This is enough to keep our target upright.

The interesting challenge to this is that front for a 2Der is different to front for a 3Der which again is different to front for a 4Der.
A 2Der has only one front. We have a whole sideways worth of front. A 4Der has a whole 360° sideways of front.

So naturally the front foot can only be directly in front of the 2D archer...
... the two front feet are evenly spaced at a triangulated angle from us 3D archers...
... the three front feet are all evenly spaced from each other at the same distance from the 4D archer.

We start to trip over ourselves when we try to think of all three front feet being evenly spaced from each other. But, it's true.

The following diagram depicts the front three feet only viewed from above.



As you can see they are all evenly spaced from each other with one lying in our left-right and the other two feet off towards the ana-kata directions.
So when I do my 4D picture I will only be able to have one leg fully in the picture as this is the only one that will lay in our plane.
I'll see what I can do about depicting the other two legs as I get further along...

In my image of the feet above there is also a darker black circle and this is where the legs actually make contact with the target itself.
This is a distorted view in this case because the target is on a lean in the back direction.
The target leans at the same angle as the back leg - which from side-on (remembering that are 360° of side-on in 4D) forms an equal opposite lean in line with the 3 front legs.

I used the term 'in line' which is difficult for us to think of three legs forming a tripod in line with each other.
Our 3D thinking says the legs of a tripod can not be in-line; but in 4D they can. They are all equally as front as each other!

In reality - and on purpose - I have made the back lean different to the angle that each front leg is angled from each other front leg.
Generally, on a 3D archer's target, the three legs will not be designed as a perfect tripod. Instead, the front of the target will lean back to favour an angle for the archer's arrows. The spread of the front legs is to give a wider sideways spread to give a stable base for the flat target. Two different purposes. The resultant footprint will be more of a squashed triangle. So I have gone with this for the 4D target as well.

What this means is that all three legs in my diagram are equally at the font and equally lean at the same angle to meet the back support leg.
I'm still working on the grid to map and draw this but I just thought I'd add some 4D insights in the meantime.

If anyone thinks that I'm am going at it wrong and need to correct something then let me know so that I don't progress too far in error...
Otherwise, I'll keep plotting away at it and present the result or more interim discoveries if there are any interesting ones.
I thought the above was interesting to discover so I thought I would share.

[quickfur]

[...] So naturally the front foot can only be directly in front of the 2D archer...
... the two front feet are evenly spaced at a triangulated angle from us 3D archers...
... the three front feet are all evenly spaced from each other at the same distance from the 4D archer.

We start to trip over ourselves when we try to think of all three front feet being evenly spaced from each other. But, it's true.
[...]

This is not hard to understand if we think in terms of maps (floor plans, if you will).

A 2D floor plan is just a line segment, so if the target is placed, say, in the middle of the line segment (of course, elevated in the 2nd dimension), then its two feet will just lie on either side (no elevation in the 2nd dimension). The archer has to stand somewhere on the line, so the foot closest to the archer will be the "front" foot, and the other will be the back.

A 3D floor plan is a 2D map, so if a target is placed in the center of the map, then you need at least 3 feet, say in a triangular arrangement. There's more flexibility in terms of where the archer can stand; for convenience, we can assume that the archer will stand somewhere on a line that passes perpendicularly through an edge of the triangle. So two legs will be "front" (i.e., nearer to the archer), and one leg will be "back".

A 4D floor plan is a 3D map. Again we may place our target in the center of the map. Then you need at least 4 feet in a tetrahedral arrangement (a tetrahedron is the simplest possible 3D solid; anything less and it will only cover a 2D area, so the target will fall over in the axis perpendicular to that 2D area). Here, something interesting happens. The archer may stand along a line perpendicular to an edge of the tetrahedron, or along a line perpendicular to a face of the tetrahedron. In the former case, you have the interesting scenario where the target has two front feet and two back feet. In the latter case (which is more analogous to the lower dimensional case) you will have 3 front feet (the vertices of the triangle facing the archer) and 1 back foot.

The latter case also has the interesting property that you can rotate the triangle of three front feet without affecting the stability of the target. So there's actually 120° of possible front feet alignment w.r.t. to the archer (i.e. 360°/3) which are all equivalent in terms of stablity.

Also, one should note the shape of the target across various dimensions: in 2D, the target is but a line segment. In 3D, the target is a circle. In 4D, the target is a sphere. Hitting the bullseye in 2D is just a matter of controlling the altitude of the arrow so that it falls at the right vertical displacement. Hitting the bullseye in 3D is more complicated: not only the arrow has to fall at the correct vertical displacement, it also needs to be centered horizontally. In 4D, another dimension is added to the mix: the arrow has to fall at the correct vertical displacement, and it needs to be centered in two horizontal displacements. That's a lot of parameters to take care of; so one can imagine that archery in 4D is a lot harder than archery in 3D!

[gonegahgah]

Here, something interesting happens. The archer may stand along a line perpendicular to an edge of the tetrahedron, or along a line perpendicular to a face of the tetrahedron. In the former case, you have the interesting scenario where the target has two front feet and two back feet. In the latter case (which is more analogous to the lower dimensional case) you will have 3 front feet (the vertices of the triangle facing the archer) and 1 back foot.

Hmmm, note to self to think about this when I get a chance...

The latter case also has the interesting property that you can rotate the triangle of three front feet without affecting the stability of the target. So there's actually 120° of possible front feet alignment w.r.t. to the archer (i.e. 360°/3) which are all equivalent in terms of stablity.

Yeh, true; cool. With the exception of what you've stated above, which I need to ponder on, the three front feet would otherwise be an equal distance from the archer; as the two front feet of our archer targets' are from us. So if there is only one back leg then the three others would tend to all be the same distance from the archer.

Also, one should note the shape of the target across various dimensions: in 2D, the target is but a line segment. In 3D, the target is a circle. In 4D, the target is a sphere. Hitting the bullseye in 2D is just a matter of controlling the altitude of the arrow so that it falls at the right vertical displacement. Hitting the bullseye in 3D is more complicated: not only the arrow has to fall at the correct vertical displacement, it also needs to be centered horizontally. In 4D, another dimension is added to the mix: the arrow has to fall at the correct vertical displacement, and it needs to be centered in two horizontal displacements. That's a lot of parameters to take care of; so one can imagine that archery in 4D is a lot harder than archery in 3D!

Although its worth noting that in all dimensions the target still leans back to better suit the flight of the arrow. So the 2Der's line shaped target is tilted back, the 3Der's circle shaped target is tilted back, and the 4Der's sphere shaped target is tilted back; but not in the sense that we think of tilting a sphere. Either this brings to our 3D mind: rotating the sphere backwards; which just gives a sphere still, or deforming the sphere backwards; which gives some sort of a skewed sphere. Neither is correct. Just as the circle remains a circle to us the sphere remains a sphere to the 4Der and is just tilted into the 4th direction.

I'm just working on some other pictures at the moment just to make sure I am understanding the principle of surface in 4D. I'll get those up as soon as I can and then get back to the target itself. I'm doing this so that I can understand better how the 4D legs attach to the 4D target. At the moment I imagine that the leg spacing between the three legs will be the same as the spacing between the two legs of our 3D target. Hopefully that will become clearer... along with how to draw the front legs in some fashion and the back leg.

