Dimensional Baby Steps

[gonegahgah]

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.

Hi QuickFur, no worries, I've just been so busy the last few days. I'll get back here later and read what has been added. Haven't had a chance. No time to even review yet atm.
I was just finishing typing the following to add which I had started to type up a few days ago:

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.

I'm tending to think this way too. I'm getting the feeling that it is more about rotating the drive mechanism within the wheel so that a particular angle of sideways is the one currently driving the forward motion. This would then give control to turning in a particular angle of the 360° of extra sideways to the current orientated angle hopefully. I'll keep plugging away and see what I can grow to understand of this along the way.

I've come to realise, QuickFur, that there is a distinction between representation and physical layout. The following could also be used to represent a sphere for a 2Der:



So every cross section of the circle is spun around the cross section's centre point into the 4th dimension to produce the sphere.
The following is just spread a bit more apart to allow more of the detail to show.



In all the images I've also shown the ground rotated into the 4th dimension and this is what makes each of these representations equivalent.

So just here I have shown the following representations:



1. Rotating circle around centre diameter line into the 4th dimension.
2. Rotating circle around a point where it meets the ground into the 4th dimension.
3. Rotating circle cross sections around their centre points into the 4th dimension.

All of these are equivalent which is shown by the ground following the same rotation pattern into the 4th dimension.

As I've realised here, with the example QuickFur provided me with of the tiger, there are other direct ways things can uniformly form shapes into the 4th dimension beside simple rotation and extrusion.

Understanding this now, which I did a couple of days ago but I've been too busy to write, then it should be easier to match what I was wanting with one of the simpler shapes that QuickFur was demonstrating for me earlier. I've got to go out so I'll try to work that out later.

[quickfur]

Finding ways of generating 4D shapes is always fun. The recent discussion about rotations make me want to consider some other simple cases as well:

Rotating cones:

1) Rotate a cone about the plane that its circular base rests on. The result is what wendy calls a "bicircular tegum", i.e., the convex hull of two circles lying in orthogonal planes. It's a kind of topological dual of the duocylinder.

2) Rotate a cone about the plane that touches its apex and parallel to its base: the result is a shape that has one torus of the duocylinder, and a kind of concave triangular torus where the other torus of the duocylinder would be.

3) Rotate a cone about a plane that bisects it from apex to base: the result is a spherical cone -- the shape of a light-cone.

4) Rotate a cone about a plane tangent to its circular base, parallel to the line from its apex to the center of its base: the result is a kind of tapered torus with equal radii (a donut with a zero-radius hole, or a pinched sphere). The tapering is from the torus to a circle that lies on a parallel hyperplane above it. If I'm not mistaken, it's the same thing as the sweep of a cone around a circle that lies in a plane extending into the 4th dimension. A kind of "cone-ated torus".

Rotating tori:

1) Rotate a torus about the plane parallel to its major radius: I believe this produces a spherated torus, or the sweep of a sphere dragged around in a circle in 4D.

2) Rotate a torus about a plane perpendicular to its major radius: I can't quite pinpoint this shape, it appears to be some kind of spherical shape, perhaps a spherated hollow sphere? It's not the same as a glome, because it's not spherical in one of its dimensions.

3) Rotate a torus about the plane that it rests on: produces what I think is the sweep of a torus with zero-radius hole around a circle in 4D. If the rotation is performed in a parallel plane displaced from the torus, then you get a kind of toroidal torus, which is the sweep of a torus around a circle in 4D. Hmm... this may be the same thing as the tiger! That's unexpected. And very cool, if I'm seeing it correctly!

Rotating other shapes:

1) Rotating a cube about a plane that bisects it, parallel to a pair of faces: cubinder (cartesian product of square and circle). Same thing happens with rotating a cube about a plane that one of its faces rest on, you just get a cubinder with a different radius.

2) Rotating a cube about a plane that bisects it across a diagonal: if I'm seeing it correctly, this produces the extrusion of two circular bipyramids (cones joined base-to-base). Its surface consists of two circular bipyramids and a weird kind of torus whose cross section is like a pair of thick L shapes joined end-to-end.

3) Rotating an octahedron about the plane that bisects it (where 4 vertices lie in ): produces a square-circle tegum (the dual of a cubinder).

I'm trying to work out what shape is produced by rotating a cube about a plane that makes a hexagonal cross-section with it, but I can't quite picture it in my head. I think its a hexagram-circle tegum with 6 small circle-triangle tegums subtracted from it, but I'm not 100% sure. Maybe wendy can confirm this.

Anyway. That's enough weird shapes for now. Now we return to our regular schedule of 4D visualization talk.

[wendy]

Suppose you have a cube in XYZ space, so a diagonal falls on the X axis. You rotate this in WX space, but you find that it fails to link up, because the simple rotation requires something like reflective YZ plane. (it corresponds to a flip or skew rotation in 3d).

If you do a 180-rotation in YZ as the WX goes through 180, a clifford rotation happens, and you get something like a swirl-prism. It's interesting, Bowers had a good look at swirl-prisms on different figures.

Every point goes around the centre, so the exact choice of axies is not a concern. All four axies will go around the centre, in four different lines, corresponds to four of the girthing hexagons of {3,4,3}. The twelve edges fall by pairs into six marginal walls, and the six faces fall by pairs into three square prisms. We now try to figure out how this fits into the {3,4,3}.

I will probably need to follow it in my viewer, it's a little complex to do straight up. I keep getting six faces, including three that make no appearnce in this view.

[gonegahgah]

There are so many levels to fathom that it is just mind blowing.
The reason I'm exploring rotation, primarily I hope, is in an attempt to understand 4D 'surfaces' to a greater degree.
Hopefully this need to understand 4D 'surfaces' will tie back to the archery example that I wish to explore for the purposes of understanding this poly-sidedness.

As an aside - but probably still relevant - one of the important aspects, I feel now, is the cross-conception across the dimensions of the modes of space.
By this I mean that 'volume', for example, is a concept greater than a straight definition of occupying 3D space as we are used to defining it.

Entering the word 'volume' at http://dictionary.reference.com has the relevant definition:
"The amount of space, measured in cubic units, that an object or substance occupies."

The key sense of this definition is: "the amount of space ... that an object or substance occupies."
It is simply measured in cubic units for us because our space has cubic proportions.

For a 4Der their space has quartic proportions so for a 4Der their 'volume' (or 4-volume) is measured in quartic units (tessaratic units).
Maybe we should be calling a tesseract a quard (pronounced kword) (ie. square, cube, quard) instead of a tesseract? Too late now but it would follow the progression.
For a 2Der their space has squared proportions so for a 2Der their 'volume' (or 2-volume) is measured in squared units.

Same goes for the concept of face.
Entering the word 'face' at http://dictionary.reference.com has the relevant defintion:
"The main side of an object, building, etc, or the front"
For uniform objects, like a cube, we refer to all the outward squares as faces because they are all the same and any of them could be looked at from front on.

So again for a 4Der the 'face' of a tesseract is any of its cube 'faces' and for a 2Der the 'face' of a square is any of its line 'faces'.

I think it is useful to parallel our thoughts to those of our 2Der and 4Der counterparts so that we can think more like them.
This may, or may not, assist us towards discovering the design of real world artifacts for our 4Der counterparts.
Hopefully it does and I feel it gives me a better sense of their lives.

Another principle I wanted to shore up, for myself, was the notion of how things 'lie' on a surface in 4D; which is a necessary extension from the concept of 'surface'.
So I'll add some pictures next to show how this turned out for me.

[quickfur]

There are so many levels to fathom that it is just mind blowing.
The reason I'm exploring rotation, primarily I hope, is in an attempt to understand 4D 'surfaces' to a greater degree.
Hopefully this need to understand 4D 'surfaces' will tie back to the archery example that I wish to explore for the purposes of understanding this poly-sidedness.

4D surfaces are basically 3D manifolds (a manifold is just an n-dimensional "cloth" that can be bent, flattened, twisted, etc.). In the case of flat objects like polytopes (the generalization of polygons & polyhedra to higher dimensions), the 4D surface consists of an assembly of polyhedra.

We may have trouble understanding how such surfaces work, but it helps to think of how a 2Der would have trouble understanding how surfaces in our 3D world work. To them, a 2D area occupies space, so they find it hard to grasp the concept of us 3D beings putting a cup on a table, say. How could you possibly put an object inside an area? It's already hard enough to understand how said object doesn't need to pass through the edges of the area, much less how it could possibly "rest on" that area.