If you think of a 2Der depicting a 3D target then, the back leg can be drawn in a single rotated frame but not either of the other two front legs at the same time as the back leg for our 3D target; unless you are no longer vertical; but verticallity is a feature that helps us to know where we are; so that's a given. It would seem to be the same for us depicting a 4Der. We can only depict one leg, in the actual present frame, at a time. We may, however, be able to depict the other legs through shadowy representations off at a rotated angle still. I wait to see.

Just as a referent to what I'm saying here is a quick diagram. I had already drawn this but I've realised that archer's targets also have a further white outer ring so I'll add that later.



But for the moment this shows how a 2Der looking at our target from the front would see the back leg only and there would appear to be no front leg to them. The target itself would look pretty much as they would expect. In this respect, I think the 4Der's target would look pretty much the same to us as our own targets face on. But, again, we would see only the back leg in our 3D frame but not any of the front legs; as they would cross 3D frames; just as our target's legs cross several 2D frames (even with rotation).

[quickfur]

[...]

Also, one should note the shape of the target across various dimensions: in 2D, the target is but a line segment. In 3D, the target is a circle. In 4D, the target is a sphere. Hitting the bullseye in 2D is just a matter of controlling the altitude of the arrow so that it falls at the right vertical displacement. Hitting the bullseye in 3D is more complicated: not only the arrow has to fall at the correct vertical displacement, it also needs to be centered horizontally. In 4D, another dimension is added to the mix: the arrow has to fall at the correct vertical displacement, and it needs to be centered in two horizontal displacements. That's a lot of parameters to take care of; so one can imagine that archery in 4D is a lot harder than archery in 3D!

Although its worth noting that in all dimensions the target still leans back to better suit the flight of the arrow. So the 2Der's line shaped target is tilted back, the 3Der's circle shaped target is tilted back, and the 4Der's sphere shaped target is tilted back; but not in the sense that we think of tilting a sphere. Either this brings to our 3D mind: rotating the sphere backwards; which just gives a sphere still, or deforming the sphere backwards; which gives some sort of a skewed sphere. Neither is correct. Just as the circle remains a circle to us the sphere remains a sphere to the 4Der and is just tilted into the 4th direction.
[...]

It may help to understand the tilting in terms of the kind of shadow it casts on the ground (assuming the sun is directly overhead).

In 2D, if the target is perfectly vertical, then it casts a point shadow on the ground. But to better suit the trajectory of the arrow, as gonegahgah has said, we'd usually like to tilt it backwards a little, so that it slightly faces upwards. In this orientation, it would cast a short line-segment shadow -- not very long, since we're not turning it all the way to face the sun, but just slightly non-vertical, so its shadow is extended from a point to a short line segment.

In 3D, a perfectly vertical target casts a line-segment shadow, where the length of the line segment is the diameter of the target. If we were to tilt it back all the way until it faces the sun, it would cast a circular shadow; but usually we only wish to tilt it back slightly. So the shadow it casts is an ellipse.

In 4D, a perfectly vertical target casts a circular shadow. A slightly tilted target casts a shadow in the shape of an oblate spheroid (i.e., a "slightly flattened sphere", or an ellipsoid in which two axes are equal, and the third is shorter than the first two). The circular shadow in the perfectly vertical case can be thought of as an oblate spheroid in which the short axis is zero-length. When the target is tilted all the way to face the sun, then it casts a spherical shadow. So the proportion of its short axis in relation to its two long axes is an indicator of the angle by which it is tilted in the 4th dimension. A 90° angle with the ground causes the short axis to be zero-length -- you get a circular shadow. As this angle is decreased, the circular shadow "inflates" into an oblate spheroid, and eventually at 0° (i.e., directly facing the sun) it becomes a sphere (the short axis becomes equal to the other two axes).

You can actually calculate the precise angle of the target just by measuring the short axis of its shadow: if D is the diameter of the target, and the length of the shadow is L, then the angle A is given by the relation:

cos A = L/D

When L is zero, we have cos A = 0, which implies A = 90°. When L = D (i.e. it's a full shadow) then cos A = 1, which means A = 0°. When L is somewhere between 0 and D, then the angle A is somewhere between 0° and 90°; the exact value of the angle is then calculated by computing the arc cosine of the ratio L/D.

So I submit that using shadows here is a good way to convey just how much the target is tilted w.r.t. the ground. It's also a conveniently physical phenomenon when the sun is directly overhead. (The above analysis works not just for 4D; it holds for all dimensions n≥2.)

[gonegahgah]

Okay, here's a first step towards looking at surfaces (and then I'll get back to the target soon).
Here are a series of pictures showing a simple square in a 2Der's world and how the 2Der views it and how we view it.



These are spread across various columns that depict different things. The first column shows both our 2Der and 3Der looking at the same square. The square is rotated into the 3rd dimension by 30° more each row below. The second column shows how the 3Der would see the square directly on for each rotation. The third column shows how the 2Der will see the square directly on for each rotation. Remember that the 3Der is viewing from a perpendicular orientation to the 2Der. The fourth column is a computer generation based upon how the 2Der perceives that the 3Der would view things. Finally this is feed directly into our 2Der's brain where they hopefully start to see 3D.















I have another two series to show but I started with this one first on purpose. It is the simplest for the 2Der to comprehend I imagine. They imagine that the square foreshortens as it rotates into the 3rd direction in a direct mathematical fashion. They can imagine that the 3Der is looking at all those lines at once and that they form a complete square before the 3Der's eyes. However, you note that I drew the 3Der's square blue and the virtual reality square red. It is hard for the 2Der to imagine that the 3Der is viewing the square from any angle but their own. So they imagine the square to be made up of the lines that make their square from their own perspective. They think it is made of red lines. Our 4Der has actually painted the 'faces' they see of the square blue (and yellow on the opposite side). This is the same trap we experience but I'll discuss that further later.

But, our 2Der is happy with this series of pictures because they think they can see how the square has a 3D surface by interpreting this picture series. Will the next two series, I'll be showing in turn, also be as straight forward for the 2Der?

[gonegahgah]

Thanks! I owe you the idea of comparing the different POVs of different dimensions in this way. I also owe 4DSpace for making me think more thoroughly about dimensional terminology, that eventually led me to realize the far-reaching consequences of objects needing non-zero n-bulk in order to exist in n-space, and what that means for 4D (and higher) visualization. Now I understand why people struggle so much with the idea that 4D (and higher) beings can see every part of 3D space, attributing it to some kind of "magical" x-ray vision or omnipresence or some such. They are thinking in terms of the "sides" of atoms that are visible to us, imagining that somehow the 4D being's sight is magically bypassing the occlusion of atoms in the back by the atoms in the front. But what's really happening is that these atoms have "sides" that are wholly unknown to us -- sides that face directions outside of 3D space. These sides are oriented such that they do not occlude each other, and that is how the 4D being can see all of them at the same time. The occluded sides are still just as occluded as before. While it is true that the 4Der sees every atom in the object, they only see a part of these atoms -- only the sides that are facing them. The sides that are obscured to us are just as obscured to the 4Der.

I woke up this morning with a sudden realisation regarding something that you said QuickFur early in the peace way back on page 3 of this thread. Again, it was something that you had immediately realised the full implications of; and only now have I grasped its further import.

My current dalliance into the exploration of surfaces must have set my mind a-working last night while I slept. (Aworking used to be an English word; but now it is obsolete! Why?)

It is basically what you have said above (and I'm sure I recall you exploring it elsewhere) but I will revisit it. When we think of a square we think of it having two faces whereas a 2Der thinks of it having four 2-faces (or line edges to us which are faces to the 2Der). When we think of a cube we think of it as having six faces but a 4Der thinks of it as having two 4-faces.

But as you've mentioned already, in 3D a square's atoms have a '3-side' as well that only we can see (seen as opposing squares) whereas the 2Der can only see the '2-sides' (seen as front, opposite and adjacent lines). So in reality our atoms need a minimal depth in the third direction to exist and a square actually has depth in our third direction.

Not that it makes any difference - except to actually allow us to see the square - but this means that what we call a square is actually a very thin square prism. Why does this matter? Well because a prism has two ends. Even if we regard it has having zero 3-length we should simply regard it as a very flat square prism; unlike a 2Der who really has just a square.

The word 'prism' tends to be tied to 3D so I'd like to swap in the word extrusion instead. In this context I'm referring to a direct extrusion which is the usual way things extrude.

So if we go up through our dimensioners they will see things as follows:
A 1Der has only a line which has two '1-faces' (or dots) but the flat extrusion of the line gives it two '2-faces' for the 2Der to refer to the line by.
A 2Der has only a square which has four '2-faces' (or lines) but the flat extrusion of the square gives it two '3-faces' for us 3Der to refer to the square by.
We 3Der's have a cube which has six '3-faces' (or squares) but the flat extrusion of the cube gives it two '4-faces' for the 4Der to refer to the cube by.

So in reality when each dimensional creature is dealing with a lower dimension 'solid' we are actually dealing with an extremely thin extrusion of it.
And by its nature, an extrusion has a front end and a back end. A long extrusion also has sides but an extremely thin extrusion only has edges for sides.

So when a 4Der sees a cube as having two 'faces' they are simply dealing with the front cube and the back cube of the very thinly extruded cube into their 4th available direction.

This explains a lot. It explains why every dimensional creature sees 'solids' of the next lower dimensional creature as having two 'faces'. This is basically just a rewording of what you have said already QuickFur. Every dimensional creature primarily notices the front and rear of an extrusion of the next lower dimensional creatures 'solids'. Hence two 'faces'.

I hope this explanation helps our 2Der to better understand how a square can have two 'faces'; just as it helps me to realise how a cube can have 'two' faces to a 4Der.

The other interesting revelation out of this is some of the things that happen when other things then direct extrusion happen - and here I explore rotation instead.
If we take a 1Der's line and - instead of extruding it - we rotate it around its centre into the 2nd direction. This produces a circle for our 2Der.
If we take the 2Der's circle and - instead of extruding it - we rotate it around a diametre into the 3rd direction. This produces a sphere for us 3Ders.
If we take our 3D sphere and - instead of extruding it - we rotate it around a mid-circle section into the 4th direction. This produces a glome for the 4Der.

[gonegahgah]

Just as an example the below chart shows how each dimension sees each step up of dot to line to square to cube:






As the greens arrows in the titles show; all of the objects are being observed from the same point-of-view.
What you can see is how, via extrusion into the extra diimension, we see a lower dimensioner's object; even if it is a very thin extrusion.

So a 2Der sees a square via its 2-face (line edge to us) and we see a square by it's 3-face (square to us).

As per below, we can also look at the square edge on, as a 2Der does, and pretty much see it as they do.
Though for them it occupies the entire width (0) of their world; whereas for us there is abundant space either side of the front edge of the square:

[quickfur]

[...] The other interesting revelation out of this is some of the things that happen when other things then direct extrusion happen - and here I explore rotation instead.
If we take a 1Der's line and - instead of extruding it - we rotate it around its centre into the 2nd direction. This produces a circle for our 2Der.
If we take the 2Der's circle and - instead of extruding it - we rotate it around a diametre into the 3rd direction. This produces a sphere for us 3Ders.
If we take our 3D sphere and - instead of extruding it - we rotate it around a mid-circle section into the 4th direction. This produces a glome for the 4Der.

Using extrusions produces prisms (e.g., extruding a hexagon makes a hexagonal prism; extruding a square makes a square prism, which is a cube). Using rotations instead produces what Garrett Jones, the founder of this site, calls rotatopes. Rotating circles and spheres produces various higher-dimensional spheres. But if we take a 2D square and rotate it around the line cutting through the center of the square, it produces a cylinder. In going to 4D, there are more interesting possibilities: rotating a sphere makes a glome, rotating a cube makes a cubinder (Cartesian product of a circle and a square). Rotating a cylinder about a square cross-section makes a spherinder (equals an extruded sphere), whereas rotating the cylinder about a circular cross-section makes a duocylinder (Cartesian product of two circles).

Applying rotation to other kinds of shapes produces all sorts of interesting things. Rotating an n-gonal prism about its n-gonal cross-section, for example, makes what I call an n-gonal "prismic cylinder", that is, the Cartesian product of a polygon with a circle. These shapes have a surface that consists of n cylinders joined end-to-end and wrapped around in the 4th direction into a ring, plus an n-gonal toroidal shape that wraps around the cylinders to close up the shape. The cubinder is one example where n=4: it consists of 4 cylinders joined end-to-end, wrapped around in a 4-membered ring, and a square torus that wraps around the round sides of the cylinders. The duocylinder also has a similar structure: its surface consists of two interlocked torus-shaped volumes that form rings in two orthogonal planes (much like the ring of n cylinders and the n-gonal torus wrap around each other in the "prismic cylinders").

This two-interlocked-rings structure is a prominent feature in 4D. It's a subset of the so-called Hopf fibration of the 4D hypersphere (glome), so convex shapes that approximate a glome tend to exhibit the same kind of structure. It's directly related to the so-called Clifford double rotations, where an object may rotate simultaneously in two orthogonal planes with independent rates of rotation.

[gonegahgah]

This two-interlocked-rings structure is a prominent feature in 4D. It's a subset of the so-called Hopf fibration of the 4D hypersphere (glome), so convex shapes that approximate a glome tend to exhibit the same kind of structure. It's directly related to the so-called Clifford double rotations, where an object may rotate simultaneously in two orthogonal planes with independent rates of rotation.

This aspect of 4D is certainly one I'll have to explore one day along with shadows that you mentioned previously. Dual rotation is very interesting. Thanks QuickFur.

For the moment I've just done up the diagrams below to depict our 4D dot, line, square and cube.
I had done these for 0D through 3D (where possible) so it's not fair to forget our 4Der world:



I've used colours to help us. Something I'm sure our 2Der friend would need to do to try to depict our 3D objects.

This gives us a good look at how a 4Der sees these things. Though obviously they will see it with a whole lot more clarity than us.
The yellow 'face' (or 4-face) of each object is its front face and the red 'face' is its back face.
I've made the front face and the back face transparent to depict that the 4Der can see the entirety of each cube face - what we would imprecisely think of as its volume.
In respect of this I must emphasize again that they only see it from towards the 4th direction in either of its opposites. So either from the yellow direction or the red direction.
If they were to look at the cube from our perspective they would be looking at it edge on and only see three, two or one squares just as we do; though for them they would be 4D squares .

I think the diagrams help us to better understand how all our objects only have two opposite faces to a 4Der and not the multiple faces that we think and know them as having.
You can see for the cube for instance that the 4Der looks at it via its yellow end or its red end. To look at the cube any other way is to look at it via some view of its 4-edges; or, in other words, from our POV.

Next thing for me to do is construct a tesseract. I hope I can show a few things with that.