In 3D, of course, this is intuitively obvious. But we in turn have trouble understanding how a 3D volume can possibly act as a surface. How can one, for example, rest a tesseract on top of a large cubical surface? Where would the tesseract go in the volume of that surface? By dimensional analogy, we can understand that a cubical volume from the 4D perspective is a two-sided surface with a cube-shaped boundary. So the tesseract can sit upon one side of it, just as a cube in our world can sit upon a large square surface. And just as we tend to rest the cube roughly in the middle of the surface, such that the bottom face of the cube lies within the square boundary of the surface, so also a 4Der would tend to place a tesseract upon a cubical desk in such a way that its bottom cube rests within the cubical confines of the desk's surface.

Many such analogies can be drawn, which are quite helpful in grasping the concept of a 3D volume that behaves like a surface.

As an aside - but probably still relevant - one of the important aspects, I feel now, is the cross-conception across the dimensions of the modes of space.
By this I mean that 'volume', for example, is a concept greater than a straight definition of occupying 3D space as we are used to defining it.

Yes, this is what I was driving at in my earlier post, upon 4DSpace's request, on dimensional terminology. In a higher dimensional space, objects of particular dimensions have a different function than in our space. A 2D object in our world is a space-dividing surface, but in 4D, it is relegated to the role of a mere margin where two surfaces may meet. A 3D object in our world occupies space, but in 4D, it is relegated to being a mere surface where two objects may meet.

Entering the word 'volume' at http://dictionary.reference.com has the relevant definition:
"The amount of space, measured in cubic units, that an object or substance occupies."

The key sense of this definition is: "the amount of space ... that an object or substance occupies."
It is simply measured in cubic units for us because our space has cubic proportions.

For a 4Der their space has quartic proportions so for a 4Der their 'volume' (or 4-volume) is measured in quartic units (tessaratic units).

Correct. And this is where it's helpful to distinguish between the absolute number of dimensions of an object, and its number of dimensions relative to the ambient space. The absolute number of dimensions is an inherent property of the object, and defines properties that are attributive to the object. A cube has 3 dimensions, no matter how high or how low the dimensionality of the surrounding space. It will not cease to have 3D volume just because you put it in a higher-dimensional space. Its function with respect to the surrounding space, though, depends on the relative difference between its number of dimensions and the dimensionality of the space.

A cube in 4D, for example, has a different function from a cube in 3D; in 3D, we may use cubes as building blocks to assemble 3D objects, say things like a Rubik's cube or some such. But in 4D, because of the presence of one more dimension in the surrounding space, the cube ceases to function as a building block, and becomes instead a surface. In that sense, it plays a role analogous to that of a polygon in 3D. In 3D, we may use polygons to assemble a polyhedral surface, for example -- 12 pentagons can be assembled to make the surface of a dodecahedron. In 4D, we use cubes and other polyhedra to make surfaces of 4D objects. None of this changes the absolute dimensionality of the cube or polyhedron -- a cube is still 3D, and a polygon is still 2D. It's just that their roles have changed because they compare differently to the dimensionality of the surrounding space.

So it's useful to have two different sets of terminology, one for describing absolute dimensionality, and another for describing relative dimensionality. Conflating the two only leads to confusion when one starts speaking of spaces of different dimensions. Wendy's Polygloss defines a consistent set of terminology that respects this difference, which is why it has utility even up to 6 to 8 dimensions. When we go up that high, our usual 3D-centric terminology utterly fails us, because it doesn't clearly differentiate between absolute dimensionality and relative dimensionality.

Maybe we should be calling a tesseract a quard (pronounced kword) (ie. square, cube, quard) instead of a tesseract? Too late now but it would follow the progression.
For a 2Der their space has squared proportions so for a 2Der their 'volume' (or 2-volume) is measured in squared units.

The mathematicians are way ahead of us. The usual terminology for n-dimensional cubes is n-cube. So a 2-cube is a square, a 3-cube is what we think of as a cube, a 4-cube is a tesseract, and so on.

In any case, I'd like to restate some of the terms I defined in the post alluded to earlier:

Relative terminology (relative to a space of n dimensions):

A facet is an (n-1)-dimensional polytope that serves as part of the surface of an n-dimensional object.

A surface is an (n-1)-dimensional manifold (not necessarily flat) that serves as part of the boundary of an n-dimensional object.

A ridge is an (n-2)-dimensional manifold where two surfaces meet. For example, two square faces in the cube meet at an edge, so the edge is a ridge of the cube. In higher dimensions, a 4-cube (i.e. tesseract) has cube-shaped facets that meet at squares -- thus, these squares are the 4-cube's ridges.

A peak is an (n-3)-dimensional manifold where three or more ridges meet. For example, the cube's corners have three edges meeting at them, so these corners are the cube's peaks. A tesseract has three cubes that meet at an edge, so the edges of a tesseract are its peaks.

The bulk of an object is the amount of n-dimensional space it occupies. Bulk is measured in terms of unitⁿ. For example, the bulk of a cube is a 3D quantity measured in, say, meters cubed. The bulk of a tesseract is a 4D quantity measured in, say, meters to the 4th power. The two are incompatible -- you can't compare meters cubed to meters to the 4th power, just as you can't compare meters squared to meters cubed. It's comparing apples and oranges.

Absolute terminology:

A vertex is a 0-dimensional point that where other elements of an object meet. For example, a cube's corners are its vertices, and a tesseract's corners are its vertices. The cube's vertices happen to be the same as its peaks; but a tesseract's peaks are not the same as its vertices. A tesseract's peaks are its edges, which are 1 dimension higher than its vertices.

An edge is a 1-dimensional line segment where other higher-dimensional elements of an object meet. A cube has 12 edges, for example, and a tesseract has 32 edges. Whereas a cube's edges are its ridges, a tesseract's edges are not ridges, but peaks.

A face is a 2-dimensional polygon where other higher-dimensional elements of an object meet. (This one is a bit debatable, since "face" could be construed to be a better used for what I use "facet" for. But then most of existing terminology is debatable, so we just have to agree on which term to use and go with that.) A cube has 6 faces, which are the same as its facets. But a tesseract's faces are not its facets; they are merely its ridges.

A cell is a 3-dimensional polyhedron where other higher-dimensional elements of an object meet. A cube's cell is the cube itself, but a tesseract's surface has 8 cells, each of which is a cube.

Length is the extent of a 1-dimensional object. For example, an edge's length is the Euclidean distance between its two vertices. Length is measured in units.

Area is the extent of a 2-dimensional object. It is measured in units².

Volume is the extent of a 3-dimensional object. It is measured in units³. Even though we like to think of volume as the unit that fills space, I prefer to let it remain 3-dimensional, and use "bulk" to refer to the unit that fills n-dimensional space -- just so we minimize the confusion that arises when people see the word "volume" and think "units cubed".

If we want to be more precise, we may prefix the dimension number to designate a specific dimension. For example, 1-bulk = length; 2-bulk = area, 3-bulk = volume.

Same goes for the concept of face.
Entering the word 'face' at http://dictionary.reference.com has the relevant defintion:
"The main side of an object, building, etc, or the front"
For uniform objects, like a cube, we refer to all the outward squares as faces because they are all the same and any of them could be looked at from front on.

So again for a 4Der the 'face' of a tesseract is any of its cube 'faces' and for a 2Der the 'face' of a square is any of its line 'faces'.

Hmm. I may have to retract my definition for "face" and use it to mean "facet". I think this is what Wendy did in her polygloss too -- face means that part of the object that's facing you, so there's no reason to fix it to 2D. So that's two votes against my definition of "face".

What do you think? Should we adopt Wendy's definition of "face" instead? We will have to find another term for 2D element, though, since we tend to refer to those a lot.

I think it is useful to parallel our thoughts to those of our 2Der and 4Der counterparts so that we can think more like them.
This may, or may not, assist us towards discovering the design of real world artifacts for our 4Der counterparts.
Hopefully it does and I feel it gives me a better sense of their lives. [...]

Absolutely. I have found, in the course of my learning to visualize 4D, that it's essential to think in terms of 3-bulk as surfaces and 2-bulk as ridges. As 3D beings, our natural tendency is to interpret 3-bulk as space-filling, and 2-bulk (area) as space-dividing surfaces, but that only leads to confusion when dealing with 4D objects. Once we wrap our mind around the idea of 3-bulk being only a surface, and 2-bulk being only a ridge, then we begin to have a true glimpse into the nature of the 4D world.

Then one begins to discover interesting things, such as the fact that 4Ders walk on floors that fill 3-bulk, and so have a much greater degree of freedom than we do. In your terms, gonegahgah, they have an additional 360° of sideways where they can move about. One also begins to note that a 4D room doesn't just have 4 walls and 4 corners; it has six walls and two different kinds of corners: twelve edge-corners where two walls meet, and eight vertex-corners where three walls meet. (That's a lot of corners!) It took me a long time to wrap my head around how a room could possibly have three walls that meet at a corner. I used to play John McIntosh's 4D maze game, and in free-movement mode, where you have full 4D freedom to move about, I often got "stuck in a corner" and didn't know how to get out, because I didn't understand what I was seeing -- three cubes meeting at an edge -- and more specifically, what that meant in terms of my relative position/orientation relative to the room.

Their doors are not mere 2D panels of wood, but 3D "blocks" (which aren't actually blocks in 4D -- we just tend to think of them that way because of our 3D-centric understanding). Door hinges don't just lie along a vertical axis, and doors don't turn around an axis; they are hinged on a 2D ridge and turn around this ridge. Because of this, a door frame in 4D doesn't just have two sides with one side hinged to the door; 4D door frames have 4 sides, and the door can be hinged on any one of these 4 sides. Unlike in 3D, where the two doorposts are not connected to each other, the 4D door frame's 4 sides are rectangles that touch each other in a cuboidal arrangement. The locking mechanism would tend to be on the opposite ridge from where the door is hinged, since that's where the lock has maximum leverage to hold the door shut against its pivoting ridge. But you can also have auxiliary locking mechanisms on the other two ridges, which may serve as additional support to keep the locked door closed. We don't have direct equivalents of this in 3D.

Lots of other fun stuff can be discovered, once we get used to "thinking in terms of volumes", as I like to put it.

[gonegahgah]

The reason I'm exploring rotation, primarily I hope, is in an attempt to understand 4D 'surfaces' to a greater degree.
Hopefully this need to understand 4D 'surfaces' will tie back to the archery example that I wish to explore for the purposes of understanding this poly-sidedness.

I was a little at a loss last night where I was originally going with all this. Now I remember...
Primarily this current very detour-ous journey is about the number of eyes required for each dimensional being. So it should eventually wend its way back to that study.

My main question I am exploring is whether a 4Der would just need the third eye for peripheral vision or whether they would actually rely on the third eye for location.
Even with one eye we can tell whether something is to our left or right but we don't align up with a target as well with just the one eye as someone with two eyes.
Likewise, I wonder if a 4Der with three eyes will line up a target better than a 4Der with just two eyes?
Also, with our two eyes we have stereoscopic vision. With three eyes would a 4Der have tri-scopic vision more suited to their environment?

These are questions I'm looking to explore to my own fuller understanding and strangely this other stuff, I'm detouring into now, applies to that fuller understanding.
So it may seem strange that surfaces may somehow apply to the number of eyes but I believe it is relevant for a fuller grasp.
Surfaces also apply to the topic of wheels as well of course but it is mainly the concept of 4D vision that I am looking to tackle via this unusual route.
So I won't focus on wheels just yet but will get back to that in due course.

We may have trouble understanding how such surfaces work, but it helps to think of how a 2Der would have trouble understanding how surfaces in our 3D world work. To them, a 2D area occupies space, so they find it hard to grasp the concept of us 3D beings putting a cup on a table, say. How could you possibly put an object inside an area? It's already hard enough to understand how said object doesn't need to pass through the edges of the area, much less how it could possibly "rest on" that area.

Exactly. This is what I am exploring, with the diagrams I am working on, to give us a better understanding from our confused view to the 4Der's truer view of 4D surfaces.

In 3D, of course, this is intuitively obvious. But we in turn have trouble understanding how a 3D volume can possibly act as a surface. How can one, for example, rest a tesseract on top of a large cubical surface? Where would the tesseract go in the volume of that surface? By dimensional analogy, we can understand that a cubical volume from the 4D perspective is a two-sided surface with a cube-shaped boundary. So the tesseract can sit upon one side of it, just as a cube in our world can sit upon a large square surface. And just as we tend to rest the cube roughly in the middle of the surface, such that the bottom face of the cube lies within the square boundary of the surface, so also a 4Der would tend to place a tesseract upon a cubical desk in such a way that its bottom cube rests within the cubical confines of the desk's surface.

Again very true however it is this confusing aspect of 'within' actually being 'upon' that I hope to show more distinctly with some diagrams.

What do you think? Should we adopt Wendy's definition of "face" instead? We will have to find another term for 2D element, though, since we tend to refer to those a lot.

Hmmm, I'm not sure. Facet certainly doesn't instantly carry a conception for most people and face does have a more accessible conception perhaps (as do bulk, ridge, etc.).
So, maybe 'face' is better. Maybe 'shape', but even that tends to have broader contexts. Maybe 'planes'. Are either of those on the right track?

Then one begins to discover interesting things, such as the fact that 4Ders walk on floors that fill 3-bulk, and so have a much greater degree of freedom than we do. In your terms, gonegahgah, they have an additional 360° of sideways where they can move about. One also begins to note that a 4D room doesn't just have 4 walls and 4 corners; it has six walls and two different kinds of corners: twelve edge-corners where two walls meet, and eight vertex-corners where three walls meet. (That's a lot of corners!) It took me a long time to wrap my head around how a room could possibly have three walls that meet at a corner. I used to play John McIntosh's 4D maze game, and in free-movement mode, where you have full 4D freedom to move about, I often got "stuck in a corner" and didn't know how to get out, because I didn't understand what I was seeing -- three cubes meeting at an edge -- and more specifically, what that meant in terms of my relative position/orientation relative to the room.

Their doors are not mere 2D panels of wood, but 3D "blocks" (which aren't actually blocks in 4D -- we just tend to think of them that way because of our 3D-centric understanding). Door hinges don't just lie along a vertical axis, and doors don't turn around an axis; they are hinged on a 2D ridge and turn around this ridge. Because of this, a door frame in 4D doesn't just have two sides with one side hinged to the door; 4D door frames have 4 sides, and the door can be hinged on any one of these 4 sides. Unlike in 3D, where the two doorposts are not connected to each other, the 4D door frame's 4 sides are rectangles that touch each other in a cuboidal arrangement. The locking mechanism would tend to be on the opposite ridge from where the door is hinged, since that's where the lock has maximum leverage to hold the door shut against its pivoting ridge. But you can also have auxiliary locking mechanisms on the other two ridges, which may serve as additional support to keep the locked door closed. We don't have direct equivalents of this in 3D.

Lots of other fun stuff can be discovered, once we get used to "thinking in terms of volumes", as I like to put it.

It gives us a greater understanding into the 4Der's life with insights like these that you're giving, thanks QuickFur. I'm not sure that my head is agreeing with '4D door frames have 4 sides, and the door can be hinged on any one of these 4 sides'. We should look at that further please to find if you are right?

As I mentioned at the beginning of the post I am looking at surfaces presently towards understanding vision. I'm just working on the diagrams, as I am able to, so I'll get those up soon hopefully.

[quickfur]

The reason I'm exploring rotation, primarily I hope, is in an attempt to understand 4D 'surfaces' to a greater degree.
Hopefully this need to understand 4D 'surfaces' will tie back to the archery example that I wish to explore for the purposes of understanding this poly-sidedness.

I was a little at a loss last night where I was originally going with all this. Now I remember...
Primarily this current very detour-ous journey is about the number of eyes required for each dimensional being. So it should eventually wend its way back to that study.

My main question I am exploring is whether a 4Der would just need the third eye for peripheral vision or whether they would actually rely on the third eye for location.
Even with one eye we can tell whether something is to our left or right but we don't align up with a target as well with just the one eye as someone with two eyes.
Likewise, I wonder if a 4Der with three eyes will line up a target better than a 4Der with just two eyes?
Also, with our two eyes we have stereoscopic vision. With three eyes would a 4Der have tri-scopic vision more suited to their environment?