I should note that, although it can't be seen, I've only made the yellow and red 'faces' transparent. The blue part of each object is actually fully solid except for its rim because a 4Der can no more see inside an object than we can. They can only see the faces (4D faces) as we do for our faces (3D faces). The rim of each blue part is transparent because the 4D 'edges' themselves form a face even though these 'edge faces' are super narrow when it comes to the 4Der handling our objects. Just like when we handle a square. The edge of the square is actually a very narrow face because it can't be 0 thick. We just think of it as 0 thick. If it were 0 thick it wouldn't be seen. I hope to show this a little better with the tesseract when I attempt to put it together.

Does this help others to understand 4D a little better?

[gonegahgah]

I've taken the liberty of spreading our 4Der's cube apart to make it easier for us to see the guts:



As I mentioned, the insides of the blue depiction are the 'solid' part of our 4Der's cube.
If we hold a square in the air it still has some real volume sadly just because it is unavoidable.
So for the 4Der holding up a cube: they will look at the cube via say it's yellow depicted face.
They will also register that it has edges, which here are depicted as the see through blue part, which helps them to identify it as a cube.
They won't be able to see inside their cube because they can't get inside the cube anymore than we could.
However they will see the entirety of the front face - which we think loosely of as occupying volume - and the entirety of the back face if they turn the cube around.
They will also see the entirety of the edges - the blue part's outside shown as see through - for all its six faces; as we think of them as but the 4Der doesn't.

[gonegahgah]

... whereas rotating the cylinder about a circular cross-section makes a duocylinder (Cartesian product of two circles).

Thanks QuickFur, now I finally understand what a duocylinder is

If you rotate the cylinder lengthwise all the way around a single point midway along its side will that also produce a duocylinder?
Also if you do a cutting operation on this, by erasing a smaller cylinder of space that is rotated around the same point, will the remainder still be a duocylinder?

I'll draw pictures when I get home if that will help to describe what I mean...

[quickfur]

... whereas rotating the cylinder about a circular cross-section makes a duocylinder (Cartesian product of two circles).

Thanks QuickFur, now I finally understand what a duocylinder is

The main thing to understand about the duocylinder is that it has a surface that consists of two toroidal 3-manifolds (torus-shaped volumes). Both manifolds are exactly the same shape, just oriented differently. One traces out a circle (a "fat circle" if you will, i.e. a torus) in the XY plane, the other in the ZW plane. These two manifolds touch each other on a single 2D sheet, that's basically a 3D cylinder wrapped around upon itself through the 4th dimension so that it forms a torus (but not like bending a pipe in 3D to form a torus, where the part of the pipe inside the "donut hole" is deformed differently from the part outside; the deformation is completely even throughout the entire sheet, and the analogue of the "donut hole" is actually a torus shape that's filled in by 1 manifold, and the inside of the cylinder forms the other hole filled in by the other manifold; both are exactly the same shape). It sounds complicated, but once you see it, it's really quite obvious.

If you rotate the cylinder lengthwise all the way around a single point midway along its side will that also produce a duocylinder?

Hmm you'll have to draw a diagram of this, I'm not 100% sure what you mean here. In 4D things rotate around a plane, so I'm not sure what you mean by rotating around a point (unless you're talking about a double rotation).

Also if you do a cutting operation on this, by erasing a smaller cylinder of space that is rotated around the same point, will the remainder still be a duocylinder?

I'll draw pictures when I get home if that will help to describe what I mean...

Please do, since what you suggest sounds interesting, but I'm not quite sure what exactly you mean.

[gonegahgah]

Please do, since what you suggest sounds interesting, but I'm not quite sure what exactly you mean.

Thanks QuickFur. I had to clarify what I was meaning myself.

Here's how I've interpreted the rotation aspect:



The 1st picture shows a line being rotated around a point. The 2nd pictures shows a square being rotated about a line. The 3rd picture shows a cube being rotated about a square. Well, you have to imagine there are more of the blue part to make up a cube but I hope it shows the idea. The lines join together to form the square that is rotated around. Is this right?

I'll try to depict a duocylinder and a spherinder tomorrow using this technique.
In the meantime here is the example picture to help explain what I'm asking about. What would this be called QuickFur?



The object is rotated into the 4th dimension around the common line (which then becomes 4 dimensional itself because of this rotation).
So I'm curious what shape the total rotation transcribes? Is it a duocylinder too?
Without the hole, is it a duocylinder?

[gonegahgah]

BTW. That is not a toilet roll. LOL.

[quickfur]

[...]
Here's how I've interpreted the rotation aspect:



The 1st picture shows a line being rotated around a point. The 2nd pictures shows a square being rotated about a line. The 3rd picture shows a cube being rotated about a square. Well, you have to imagine there are more of the blue part to make up a cube but I hope it shows the idea. The lines join together to form the square that is rotated around. Is this right?

Correct, the result of the third rotation is a cubinder (Cartesian product of circle and square).

[...]
In the meantime here is the example picture to help explain what I'm asking about. What would this be called QuickFur?



The object is rotated into the 4th dimension around the common line (which then becomes 4 dimensional itself because of this rotation).

Hmm. This is actually ambiguous, because 4D rotations happen about a plane, not a line. Your diagram appears to depict a 3D rotation, but if we try to interpret it in a 4D sense, there are at least two possibilities:

1) The rotation happens around the plane that the cylinder is resting on. Because the rotation is off-center, this appears to produce a shape that's the extrusion of a torus with minor radius 0. (I.e., a donut shape where the hole has zero radius, so it's like a sphere pinched in the middle.) The hollow part inside the cylinder causes the torus ends of the result to be hollow, too, so the entire shape will have a hollow part inside. I'm not too good with shapes with holes; you probably want to ask wendy about this one. If I'm seeing it correctly, the hole should be in the shape of an extruded torus, and goes right through the object (but doesn't divide it).

2) The rotation happens around the plane that bisects the cylinder. I'm not too sure about this one, because the rotation is off-center, but it looks like this should produce a duocylinder as well. The hollow in the cylinder causes the resulting duocylinder to have a hole that wraps around one of its bounding torus surfaces.

4D objects can have two kinds of holes: (1) the kind produced by bending a 4D tube (say a long spherinder) around so that its ends touch, thus producing the sweep of a glome around a circle: so there is a hole that goes through the middle. (2) the kind produced above, where the hole is embedded inside the object. I'm not sure how to describe this; wendy would know better.

[gonegahgah]

Hmm. This is actually ambiguous, because 4D rotations happen about a plane, not a line. Your diagram appears to depict a 3D rotation, but if we try to interpret it in a 4D sense, there are at least two possibilities:

Thanks for pointing out the ambiguity QuickFur. After examining what I was doing yesterday morning I realised how I was looking at it wrong way. Then this morning I realised how I was still looking at it the wrong way. Finally, this should be what I am actually after, if i may.....

Well first off to explain. The following is how we can produce a donut from a circle with a hole:



The rotation occurs around the line drawn in a circle following the middle of the drawn space. This produces a shape such as follows:

(Picture courtesy of google images).

So that is, finally, what I think I am after. So here is what I am hoping you can name for me QuickFur. What does this describe please?



All those circles are actually filled inbetween to form a cylinder with a hole along the centre.
The rotation occurs in the 4th dimension. I would imagine now that the plane being rotated around is a cylinder wall.

This is still a sort of rotation but not really in the usual manner we've been describing. I can only think to call it donutification but maybe there is a better term?

Is this one of the already named 4-solids QuickFur?

[quickfur]

[...]Well first off to explain. The following is how we can produce a donut from a circle with a hole:



The rotation occurs around the line drawn in a circle following the middle of the drawn space. This produces a shape such as follows:

(Picture courtesy of google images).