I'm interested in that too, though at the moment I don't really have a strong inclination towards a specific number of eyes. Clearly, at least two eyes are needed in order to have stereoscopic vision, but beyond that, I'm not so sure. Having more eyes increases peripheral vision, but that alone doesn't really tell us how many more eyes will be good. We may choose 3 as the minimum to span a 2D plane, but that is arbitrary. We might as well settle with, say, 17 eyes. One may argue that's too many, but then we don't have a good reason why the number should be small, either.

Anyway, my point is that it would be good to find more definite criteria that would help narrow down the ideal number of eyes to a small set of possibilities.

[...]

What do you think? Should we adopt Wendy's definition of "face" instead? We will have to find another term for 2D element, though, since we tend to refer to those a lot.

Hmmm, I'm not sure. Facet certainly doesn't instantly carry a conception for most people and face does have a more accessible conception perhaps (as do bulk, ridge, etc.).
So, maybe 'face' is better. Maybe 'shape', but even that tends to have broader contexts. Maybe 'planes'. Are either of those on the right track?

I chose "facet" by analogy with the facets of a cut diamond: a polygonal cut that lies on the surface of the gem; polytopes are sorta like higher dimensional "gems" with many facets.

But based on the recent discussions, and on re-reading Wendy's little monograph on walls, I'm starting to lean more towards using "face" as the (n-1)-dimensional element instead of the 2D element. Wendy's statement that a face is something that faces you, is a very compelling argument for this. Speaking of a tesseract's faces while referring to what are only its ridges, only fosters confusion on the part of the audience, since these 2D elements do not face anything, and amount to a mere margin where the "real faces", i.e., the 8 cubes, meet.

I don't like "shape" because ... well, it's too broad. A tesseract is a shape. Saying that the shape (tesseract) consists of 8 shapes (cells) doesn't help clarify anything, IMO. And "plane" already has an existing meaning: an n-dimensional flat surface with indefinite extent. Reusing the same word to mean an n-dimensional element of a polytope only overloads an already badly overloaded word. Currently it is unclear whether "plane" means 2D plane, or (n-1)-dimensional (hyper)plane. I believe Wendy uses "plane" in the (n-1)-D sense, but I often use it in the 2D sense. This can be very confusing to someone not in the know.

Sometimes I wonder if we really should just grit our teeth and adopt Polygloss terminology instead of confusing ourselves needlessly with the current conventional bad terminology riddled with inconsistencies. It's sorta like trying to appreciate poetry written in another language. You really want to read it in its original language to get a full appreciation for it, instead of a mangled badly-done translation into Pig Latin. Polygloss terminology may sound foreign to the unaccustomed, but that little learning curve seems a small price to pay for being able to accurately and effectively describe things in higher dimensions without the tarpit-riddled minefield that is the current conventional terminology.

[...]
Their doors are not mere 2D panels of wood, but 3D "blocks" (which aren't actually blocks in 4D -- we just tend to think of them that way because of our 3D-centric understanding). Door hinges don't just lie along a vertical axis, and doors don't turn around an axis; they are hinged on a 2D ridge and turn around this ridge. Because of this, a door frame in 4D doesn't just have two sides with one side hinged to the door; 4D door frames have 4 sides, and the door can be hinged on any one of these 4 sides. Unlike in 3D, where the two doorposts are not connected to each other, the 4D door frame's 4 sides are rectangles that touch each other in a cuboidal arrangement. The locking mechanism would tend to be on the opposite ridge from where the door is hinged, since that's where the lock has maximum leverage to hold the door shut against its pivoting ridge. But you can also have auxiliary locking mechanisms on the other two ridges, which may serve as additional support to keep the locked door closed. We don't have direct equivalents of this in 3D.

Lots of other fun stuff can be discovered, once we get used to "thinking in terms of volumes", as I like to put it.

It gives us a greater understanding into the 4Der's life with insights like these that you're giving, thanks QuickFur. I'm not sure that my head is agreeing with '4D door frames have 4 sides, and the door can be hinged on any one of these 4 sides'. We should look at that further please to find if you are right?

I think the way I said it may have been ambiguous. I meant that we can choose any of the 4 sides of the door frame to put the hinges, but obviously the door can't just randomly switch its hinges to another side (not without disassembly and reassembly, anyway).

As for door frames having 4 sides, it helps to think of how doors and doorways work in 3D. A doorway is basically a gap in a wall to allow passage to the other side. For simplicity, we may assume the wall takes the shape of a square:



For the purposes of this illustration, we'll just assume this square wall is all by itself. In a realistic situation, of course, it would be joined to other walls, but let's keep things simple.

Before we introduce the doorway, let's take a moment to consider how a 2Der would view this wall. The 2Der's viewpoint is from the side, say from the left side of the wall. It sees the left edge of the wall (let's ignore the gray flooring for now). From its point of view, any doorway through this solid object must be cut through from the left edge ot the right, so that it can pass through this obstacle. But is this how 3D doorways work?

Not at all:



This totally makes sense from our 3D point of view: the doorway is usually cut from the middle of the wall, making a roughly rectangular gap in it. This allows us passage to the other side of the wall. But from the 2Der's point of view, this doorway makes no sense at all. How on earth is it even a doorway, since (from the 2D perspective) it's embedded inside the wall?! How does one pass through it anyway? To us, it's obvious, but to the 2Der, this seems like a really strange thing to be called a "doorway". Where's the "way"?! It seems like just a hollow inside the wall.

The 2Der isn't any more impressed when we now attach a door:



It looks like we have just filled in part of the hollow we made with a trapezoidal object. How is such a thing even remotely like a door?! A 2Der would find this absolutely astounding. Keeping this in mind, now let's turn to the 4D case.



I'm doing a projection from 4D to 3D here, and because our disadvantaged 3D viewpoint requires transparency in order to see the what's inside the cube-shaped wall, I decided to omit the brick texture (it would utterly clutter the image and make it impossible to see anything). For reference, the grey block represents the projection of the floor. I clipped the whole thing to be within a cubical volume, just so we can draw a useful analogy with the square images of the 3D doorway above, that's why the floor is finite (in the 3D model, it's an infinite plane, but an infinite hyperplane is kinda hard to represent in a 2D image! So imagine that the sides and bottom of the grey block extends outwards and downwards indefinitely. The top face is actually "at infinity" because it's on the 4D horizon, so it actually already represents the part of the hyperplane that extends forward (from the 4D POV) indefinitely).

Now, from our 3D-centric point of view, a doorway through this cubical wall would have to cut through two opposite faces, since otherwise how could one pass through it? From a 4Der's point of view, though, that would be silly, since all that's needed is to cut a cuboidal hole from the center of the wall, something like this:



OK, this image is rather unclear, because the little cuboid in the middle could actually represents a hole in the cube, not a separate solid object. To make it clear what's happening here, here's what the cube looks like without transparency and viewed from the bottom:



I hope it's clear that this is a cuboidal hole cut from the middle of the cube.

In any case, we would be quite baffled by how cutting such a hole in the cube could possibly equal making a doorway through the cubical wall. So we have to understand that, just as in 3D our doorways are cut not from the edge of the wall but somewhere along its length, so in 4D the doorways are cut not from the ridges of the wall, but from somewhere in its volume. A 3D doorway is of no help to a 2Der attempting to cross the wall, as it were, from the side; one needs to move in the 3rd direction for the doorway to provide passage through the wall. Similarly, a 4D doorway is of no use to us, and indeed, makes no sense to us if we look at it from a 3D-centric point of view. One needs to be walking along the 4th axis in order to cross to the other side of the cubical wall.

Now, looking at the 3D case again:



Notice that the doorway has two vertical edges, one on the left, and one on the right. So we may hinge the door either on the left or on the right. In this case, we hinged it to the right.

Now in 4D, the hinge must cover a 2D area, since rotations in 4D happen around a plane, not an axis. So any of the 4 lateral faces of the cuboidal hole would do. Here's one of the possibilities:



I chose to color the door green 'cos it's more visible with a high-contrast color. I also increased the transparency of the other objects in the scene so that the door can be seen more clearly. Here, it appears as a frustum shape, but it's just a cuboid seen from an angle -- I left the door slightly ajar so that it's clear that it's not just the cuboidal hole. The hinge is on the right face of the cuboidal hole.

When the door is fully shut, then the door appears as a cuboid of exactly the same proportions as the hole in the cube, so it blocks the 4Der from passing over to the other side of the wall. As seen above, it's slightly ajar, so the 4Der could see around its edges and get a glimpse of what's on the other side. A fully opened door would not be visible, because it would have swung over to the far side of the wall in the 4th direction and would be obscured by the part of the wall around the doorway. Then the 4Der can walk through the doorway along the 4th direction. If you can imagine it, the 4Der's coordinates would be initially (0,0,0,5), where (0,0,0,0) corresponds with the center of the image where the green door is. The red cubical wall extends around this doorway, with the outer corners at (±1,±1,±1,0). The 4Der walking through the doorway would thus pass through (0,0,0,4), then (0,0,0,3), then (0,0,0,2), then (0,0,0,1), at which point she is standing right in front of the doorway. The wall is still 1 unit away. Then she keeps walking and passes (0,0,0,0) -- where the doorway represents a convenient hole in the cube so that her movement is not blocked by the wall, as it would be if she were to walk, say, from (1/2,0,0,1) to (1/2,0,0,0). Thus, she passes to the other side at (0,0,0,-1).

If the door were shut, then the 4Der would encounter an obstacle when trying to move from (0,0,0,1) to (0,0,0,0), because the door would be at (0,0,0,0).

Whew! These images took quite a while to make. I hope you enjoyed this lesson on the doorways of the dimensions.

[gonegahgah]

Thanks for the pictures QuickFur. They look great. They do take awhile to make; I know.

Even with the diagrams I've had to unthink 3D to grasp them. I've also thought of some diagrams to draw up to further explain it too hopefully.
But, first I'll try to complete these surface explanation diagrams..

[quickfur]

Finding ways of generating 4D shapes is always fun. The recent discussion about rotations make me want to consider some other simple cases as well:
[...]
3) Rotating an octahedron about the plane that bisects it (where 4 vertices lie in ): produces a square-circle tegum (the dual of a cubinder).
[...]

Just thought of another interesting shape one can generate from an octahedron: rotate the octahedron around the plane parallel to its axis and that bisects a pair of edges (basically a bisection of the octahedron but instead of cutting along the edges, cut through the middle of four faces). This produces a shape whose surface consists of 4 cones and two triangular tori joined along a cylindrical ridge. If I'm not mistaken, this should be a cylinder bipyramid.

There seems to be a pattern to these shapes... it seems that the relationship of the cylinder bipyramid to the octahedron is that the cylinder is generated by rotating a square in 3D, and since the octahedron is a square bipyramid, rotating it is equivalent to making a cylinder bipyramid. IOW, the operations seem to be associative: start with a square, and you can either make a bipyramid first then rotate it, or you can rotate it then make a bipyramid, and you get the same shape. Perhaps one can invent a "shape algebra" based on this property.

[gonegahgah]

Here we go. The first set of pictures that I have just finished.



This first pair of images is a look at how a 2Der would think of our feet compared to theirs. The darker red dots are where the foot connects to the leg.
In the left diagram they understand that their feet can only face forwards and the race of people - their opposites - feet can only face opposite to theirs.
When they try to think of our feet they can only really think of it in relation to a square instead of their usual line surface.
In the right diagram they visualise that our feet are inside the square.
If they think about it they will realise that not only can our feet face forward and backwards but they can face in any of the 3rd direction available to us.
They would imagine our feet inside their square with feet facing forward, facing back, and facing in directions towards upwards and downwards to match how they think of a square. There is not too much problem with this; just yet.

Here's another set of pictures. I've remodeled our 2Der so that they are a little bit more like what we might see.
We can now see their bloody insides and they now have a yellow outfit that they must apparently stick on their front and backs somehow.



The first image, on the left, again shows the world as the 2Der knows it.
When we get to the second image, in the middle, we get a picture of how our 2Der might imagine how we walk inside their square.
There are some fundamental misconceptions in that view because upwards - which should be the same no matter how we are turned - is instead depicted to match the feet.
But, the 2Der has no 3-space to imagine us in so they find it difficult to not think that we are somehow inside their square; or at least partially inside it with our feet touching the inside somehow.

Of course the picture on the right is more how we understand things but I still have it sideways to reflect the 2Der's concept of a rectangle - being used to represent a road.
They need to accept that our feet can point any direction in the square's area while we are still standing upright.
They don't have enough dimensions to deal with sideways and upways all at once and this leads to confusion about how we can be inside their squares but on the surface.

That took awhile to create partly because I recreated my 2Ders. I'll be recreating the 3Ders as well so the next set of figures might take a little while too...

[gonegahgah]

While I'm looking at the 2Der's view of our surfaces I should also look at the 2Der's view of our feet.
This is bypassing some of the other diagrams but it has helped me to understand the 4D surfaces better now.



Here we can see slices of a 3Der's foot. The 2Der tends to think of the footprint as resting on the ground and the ground to them is a line.
So naturally enough they will tend to think of our feet as lines lying on the ground but there must be many lines. They don't really have a concept for side-by-side.
They only really think of in-front-of or behind, or above and below.
In these pictures we can see that there are different ways to construct our 3D foot from line slices.



If the 2Der thinks of our ground as an 3-area (or 2-volume) then they will picture our foot tread inside of a 3-area.
They could picture our foot as made of many lines with each line being inside the 3-area.
Their mind says that the footprint is inside the 3-area but theoretically they know it must lie on the 3-area; not in it.
To help themselves understand they rotate the footprint in their 3-area keeping the lines all to the same orientation.
The lines change for them as the footprint rotates into our sideways and no matter which orientation the foot is the lines always make full contact with the 3-ground.

This is their first clue that the footprint itself is continuous and not made of lines.
Overall the separate lines define the same shape and all those footprint lines are supposed to be in contact with the ground at the same time.



The 2Der may tend to construct the footprint out of horizontal lines or out of vertical lines so they can stack them above each other to see them all at once. They may like to use horizontal lines to give themselves more of a feel of the foot lying on the ground. This then presents a dilemma for them.



This is because each line of the foot is thought of as being above the lower line; and yet there has to be ground under each line as shown in the picture on the right.
Otherwise they would think that only the bottom line actually makes contact with the ground and they would wonder at the funny shape of our foot.
How can this strange shape meet the ground with a gaping arch in it?

We naturally understand how our foot works but it is a struggle for the 2Der to understand how our foot doesn't appear to be uniformly touching the ground at all points.
They imagine that we would collapse at the arch area as only tiny parts of a line are holding us up for those sections.

It is the continuity and the uniformity of contact that I have finally come to understand.
This has allowed me to finally understand that a 4D wheel can have any 3-volumetric shaped tread print all around its rim just as long as there is enough volume present to support the weight of the vehicle.

My brain was railing against that idea but now I understand. These were some of the pictures that I hadn't got to yet: that is how a 4D shape can be sliced up by parallel planes at any spherical angle. And just as all those lines in the foot make contact with the ground; so do all the plane slices of a 4D wheel tread, forming in total a 3-volume when connected, also all make contact with the ground no matter what spherical rotation the planes are drawn at. Each plane makes contact with the 4-ground just as each line in the footprint makes contact with the 3-ground. I'll still attempt to draw those pictures sometime.

Probably you would tend have shapes that form uniform treads around the wheel rather then something wobbly. QuickFur has depicted various examples in his intro on 4D wheels.
So it's good that I appreciate that now, finally, so hopefully we can get back to the archery somewhen.

[quickfur]

Alright, bringing this discussion back to this topic, where it belongs (how did we manage to get from 4D road markings to 3Ders rotating in 4D?!).

I suppose they should be the same. Though I do still prefer the left approach, even if it suffers from crowding problems; it seems a bit counterintuitive that objects laid out in the lateral dimensions would be represented by a vertical rotation.

Just like the inverted glasses you described I feel it would become intuitive quite quickly. It's like the equivalent representation for the 2Der's rotated slices:

But the 2Der doesn't have the option to represent the the left option; they can only represent the right option; so they are stuck with that.
Or, alternatively, they could place everything, that is off in our 3rd dimensional, behind and in front of what they have in their 2D plane; which would be very crowded.
The same goes for us which is why the sky and underground hopefully afford the more intuitive space to place objects off in the 4th direction.
We do, unlike the 2Der have the option to use either representation but utilising the emptier up-down direction eliminates some of the crowding as for the 2Der's example.
The intro, and other examples, here use that same trick at times of using vertical to represent the ana-kata directions.
It's always a juggle to understand statically but I feel that immersion would make it easier and soon very intuitive.