So that is, finally, what I think I am after. So here is what I am hoping you can name for me QuickFur. What does this describe please?

OK, so this isn't really rotation, it's some kind of eversion (turning inside-out). Essentially, you have a disc with a circular hole punched through it, and then you're turning it inside-out so that it traces out a donut shape. Am I understanding you correctly?

All those circles are actually filled inbetween to form a cylinder with a hole along the centre.
The rotation occurs in the 4th dimension. I would imagine now that the plane being rotated around is a cylinder wall.

This is still a sort of rotation but not really in the usual manner we've been describing. I can only think to call it donutification but maybe there is a better term?

Is this one of the already named 4-solids QuickFur?

I don't know what to call it, but it's certainly not rotation. By definition, rotation does not deform the object, but the kind of operation you're describing does deform the object, and that not in a linear way, so it's not rotation. I think "eversion" best describes it.

Now, as to what the resulting shape would be, I'm not 100% sure, but you may have discovered what Marek & co. in the Toratope section of this forum called the "tiger". Yeah, it's a weird name. I'm not sure how it came about... I think they were exploring different ways of constructing rounded shapes, and trying to put concrete equations to the various constructions they generated. When they came to a particular construction, they had some difficulty finding equations for it -- the metaphor is like hunting for a tiger in a forest of candidate equations, hence the name "tiger". I have to admit I'm not 100% familiar with the tiger, but what you describe sounds like the construction for it.

[quickfur]

Here's the definition of the "tiger": http://teamikaria.com/hddb/wiki/Tiger

And I found the etymology of "tiger". Apparently it's a Japanese pun. Who would have thought...

And this topic is a detailed exploration of the equations that describe the tiger.

[gonegahgah]

The 'tiger' is a cool shape but sadly it is not what I am after I suspect. When you mentioned eversion it told me I was going wrong. I suspect that I need to look at, again, the original way I posted the question because it may have been the right question afterall.

First off let me reach out to our 2Der again for reference. Normally we would think of a 2Der creating a sphere by imagining rotating a circle around a line across its diameter.
This would be a depiction of that:



However, doing it this way may be difficult for the 2Der to understand conceptually. Instead they might either do the slice method of a growing and then shrinking circle. Alternatively they might do, what I have done in a similar fashion, and that is draw several circles above each other and imagine that they are rotated around their centre points to different angles of equal amounts and that combined these form a sphere.

But, perhaps there is another way.
Although the 2Der would have difficulty drawing this; wouldn't the following be a valid representation of a sphere as well?



What I have done is rotate the 2Der's world around a point in their plane - the black dot at the centre - to represent the rotation into our left-right dimension.
Pretty much the approach that my rotational method uses.

The same process could even be done with a square with the result being a cylinder standing on its circle base.

If this is acceptable, then we should be able to rotate our 3D world around a line in a similar fashion. If so then the original question below was valid?



I may be able to represent it in a different way later but does this make the question make more sense and remove the ambiguity?

[wendy]

If you are wrangling rotations in four dimensions.

1. The wheel-rotation.

Consider the cartoon character Thor from the B.C. cartoons http://en.wikipedia.org/wiki/B.C._%28comic_strip%29. Thor stands on the axle of the wheel, astride the wheel, rather as one might stradle a bicycle.

The relevant point here is that space, divided by up and forward (the hedrix or 2-space of the wheel), gives the axle-space or across-space (the N-2 space around the wheel). In 3-dimensions, the across-space is 1-D, which gives 'left and right'.

In 4-dimensions, the around-space is 2d, so even this space would support rotation. The effect of rotation here is that the 'left-right' axis would rotate! What this means, is that if you're rolling along on your wheel, and your axle is rotating around the wheel, it would be harder to turn (since the across-clock-face is turning, and if you're trying to turn to '3-oclock', it's no comfort if the 3-oclock becomes 4-oclock or whatever. This is why, in higher dimensions, it's not a really good idea to imagine wheels having rotations in the across-space.

On the other hand, we might note that owls have such large eyes, that in order to turn the eye, a head-motion is called for. In this case, the whole head turns over the 'across-space' (left-right), so a proper rotation in the across-space might assist some things with a fixed angle-of-turn to effect turns (ie turn so up is 3-oclock + turn).

2. Planets.

Unlike wheel rotation, the nature of planets tend to a clifford-state, which equates to equal turning in both the wheel-space and its around-space. All points go around the centre of mass, this is a tendency caused by the tidal equalibrium of energy.

3. General rotations.

The general rotation in four dimensions, is for un-equal speeds in two orthoganals. Looking at an axis from each of the un-equal speeds, one gets a 2-space occupied by a lissajous or bowditch curve. Although these can in theory, cover all of this 2-space, the distance between the point and the two orthogonal axies does not change: it in effect, orbits both axies at different speeds. It is restricted to what is called a torus (for want of better word). A 'tiger' is a spherated torus of this sense.

When such a torus is unwrapped, it gives a rectangle, of sides sin and cos of the climata (angle between the axles). The movement moves at constant but different speeds along both sides, so it gets ruled into a set of parallel lines. When the speeds are equal, the line is a diagonal (or "broken" diagonal) of the same length as the axies. Only the various diagonals and broken diagonals are great circles around the centre.

4. The sun and climata.

For a planet rotating at constant clifford speeds, the rotations all follow 'broken diagonals' in the same sense (eg bottom-left to top-right). The sun in the sky moves on a 'broken diagonal' of the opposite sense (ie bottom-right to top-left), and is restricted to one torus (the tropics). This torus is then orbits at the same distance, exactly two axies, the distances between the torus and these axies being half the angle that the left-diagonals and right-diagonals cross.

Because rings on these torus all have the same climate, (varying as from 0-N to 90-N on the earth), it is probably appropriate to call these angles etc the 'climata' or climate-defining lattitude.

The three coordinates one might generate by navigation and the seasons, are then (1) longitude, or the effect of rotation, (2), climata, the progression between very hot climes, and very cold ones (ie equatorial to polar in 3d-speak). and (3), annular, or the progressions of months, representing the progression of the sun along the zodiac, maps onto every climate-torus.

5. The space of rotations.

We see that the set of left-turning curves form a zenith-sphere, where every instance of a circle in a clifford-parallel is a sphere. One can, for each left-set, construct a right-set, which contains exactly one common element. For example, on the 4-planet, the normal rotation might be a left-set, there is exactly one great arrow (circle + direction), that is right-parallel to the zodiac (or sun-trace). This corresponds to the 0-N line.

All rotation-modes in 4d can be represented as a sum of opposite clifford rotations, or in the notation of quarterions, left and right multiples. One might note for the main axis, one has a speed L+R, and for the orthogonal axis, L-R. Then it is a superposition of a L-clifford rotation of speed L, and a right-clifford rotation of speed R.

For the great arrows themselves, we have L=R, so the general positions is to have a prism-product of two 3-sphere-surfaces, where every point X, Y represents a great arrow, -X, Y and X, -Y represent the orthogonal in L and R rotations, while -X, -Y represents the reverse rotation of the sphere.

6. The wheel-axies of polytopes

Relative to this axis, we can plot (say), the simple wheel-rotation symmetries of the various polytopes.

For the {3,3,5}, there are 144 pentagonal points, 400 trigonal points, and 900 digonal points. These various points lie at the vertices of x3o5o&x3o5o, o3o5x&o3o5x, and o3x5o&o3x5o, the double-prism products of the icosa, dodeca and icosadodeca, when full symmetry is maintained in 6D.