I suppose with the vertical layout the 4th direction will feel sorta like "another up/down", since objects that are extended into 4D will have their extended parts show up above/below the current 3D view as "shadows", which can be "rotated into reality" with either type of rotation.

As I've mentioned previously, a 2Der doesn't really get a sense of left or right as we do; only that they are opposite. You could transpose them and they wouldn't know.
The same goes for us with ana-kata. Ana could be rotated skywards or groundwards; it really makes no difference. Maybe it would even be useful to be able to switch?

Left and right retain their distinction even though the 2Der has no such concept of left or right. A left-handed mirror image is distinct from a right-handed mirror image. A right shoe is distinct from a left shoe. For a 2Der, this would be hard to grasp, but it's a hard reality of 3D.

Similarly, ana/kata retain their distinction even though we have a hard time imagining it. There can be chiral objects in 4D which are not equivalent to their mirror image; such things also exhibit a left/right (or rather, ana/kata) variant. So it's important to maintain this distinction.

However, you're right that the names ana/kata are arbitrarily assigned; it's really just a single new axis that stretches off in two opposite directions; which name you give to which direction is arbitrary. As long as they are named differently you're OK. Practically speaking, one may invent a 4D analogue of a "right hand rule" that tells you, given the first 3 axes, which way ana and kata should point. As long as everyone is consistent in assigning directions, everything will work correctly.

Now, one reason I prefer the horizontal layout for ana/kata rather than skywards/groundwards, for the very reason that, being two opposite lateral directions, ana/kata are equivalent via a 180° rotation. Whereas if they are represented vertically, then one of them seems to be "preferred" because of the direction of gravity, contrary to the fact that they are actually perpendicular to gravity and behave like lateral directions.

This isn't a bulletproof argument, however. One argument for doing skywards/groundwards, besides the reason you cited (less crowding), is the player's visualization of the 3D floor. It's easier to imagine the additional axis as "another vertical direction", and therefore the new parts of the floor that come into view when you rotate ana/kata as a "pseudo-vertical extension" of the floor, so that the apparently-square piece of floor becomes conceptually a cubical piece of floor. It's much harder to imagine the 3D floor as a lateral extrusion of what we already perceive as a horizontal floor -- our brain's 3D centric mode of thought has no more room to fit in another lateral dimension where the floor might be extruded. So the sky/ground layout makes it slightly easier to come to terms with the 3D-ness of the floor, and even to have a somewhat accurate mental understanding of it as being cubical in shape.

Actually I was wrong. Either one of the rotations will give you full access to the entire 3D floor plan. It's just a matter of how it changes your 4D orientation. One changes what you're looking at, and the other only changes your orientation.

Sorry, I had thought you had already made that correction. Either view gives you full access to left and right still. This allows you, in the positioning view selection, to wander around an object, that intersects your 3D plane, to try to find the opening. Also, once you have an object in you view, you can still use left and right and the rotation to adjust how far you are ana/kata-wards. The location view just allows you to more easily bring objects off in the 4th dimension into your 3D plane in the first place.

So I was wrong... only one of the rotations is enough to give full access to the 3D floor. But it might be nice to still have both options.

The usefulness of having both options was a surprising but pleasant discovery for me when I realised it. It sort of made some kind of sense, only after this emerged, that this would be the case for us 3Ders when navigating a 4D world. I wouldn't have realised it until I drew up the alternate view selections. It was one of those 'aha' moments for me.

I like what one prominent indie game developer, Jonathon Blow the creator of Braid, said and that was that a good way to design a game is to design the laws of the game's universe and then discover what these laws mean. He said it much better than that. I'll try to find and quote Jonathon more correctly. Here we have a ready made universe, by extrapolation, and what we are doing is discovering what it's laws mean in terms of our interaction with it and a 4Ders interaction with it.

Quite so. In fact, most of mathematics may be considered as exploring the consequences of a postulated set of rules (i.e. what theorems result from a certain set of axioms).

I don't know about ready-made universe, though. When I first started out with 4D visualization, I thought that one could simply add an extra dimension to the laws of physics, and everything will work as before, at least in an analogical way. I think a lot of us on this forum thought so too. But eventually we discovered that certain things didn't carry through: planets can't have stable orbits, the 4D Schroedinger equation for the hydrogen atom has no minima so atoms as we know them can't exist, etc.. In order to get a 4D world that's intuitively "closer" to how we imagine a world should work, then, requires postulating new, foreign laws in the 4D universe. To have a workable hydrogen atom, for example, one method is to modify the laws of electromagnetism... which has other side-effects, like violating causality at the microscopic scale. Wendy suggested a counteracting force to gravity so that planetary orbits can be stable, but this morning I suddenly realized that it has other consequences too.

In other words, it's not just a matter of exploring the consequences of having a 4th spatial dimension, but also a matter of inventing our own rules (or discovering what these rules must be) so that it will be possible to have an "intuitive" 4D world, and studying the consequences of these rules. Different people may use a different set of rules (since 4D itself doesn't dictate one over the other necessarily), with different, possibly incompatible, results, and it can become hard to decide which set of rules to adopt.

[gonegahgah]

Alright, bringing this discussion back to this topic, where it belongs (how did we manage to get from 4D road markings to 3Ders rotating in 4D?!).

I was beginning to wonder that myself. Thank you for bringing it back over here.

I suppose with the vertical layout the 4th direction will feel sorta like "another up/down", since objects that are extended into 4D will have their extended parts show up above/below the current 3D view as "shadows", which can be "rotated into reality" with either type of rotation.

Kind of sort of. But with the same representation as if they were sideways. If we are close to a corner and scan across the wall the far part of the wall will be further away and smaller. The same would occur where a 2Der scans across our 3D wall where the far part of the wall would drop out of the sky and be smaller. So if a building is off in 4D then it will look how it would look if we could turn to look at it, but be in the sky. So although the rotation comes out of the sky it is still rotating in from one of the sides.
To represent this we just draw a sideways slice and bring it back to our plane and rotate it up the same number of degrees (but leave out some things and make it colour shadowed).

Left and right retain their distinction even though the 2Der has no such concept of left or right. A left-handed mirror image is distinct from a right-handed mirror image. A right shoe is distinct from a left shoe. For a 2Der, this would be hard to grasp, but it's a hard reality of 3D.
Similarly, ana/kata retain their distinction even though we have a hard time imagining it. There can be chiral objects in 4D which are not equivalent to their mirror image; such things also exhibit a left/right (or rather, ana/kata) variant. So it's important to maintain this distinction.

Absolutely, left is different from something that is right. I think our body mostly processes left handed molecules with a few right handed molecules thrown in. I think I recall that sugar is an example of this. But, it's just like those special glasses you spoke of. If it were possible to cross the existing left-right wiring of our brain (maybe by turning us 180° in the 4th dimension) then we would probably be none the wiser except when we noticed that everything is back-to-front to where we remember it to be; unless we could reverse those memories as well. There certainly is a real direction we call left and a real direction we call right but understanding which is left and right is a product of our brain.

However, you're right that the names ana/kata are arbitrarily assigned; it's really just a single new axis that stretches off in two opposite directions; which name you give to which direction is arbitrary. As long as they are named differently you're OK. Practically speaking, one may invent a 4D analogue of a "right hand rule" that tells you, given the first 3 axes, which way ana and kata should point. As long as everyone is consistent in assigning directions, everything will work correctly.

But, it's not just the names that can be arbitrary. Look at the fellows on here who had a condition called VSI - that seems to be linked to some people with aspergers - which meant that their brains would switch left and right and suddenly think everything was opposite to where it should be. For them that is reality. Right may at one moment agree with us and at other moments disagree. I really don't think that the universe flips around for them and I personally don't believe that this is a 4Der trait but I do believe it shows the power of our brains.

Now, one reason I prefer the horizontal layout for ana/kata rather than skywards/groundwards, for the very reason that, being two opposite lateral directions, ana/kata are equivalent via a 180° rotation. Whereas if they are represented vertically, then one of them seems to be "preferred" because of the direction of gravity, contrary to the fact that they are actually perpendicular to gravity and behave like lateral directions.

Hopefully we can test out these various approaches one day if we, or someone, can get some sort of software up and running to demonstrate these. I'm hoping so.

This isn't a bulletproof argument, however. One argument for doing skywards/groundwards, besides the reason you cited (less crowding), is the player's visualization of the 3D floor. It's easier to imagine the additional axis as "another vertical direction", and therefore the new parts of the floor that come into view when you rotate ana/kata as a "pseudo-vertical extension" of the floor, so that the apparently-square piece of floor becomes conceptually a cubical piece of floor. It's much harder to imagine the 3D floor as a lateral extrusion of what we already perceive as a horizontal floor -- our brain's 3D centric mode of thought has no more room to fit in another lateral dimension where the floor might be extruded. So the sky/ground layout makes it slightly easier to come to terms with the 3D-ness of the floor, and even to have a somewhat accurate mental understanding of it as being cubical in shape.

I'm actually hoping that the player will develop a whole sense of the additional sideways freedom with this approach. I have this feeling that it will become evident how 4D actually works if someone plays a game using this approach for long enough; as long as the definition of 4D is correct. This approach has helped me tremendously to understand 4D better.

I don't know about ready-made universe, though. When I first started out with 4D visualization, I thought that one could simply add an extra dimension to the laws of physics, and everything will work as before, at least in an analogical way. I think a lot of us on this forum thought so too. But eventually we discovered that certain things didn't carry through: planets can't have stable orbits, the 4D Schroedinger equation for the hydrogen atom has no minima so atoms as we know them can't exist, etc.. In order to get a 4D world that's intuitively "closer" to how we imagine a world should work, then, requires postulating new, foreign laws in the 4D universe. To have a workable hydrogen atom, for example, one method is to modify the laws of electromagnetism... which has other side-effects, like violating causality at the microscopic scale. Wendy suggested a counteracting force to gravity so that planetary orbits can be stable, but this morning I suddenly realized that it has other consequences too.

In other words, it's not just a matter of exploring the consequences of having a 4th spatial dimension, but also a matter of inventing our own rules (or discovering what these rules must be) so that it will be possible to have an "intuitive" 4D world, and studying the consequences of these rules. Different people may use a different set of rules (since 4D itself doesn't dictate one over the other necessarily), with different, possibly incompatible, results, and it can become hard to decide which set of rules to adopt.

It is true that different definitions may create different likely universes. After all, the scientists are still trying to work out the rules of our universe!
I wonder if we invent any tools in the 4D world if we can get a patent for them? Lol.

[wendy]

The basic theory in the PG is that words like 'face' etc, already have trans-dimensional uses, one of which is geometric. The common use is the trans-dimensional ones, so that's what's retained. The words are horribly loaded in normal speak, and if you start using face as 2d element (hedron), then you start getting ideas like 4space's notion that a 'face' divides space. That's why i removed the fixed dimensions from these.

The simplest notion is to use 3-space and 4-space, but the apparentness of 3-space dividing 4-space is not as strong as a plane divididing space (in every dimension), is because like space, a plane moves with the dimension of all-space.

This means to produce fixed dimensional words, we need to invent a series of names, with distinctions like the chemists do with nitric acid vs nitrous acid. It helps in people who look at /telescope/ and say /tele/=far + /scope/ = i see: i see far. It's close enough to the real meaning. This is what i tried in the PG.

The basic roots come from 'polyhedron' = "many 2d patches" is read as /poly = many/ + /hedr = 2d/ + /-on = patches/.

The thing is completely regular. chor- is 3d (polychoron), etc.

One can then describe a great variety of things by using the -id, -on, -ix, and -ous extensions.

A hedrid is a 2d solid. A hedrous thing is a 2d nebelous or cloudy thing in the shape of a 2d thing. A hedrix is a closed surface, like a plane or a sphere. A hedron is a patch on a plane, or a bounded hedrid that is part of a bigger thing (a patch).

Once one learns this, then the roots like chorid and latrous (snakes are latrous, or linear, without necessarily being staked out on a line).

Cells and walls and sills, all have double letters, apply, as in common speak, to tilings. A wall divides two cells, and a sill bounds walls. The cell can be either eg 2d (as in the game of life, or in Dungeons and Dragons, or other war-games), or 3d. The basic idea of 'foam' or solid mass of cells is envisiaged: the meaning shift is quite apparent in quickfur's render of polychora as a foam of polyhedra.

[quickfur]

The basic theory in the PG is that words like 'face' etc, already have trans-dimensional uses, one of which is geometric. The common use is the trans-dimensional ones, so that's what's retained. The words are horribly loaded in normal speak, and if you start using face as 2d element (hedron), then you start getting ideas like 4space's notion that a 'face' divides space. That's why i removed the fixed dimensions from these.

[... snip bunch of good stuff ... ]

Thanks, wendy, for this nice intro to the PG's conventions. I think I shall spend some time to write up an article arguing for the adoption of the PG as a better and more consistent way of describing higher-dimensions. It will probably take some time, as the current CRF project is soaking up most of my free time, and real-life duties puts a cap on how much free time one can squeeze from a day. But I hope to include a kind of layman's introduction to the PG that we can point newbies at, so that PG adoption will (hopefully) have much less resistance.

[gonegahgah]

Thanks, wendy, for this nice intro to the PG's conventions. I think I shall spend some time to write up an article arguing for the adoption of the PG as a better and more consistent way of describing higher-dimensions. It will probably take some time, as the current CRF project is soaking up most of my free time, and real-life duties puts a cap on how much free time one can squeeze from a day. But I hope to include a kind of layman's introduction to the PG that we can point newbies at, so that PG adoption will (hopefully) have much less resistance.

I think that's a great idea. Thanks, QuickFur. What is CRF?

[quickfur]

[...] What is CRF?

CRF stands for Convex Regular-Faced (though I suppose using Polygloss terminology it would have to be CRH: convex regular hedrous). That is, the object's 2D elements must be regular polygons, and the object itself must be convex. It's one of the possible generalizations of the Johnson solids to 4D .

Currently the list of 4D CRF polychora is not yet fully known, though some subclasses have been enumerated (esp. Richard Klitzing's list of segmentochora, which are CRFs that can be inscribed in a 4D sphere and whose vertices lie solely on two parallel hyperplanes). We have been searching for more instances of them over in the Polytopes section of this forum.

[Hugh]

Look at the fellows on here who had a condition called VSI - that seems to be linked to some people with aspergers - which meant that their brains would switch left and right and suddenly think everything was opposite to where it should be. For them that is reality. Right may at one moment agree with us and at other moments disagree. I really don't think that the universe flips around for them and I personally don't believe that this is a 4Der trait but I do believe it shows the power of our brains.

Hi gonehahgah, is it VRIs (Visual Reorientation Illusions) you're referring to? I've never heard of an aspergers link to it though.

The reality for people who experience VRIs is that the whole universe is perceived to flip around either 90 or 180 degrees in an instant. Perfectly healthy astronauts in space experience perceived ceiling/floor VRIs frequently, whereas here on Earth, we usually only experience the perceived N/S/E/W directions interchanging.

It's fascinating to hear you and quickfur discussing 3D and 4D rotations and what would be experienced by someone within them as you are working through this thread.

I'd like to add a few points to the discussion as well.

As each discussion has gone earler with the 2Der trying to understand 3D and 3Ders trying to understand 4D, it has been apparent how limited that each viewpoint is in understanding the totality of that higher dimension.

We look around us and perceive a 3D world. Is it really only 3D? Is it possible that there are higher dimensions?

The math works out so much better when we add in the existence of higher dimensions to explain the physics of how the universe works (see the Michio Kaku quote in the VRI thread about this).

We think of what a 2D being sees around itself (a 1D infinitely thin line) and realize how limited that it would be in projecting what is actually around it.

Could it be that we see only a limited 3D slice around ourself of a higher dimensional universe?

If there was a 4D universe around us we'd only see a 3D slice of it right?

If we are actually 4D ourselves, how would we view that 4D space around us?

3D stays 3D in 4D as well, this is important to think about.

If a 4D being has a 3D viewpoint, it would see 3D around itself, in 3D.

So where do the other directions come in... ana and kata?

Well, my way of thinking is this... A 4D being sees a 3D slice of the 4D universe all around itself (it's limited in vision just like the 2D being is)...

So the extra directions allow it to see that 3D slice around itself from the other directions that are available.

The 4D being's body, including its eyes, exist in 4D, not just in 3D, so when it sees the same 3D slice from a 90 or 180 degree different direction, it's possible because those directions are available to it to look in.

All this talk about a 4D being seeing 3D as "flat" is, in my way of thinking, the wrong direction to think in... it would see 3D as it is, in full 3D, as we see the world all around us.