For the group {3,4,3}, the crossing points correspond to 64 bi-cube vertices[3], 144 bi-CO vertices [2], and 36 bi-octahedra vertices [4]. There is a subgroup of this, of order 2, which eliminates the 144-biCO vertices, which is 36 bi-octahedra and 32 (as 2×16), corresponding to the prism of two tetrahedra, and its central invert.

Because each great circle represents two great arrows (X,Y vs -X,-Y), one has to halve the numbers above for the static axies.

The simplex in four dimensions, does not have this kind of symmetry, opposite a 3-rotation is a 2-rotation. It's considerably more complex to map these against the rotational phase-space.

7. Planet-rotations and phase-space.

If one supposes that the rotation-space is actually a 6-sphere, then every possible rotation is represented by a single rotation. Suppose that one takes a unit circle through x,y from X,Y. Then every possible ratio of y/x would cross this circle, and because it is a single circlet on the 6-sphere, every possible phase and variation of rotation is a single point in 6d. (the actual intensity of rotation plots radially). Now, for any state of rotation, one would tend to steer for planet-rotation, to the state of equal density, ie X=0 or Y=0, but for steering something, one might head to Y=X or Y=-X (wheel rotation).

[quickfur]

[...] But, perhaps there is another way.
Although the 2Der would have difficulty drawing this; wouldn't the following be a valid representation of a sphere as well?



What I have done is rotate the 2Der's world around a point in their plane - the black dot at the centre - to represent the rotation into our left-right dimension.
Pretty much the approach that my rotational method uses.

Hmm. While I understand the general approach you're driving at, I'm not sure if this diagram is necessarily a good representation of how a sphere is generated. For one thing, it looks like the circle is rotating about a point on its edge, which, if we do this in 3D in a plane perpendicular to the circle, produces not a sphere, but a torus of equal radii (i.e. the "pinched sphere" shape I referred to in an earlier post, basically a donut where the hole has shrunk to a point -- btw, I made a mistake in my earlier post, the torus doesn't have minor radius 0, just that both major and minor radii are equal so the radius of the hole is zero).

But supposing that we translate the circles so that the stationary axis of the rotation passes through the center, then what we have essentially is that we're rotating a circle about its axis as before, except that when we draw the diagram, we turn the circle in each rotated position 90° so that it lies flat on the plane again. So essentially, you have an implicit 90° turn there.

What this amounts to is that the rotational axis in 3D is mapped to the center of rotation in 2D, so what we depict as a point in the diagram is actually an axis in 3D. Using this interpretation, then, let's look at your diagram again:

[...]

[...]

If we assume the analogous thing happens as I described above (translate the cylinder so that it's centered on the rotational axis, which is implicitly a rotational plane in 4D), then this is the same thing as rotating a cylinder around a plane that bisects the cylinder. The result is a spherinder with a hole dug through the middle (basically a spheridrical tube with a hollowed out interior, which opens out at both ends).

If we forego the translation so that the cylinder is not centered on the rotational axis (implicitly a rotational plane in 4D), then the result is the extrusion of a torus with a zero-radius hole, as I've described before, with a toroidal hole carved through it.

The other thing to note, though, is that your 3D diagram is not completely analogous with your 2D diagram, because in the 2D diagram you have the circle rotating around the point (IOW, the axis perpendicular to the circle's edge), but in the 3D diagram, your drawn axis is tangent to the circular cross-section of the cylinder. So it's not fully clear which kind of rotation is meant here -- I just made an educated guess.

Anyhow, it may be helpful to enumerate the possible shapes that are generated by rotating a cylinder through 4D. First, let's consider the ones where the plane of rotation passes through the center of the cylinder. To avoid confusion, let's say the cylinder's lids lie parallel to the XY plane, and the axis of the cylinder (the line that connects the center of the lids) lies on the Z axis. So we have the following cases:

- Rotations in XY, YZ, and XZ planes: these are just 3D rotations, so you only get 3D shapes out of them.
- Rotate in XW plane: spherinder (stationary plane is YZ)
- Rotate in YW plane: spherinder (stationary plane is XZ -- same result because cylinder is symmetrical in the XY plane)
- Rotate in ZW plane: duocylinder (stationary plane is XY)

Then the off-center cases, where the stationary plane of the rotation is tangent to the cylinder:

- Rotations in XY, YZ, and XZ planes: various 3D objects;
- Rotate in XW plane: extrusion of torus with zero-radius hole;
- Rotate in YW plane: same as previous case since cylinder is symmetrical in XY plane;
- Rotate in ZW plane: duocylinder (the off-center displacement doesn't make a difference in this case because the displacement lies in the stationary plane -- to understand why this makes no difference, consider translating a circle along the rotational axis and then rotating it: the result is the same as rotating it first then translating it).

Now just for curiosity's sake, let's also look at the cases where the cylinder is displaced even further, so that the stationary plane does not touch it:
- Rotate in XW plane: extrusion of torus (the torus hole has non-zero radius);
- Rotate in YW plane: same as previous;
- Rotate in ZW plane: extrusion of duocylinder's ridge (a kind of toroidal prism) (? -- not 100% sure about this one).

If the cylinder has a hole bored through the middle, like in your diagram, then the following modifications occur:
- Spherinder becomes a spherinder with a hole bored through it (basically the spherindrical analogue of your bored cylinder);
- Duocylinder becomes a duocylinder with a toroidal hole bored through it;
- Extrusion of torus with zero-radius hole becomes the same shape with a toroidal hole bored through it (the toroidal hole is a different shape from the previous case; in this case, the shape of the hole is basically the extrusion of a 3D torus).
- Extrusion of torus (with non-zero radius hole): similar to previous case; the shape of the hole is just the extrusion of a torus with a smaller minor radius.

Hope this helps.

[quickfur]

[...]In 4-dimensions, the around-space is 2d, so even this space would support rotation. The effect of rotation here is that the 'left-right' axis would rotate! What this means, is that if you're rolling along on your wheel, and your axle is rotating around the wheel, it would be harder to turn (since the across-clock-face is turning, and if you're trying to turn to '3-oclock', it's no comfort if the 3-oclock becomes 4-oclock or whatever. This is why, in higher dimensions, it's not a really good idea to imagine wheels having rotations in the across-space.

Yeah, I've come to the same conclusion, that you probably want your wheel to be in the shape of the cartesian product of some polygon with a circle. The polygonal cross-sections allow you to thread the wheel through a polygonal hole which fixes its orientation relative to the vehicle.

The polygonal hole may be made a part of another mechanism that allows the driver to shift the left-right axis when desired, while keeping it fixed for the most part. In this way, one only needs an axle mechanism that allows turning in a single axis (just left/right); if turning in any of the other 360° of directions is desired, one first rotates the polygonal hole so that the turning axis aligns with the desired direction, then execute the turn itself.

[...]
The general rotation in four dimensions, is for un-equal speeds in two orthoganals. Looking at an axis from each of the un-equal speeds, one gets a 2-space occupied by a lissajous or bowditch curve. Although these can in theory, cover all of this 2-space, the distance between the point and the two orthogonal axies does not change: it in effect, orbits both axies at different speeds. It is restricted to what is called a torus (for want of better word).

This torus is just the margin of the duocylinder, right?

A 'tiger' is a spherated torus of this sense.

I still have some trouble understanding the spheration here. Is this equivalent to taking the set of points that are within a certain distance from the margin of the duocylinder?

[...] For a planet rotating at constant clifford speeds, the rotations all follow 'broken diagonals' in the same sense (eg bottom-left to top-right). The sun in the sky moves on a 'broken diagonal' of the opposite sense (ie bottom-right to top-left), and is restricted to one torus (the tropics). This torus is then orbits at the same distance, exactly two axies, the distances between the torus and these axies being half the angle that the left-diagonals and right-diagonals cross.

Because rings on these torus all have the same climate, (varying as from 0-N to 90-N on the earth), it is probably appropriate to call these angles etc the 'climata' or climate-defining lattitude.

OK, this sounds the same as the geocentric model I posted about in the other thread. The orbit of the planet basically divides its surface into regions delineated by concentric tori, starting from the great circle representing the "tropics" and converging on the orthogonal circle which basically has arctic climate.

The three coordinates one might generate by navigation and the seasons, are then (1) longitude, or the effect of rotation, (2), climata, the progression between very hot climes, and very cold ones (ie equatorial to polar in 3d-speak). and (3), annular, or the progressions of months, representing the progression of the sun along the zodiac, maps onto every climate-torus.
[...]

Hmm, how does (3) factor into navigation? If the ecliptic is close to the celestial equator (but likely not equal, and probably not intersecting either, unlike the 3D case), how would it help fix a cardinal direction on the surface of the planet?

[wendy]

There are several senses of torus, but the sense when referring to the sphere, is that of the margin of the duocylinder.

Spheration is pretty much when you replace each point of a figure with a sphere. Points become spheres, lines become cylinders, etc. Spherating an object simply includes inside the figure, any point that lies at r from the figure.

An example is the http://atomium.be/ . One sees here a cube with its lines crossing at a central vertex. The vertices are spherated at a value of R, which generates large spheres. The lines are spherated with r which generates thinner spheres. If you in 3d spherate a circle with 'r', you get a torus.

The tiger is the set of points, at distance 'r' from a point on the margin of a duo-cylinder. The 'at distance r' is the spheration, since 'at distance r' from a point gives a sphere.

The tiger is a "bi-glomolatric prism spherate" in PG talk. A glomolatrix is a sphere-surface. The prism-product creates a 2d surface, being a torus in the spheric sense. The spheration "fattens" out this into a solid, more or less by 'multiplying it' by a circle, radially at each point, like a regular torus is a curcle multipled radially by a smaller circle.

In navigation. Consider first, in 3D, that lattitude runs from +90 to -90. In four dimensions, one could imagine this 3-sphere as a section of the 4-sphere, and the pole + hemisphere is rotated orthogonal to the equator around the full polar circle. Applying this to a line of lattitude as a diameter of the circle, the full circle is generated by rotating the lattitude-line around its midpoint. The '90' points now sweep out full circles etc. To make the zenith-sphere, you then pull the 90-circle into the pole opposite the 0-point.

The seasons come by imagining that either of these serve as 'hands' on the season-clock. What's ever at twelve makes summer, at 3 autumn, 6 winter, 9 spring. In three dimensions, it's a line, with a N and S end, the ends point at S and S+6, eg 3 and 9 or 2 and 8. In 4D, the hand is a full disk, and any point of the sphere maps onto some ray of this disk. So when the disk is at '5', there are points at '6' (mid winter), and 3 (midspring), and every point of the circle. So you have 'time zones' and 'season-zones'.

[quickfur]

[...] Spheration is pretty much when you replace each point of a figure with a sphere. Points become spheres, lines become cylinders, etc. Spherating an object simply includes inside the figure, any point that lies at r from the figure.

Ah, I see, thanks. I think I was confusing spheration with some other kind of operation that people were discussing some years ago that was used to generate torus-like figures.

[...] The tiger is the set of points, at distance 'r' from a point on the margin of a duo-cylinder.

So essentially a tiger is just a "fat" duocylinder margin? What of the equations given on the wiki? They seem to describe some kind of parametrization that involves revolutions within the "fat duocylinder margin" bulk.

[...] In navigation. Consider first, in 3D, that lattitude runs from +90 to -90. In four dimensions, one could imagine this 3-sphere as a section of the 4-sphere, and the pole + hemisphere is rotated orthogonal to the equator around the full polar circle. [...]

Hmm. We appear to be using different approaches here. I'm trying to figure out how native inhabitants of the 4D planet would figure out their cardinal directions phenominologically. The rising and setting of the sun is one reference that can be used to fix a pair of directions (east/west). The celestial equator (projection of one of the planet's rotation planes onto the celestial sphere) may also be used. I may be mistaken, but I think that if the planet is undergoing double rotation, then the constellation over the course of a night should rotate around the two orthogonal circles by about 180°, so by tracing the paths of the right constellations across the sky, one can determine where the two circles are. This then fixes two pairs of directions; east/west (which should be reasonably close to the ecliptic, though not the same) and an orthogonal pair of directions which may be called north/south (as they indicate the most direct paths to the "arctic circle" -- not the same as the arctic circle on Earth, but a great circle analogous to our N/S poles). The last pair of directions then will be perpendicular to the plane defined by these two pairs of directions.

This approach sections the planet into concentric torus-shaped "latitudes", which begin from the equatorial great circle and converges on the orthogonal circle (the "arctic"). Each toroidal region represents places of similar climate. Dividing the planet into spherical latitudes is convenient for a mathematical description of the planet, but I doubt that would be how navigational technique would arise in a native culture.

Or am I off-base here?

[wendy]

You're spot on with the navigation. You can't assume that because people use spherical coordinates in 3d, it works in 4d as well.

East/West isn't a direction in the sense that it is in 3d. The E/W follows the rising and setting of the stars. That is, for a person there is a line around the horizon where things rise on one side, and sets on the other. The EW axies don't all point the same way. If you're into complex numbers, you can follow the individual EW axies by considering a = W+iX ,b = Y+iZ, then you look at wb , wa. where w is cis (a. t), a is a constant speed.

Lattitude is likewise not a direction, but a set of clifford-parallels in the opposite sense to Longitude (EW). The actual alignment is set by the zodiac (which crosses longitude at the tilt angle). The second set is the range where the right-parallels of the zodiac cross the left-parallels of the rotation. These converge exactly at the 'equator (0C)', and opposite at the 'poles' (90C). These torii define climate, hence the 'c'. You can use N/S for these, i suppose, since S derives from 'sun'. What we call the equator (equaliser) would be the south-circle, and the poles are the north-circle. It's kind of like S and N pole, but the climates are pushed so that the distances from the N pole are doubled, and the equator falls on the S pole.

The third set of axies. When the sun is in a point of the zodiac, it lies over a particular great-arrow. Through the year, the sun moves to subseqent great-arrows on the tropic-torus, returning to the original one after a (tropical) year. One can construct a single line from a point to the hot-clime (souþ circle), and cold-clime (north-circle). All points along this line has the same time of day, and the same time of year, and therefore the same annulum and EW axies.

This means that there is a third axis, which crosses the time axis at the double-angle of climata, which defines where in the year-circle you are. At the south-circle, the time-of-day and time-of-year both run forward. On the north-circle, the time-of-day runs forward, but the time-of-year runs backwards.

Spherical coordinates only work in 3d, because the rotation of (radius, lattitude, longitude) gives rotation only in longitude. In 3d, then the coordinates would consist of a rather non-linear (radius, climata, lattitude, longitude), with rotation gives longitude (or lattitude).

[wendy]

The product that generates holed polytopes, is in general, the comb product.

In practice, the comb is the regular product by repetition of surface. This involves generally a loss of dimension. For tilings, the actual tiling is a virtual surface, and there is no net loss of dimension, so eg a square-lattice (surface of a polyhedron), × a horogon (line-segments = surface of a polygon), gives the cubic lattice.

Another implementation is to make the product loose the dimension radially. You can get the result by either sweeping a small circle of r=1 in the XZ plane, at x=2,0,0 around the z axis. The rotation sweeps out for each point of the XY plane, a copy of the XZ circle, and in the XZ space, there is a copy of the circle in the XY plane, for each point of the XZ circle. You can even multiply polygons, to get a (non-uniform) tiling of squares over a torus.

The notion was that i posited that if you take a tripple-polygon product in four dimensions, it could be done in more distinct ways than it really can. This is because one can do topological transforms that pass through or near the tiger.

The tiger might be constructed as a tripple-polygon product, with two undeformed ones and one radially deformed one. For example, if you multiply two dodecagons in 4d, you get the square margins of the bi-dodecagonal prism. This surface, if equal-edged, falls on the vertices of a 45-degree torus. The next process is to replace each square with a new radial figure. At a vertex, you can replace the vertex by say a (distorted) pentagon in that plane. One can replace each vertex of the square by such a figure, rotated around the axies. This taken together replaces the square with a square-pentagon prism, of which we have a packing of 144 around the figure. All of the pentagon-prism faces are left inside, so we just have a ring of five distorted cubes around the square.

This in effect multiplies (repeats) a pentagon at each point of the surface, so we get the product of the surface of a pentagon (five line segments), by the surface of the thing (144 squares), to give 720 cube-likes, which folds out to a 5×12×12 cubic grid.

[quickfur]

[...] East/West isn't a direction in the sense that it is in 3d. The E/W follows the rising and setting of the stars. That is, for a person there is a line around the horizon where things rise on one side, and sets on the other. The EW axies don't all point the same way. If you're into complex numbers, you can follow the individual EW axies by considering a = W+iX ,b = Y+iZ, then you look at wb , wa. where w is cis (a. t), a is a constant speed.

Hmm. Can the celestial equator be deduced from observing the paths of the constellations, though? One can then define east/west as lying perpendicular to the clifford parallel that intersects one's position.

Lattitude is likewise not a direction, but a set of clifford-parallels in the opposite sense to Longitude (EW).

Yep.

The actual alignment is set by the zodiac (which crosses longitude at the tilt angle). The second set is the range where the right-parallels of the zodiac cross the left-parallels of the rotation. These converge exactly at the 'equator (0C)', and opposite at the 'poles' (90C). These torii define climate, hence the 'c'. You can use N/S for these, i suppose, since S derives from 'sun'. What we call the equator (equaliser) would be the south-circle, and the poles are the north-circle. It's kind of like S and N pole, but the climates are pushed so that the distances from the N pole are doubled, and the equator falls on the S pole.

Yeah I had the etymology of "south" in mind when naming these directions. So unlike 3D where north is directed at the rotational pole and south is directed at its antipode (or vice versa), in 4D "south" actually literally means "towards the sun" (i.e. a geodesic, or most direct path, from one's current location to the 0C circle (the "equator"). "North" is directed at the orthogonal circle, the 90C circle.

One interesting consequence is that if you're standing on the 0C or 90C circle, then "north" (respectively "south") is ambiguous, just as "south" is ambiguous in 3D when you're standing on the north pole, and "north" is ambiguous when you're standing on the south pole. The difference, though, is that in 3D, the poles lie in generally inhospitable areas, and so people are less likely to run into this ambiguity problem in everyday life, whereas in 4D, the 0C pole lies in the middle of the tropics, so tropical dwellers are bound to be well-acquianted with the problem. I wonder what solutions a tropic-dwelling culture would invent to be able to deal with this ambiguity in a practical way.

The third set of axies. When the sun is in a point of the zodiac, it lies over a particular great-arrow. Through the year, the sun moves to subseqent great-arrows on the tropic-torus, returning to the original one after a (tropical) year. One can construct a single line from a point to the hot-clime (souþ circle), and cold-clime (north-circle). All points along this line has the same time of day, and the same time of year, and therefore the same annulum and EW axies.

Yes, this is what I had in mind. But I suppose the trouble is that outside of this line, one may have different EW axes, depending on the relative orientations of the plane of the ecliptic with the plane of the celestial equator. (And it seems that this is not just a simple angle, since two 2D planes in 4D have a much larger set of possible relative orientations.)

This means that there is a third axis, which crosses the time axis at the double-angle of climata, which defines where in the year-circle you are. At the south-circle, the time-of-day and time-of-year both run forward. On the north-circle, the time-of-day runs forward, but the time-of-year runs backwards. [...]

Huh, this is an interesting effect. This might be cause for great cultural misunderstandings between arctic dwellers and tropic dwellers. Much hilarity would ensue.

[quickfur]

It seems we have inadvertently derailed the discussion about gonegahgah's method of 4D visualization. Here's my feeble attempt at bringing things back on track.

I've taken the liberty of spreading our 4Der's cube apart to make it easier for us to see the guts:



As I mentioned, the insides of the blue depiction are the 'solid' part of our 4Der's cube.

Here I'd add that if the 4Der were to look at the cube from the side, as from our POV, then they would see up to 3 faces at a time, but even then it's not exactly the same as we would see it. We can only see the blue part -- at the most 3 squares on the surface of the blue cube depicted above. But the 4Der's retina have an additional width dimension, so they actually see each square as a red-blue-yellow sequence. Of course, in practice this sequence is imperceptibly thin, so it's close enough to how we perceive the cube, but it's not quite the same.

The analogy is when we look at a square edge-on, we see the width of 3 atoms (the square is a thin cube) -- to use another of gonegahgah's images:


The 2Der, however, can only see the (edge of the) middle slice of atoms, just a 1 atom wide line segment of the square's edge. We can paint the left slice red and the right slice yellow, and it would register in our 2D retina as red/yellow "fringes" around the line that is the square's projection onto our retina. But this red/yellow layer is imperceptible to the 2Der.

So the red/yellow cubes in the first image will show up as red/yellow "fringes" around the square face(s) if the 4Der would look from our POV, and the square face itself shows up as a square image in their retina. But we can only see the square itself. If we imagine the 4Der's retina as a 3D array of light-sensitive cells, then the square face may be show up, for example, as a grey square sandwiched between a red square and a yellow square, each just 1 pixel thick. But if we look from the same POV, our 2D retina, which has only a 2D array of light-sensitive cells, can only register the grey square, not the red/yellow squares.

To us, a grey square in our field of vision fills up a significant area of our retina: a 2D area occupying a significant percentage of the total area of our retina. But the red/grey/yellow sandwich image that forms in the 4Der's retina, though it also fills up a 2D area (well, actually the volume of a very thin cuboid), that area is only a tiny fraction of the total 3D volume of their retina. So the square image is to their perception, as a thin line is to our perception. Just as we prefer to look at a square from an angle that it would occupy more than just the space of a line in our field of vision, so the 4Der would prefer to look at a cube from an angle that would occupy more than just the space of a 2D area in their field of vision. When we look at the square from such an angle, we would not see the 3-atom sandwich of the square, but just one of its two faces; from the analogous POV of the 4Der looking at the cube, they wouldn't see the red/grey/yellow sandwich, but just the yellow cube "face" (or the red cube if they turn it around).