Like Aale de Winkel said earlier, 4D vision isn't x-ray vision, it's just seeing the light rays coming from more directions.

So if we think that if we're 4D, and we're looking at a 4D table in front of us, but that we are limited in our vision and only see the table in 3D in front of us (a 3D slice of it), then the fact that we can use VRIs to cognitively see it from a different 90 or 180 degree direction, to me, shows that those directions that the light is also coming from actually do exist.

Like I've said, I truly wish that I could show each of you exactly what a VRI is, and let you perceive it fully, to see the potential that it has in explaining the possible existence of higher dimensional space.

[Keiji]

On the subject of VRIs, I used to have a lot of trouble understanding them when I first read about them on this forum - but by now they happen very often, usually when I'm in my room and it's dark, I'll suddenly feel like I'm in a different room or the house is pointing a different way, and have to reorient myself.

I also play Minecraft a lot, and I'm not sure if reorientation in 3D video games is relevant here, but in my latest world, I used to have a staircase where, every single time I travelled up or down it, I would rotate round 540 degrees, and think I rotated 720 degrees. The result was that the route to spawn on the surface was in one direction, while the route to spawn in my tunnel was in the completely opposite direction! Even after realising that, I still never interpreted it right, and found it very difficult to imagine both routes going in the same direction. Eventually I rebuilt the staircase though, so it doesn't happen any more.

[Hugh]

On the subject of VRIs, I used to have a lot of trouble understanding them when I first read about them on this forum - but by now they happen very often, usually when I'm in my room and it's dark, I'll suddenly feel like I'm in a different room or the house is pointing a different way, and have to reorient myself.

I also play Minecraft a lot, and I'm not sure if reorientation in 3D video games is relevant here, but in my latest world, I used to have a staircase where, every single time I travelled up or down it, I would rotate round 540 degrees, and think I rotated 720 degrees. The result was that the route to spawn on the surface was in one direction, while the route to spawn in my tunnel was in the completely opposite direction! Even after realising that, I still never interpreted it right, and found it very difficult to imagine both routes going in the same direction. Eventually I rebuilt the staircase though, so it doesn't happen any more.

Hi Keiji, that's great to hear that you are becoming more aware of the occurrence of VRIs! I'm going to quote and respond to your post in the VRI thread. I don't want to be perceived as "highjacking" this thread and taking it in a different direction than was originally intended.

That having been said, if my earlier points about what a 4D being sees or what a 3D viewpoint of a 4D being perceives in a 4D environment want to be discussed, then that's more in line with the thread's direction...

[gonegahgah]

Hi gonehahgah, is it VRIs (Visual Reorientation Illusions) you're referring to? I've never heard of an aspergers link to it though.

Hi Hugh. My apologies on both fronts. Yes, that's what I meant. Sorry, I made a mistaken assumption about the association with aspergers.

The reality for people who experience VRIs is that the whole universe is perceived to flip around either 90 or 180 degrees in an instant. Perfectly healthy astronauts in space experience perceived ceiling/floor VRIs frequently, whereas here on Earth, we usually only experience the perceived N/S/E/W directions interchanging.

That's cool, I was simplifying the experience. You did mention the four orientations; not just two.

It's fascinating to hear you and quickfur discussing 3D and 4D rotations and what would be experienced by someone within them as you are working through this thread.

I think things might be on a bit of a roll now. So hopefully the new upcoming pictures will start to give a better picture of what I'm driving for.

The math works out so much better when we add in the existence of higher dimensions to explain the physics of how the universe works (see the Michio Kaku quote in the VRI thread about this).

So far the math hasn't actually given us a higher dimensional universe in the ways that string theory is striving to do so. I'm actually a believer in a far simpler universe with a single base force (which I think of as true gravity). However, the math of that simpler universe may not so readily conform to our desire for elegant math equations. This is because I give much more credence to circular motion and its play on things - than does our science - but circular motion does not produce pretty math. Take for example: opposite = hypotenuse x sine(angle). There is no direct formula for sine. It can't be made into a simple formula. There are math problems where you can't use things like sine() and there is no way to produce a direct formula for the problem. The pattern for those problems is unable to be formulised. Not something that is very attractive but I hold it as the real truth. But, that's by the way. Just adding my thoughts on the subject.

We think of what a 2D being sees around itself (a 1D infinitely thin line) and realize how limited that it would be in projecting what is actually around it.
Could it be that we see only a limited 3D slice around ourself of a higher dimensional universe?
If there was a 4D universe around us we'd only see a 3D slice of it right?

That is an interesting thought but I would tend to say that by direct analogy, to the 2Der seeing a 1D line before them, we only see a 2D plane before us.

If we are actually 4D ourselves, how would we view that 4D space around us?
3D stays 3D in 4D as well, this is important to think about.
If a 4D being has a 3D viewpoint, it would see 3D around itself, in 3D.

But 'around' is different to 'before'. A 2Der sees a 1D world before their eyes but a 2D world around themselves.
We see a 2D world before our eyes but a 3D world around ourselves.

So where do the other directions come in... ana and kata?
Well, my way of thinking is this... A 4D being sees a 3D slice of the 4D universe all around itself (it's limited in vision just like the 2D being is)...
So the extra directions allow it to see that 3D slice around itself from the other directions that are available.

This is where I've been saying that a 4Der doesn't just have ana-kata in my approach; a 4Der has towards ana and towards kata. They have a whole 180° towards both of these.
So where you mention flipping; I mention rotation.
If you were seeing in the space, that I could fit 16 people, 64 people instead, when I can only see 16, then I would think you were seeing 4D as I picture it to be.
My picture of 4D is that things need to be able to move in and out of our 3D space without our understanding why; not just our world flip around as it is.
As in the intro, accessible from the links at the top, if a 4Der put a 4D fence post sideways through our world we would only see a floating cube and not the whole post.

Do you see objects floating in the air for no apparent reason? This would be my test of if your 4D matched the type of 4D I'm espousing.

All this talk about a 4D being seeing 3D as "flat" is, in my way of thinking, the wrong direction to think in... it would see 3D as it is, in full 3D, as we see the world all around us.

With your model; yes. With my model; no. Hopefully this will become a little more 'intuitive' with some of the next pictures.

Like Aale de Winkel said earlier, 4D vision isn't x-ray vision, it's just seeing the light rays coming from more directions.

But, that doesn't mean that they flip directions. When we move our vision the image changes smoothly.
So in my model the 4Der's view would change smoothly as they turn their head in any direction. It doesn't suddenly flip.

So if we think that if we're 4D, and we're looking at a 4D table in front of us, but that we are limited in our vision and only see the table in 3D in front of us (a 3D slice of it), then the fact that we can use VRIs to cognitively see it from a different 90 or 180 degree direction, to me, shows that those directions that the light is also coming from actually do exist.
Like I've said, I truly wish that I could show each of you exactly what a VRI is, and let you perceive it fully, to see the potential that it has in explaining the possible existence of higher dimensional space.

Hmmm, it would probably be very confusing to start off with. If what you have is a form of 4D vision then it is, by my model, a very limited and rigidly aligned model that can't escape the bounds of our 3D space. The model I'm looking at doesn't have the rigid alignment to our 3D space; and movement can exist out of our 3D space. Hopefully, as I say, some of the following pictures will give a better impression of what I'm defining as 4D vision.

[Hugh]

Hi gonegahgah, thank you for your reply.

I responded to it in the VRI thread, because it deals with VRI discussion and I didn't want to highjack your thread here...

Thanks for all your work, pictures and ideas you're putting up in here!

[gonegahgah]

I thought I would post this here, instead of the in the '4D Road Markings' thread as it doesn't directly relate to roads though it is put of my thought chain.

The following is a picture depicting our 3D Space as slices.
Down the centre is the 2Der's space which extends forward and backwards only; but includes upwards/downwards of course.
I've broken our 3D space up into slices along the full extent of the 2Der's space.
An infinite array of 2D left-right slices - which I could only show a few of - go to form our 3D space.
Unlike the 2Der we see all of the slices as connected as a continuous whole.



The fascinating aspect of the 4D world is that each of our sideways slices - only having left and right - become a full 360° of sideways for the 4Der.
The picture below represents the size equivalent, for every single left-right sideways slice I have depicted on the previous diagram, that the 4Der will have.



There is one of these full 360° of sideways, in the 4D space, per every single one of our left-right slices - drawn or not drawn.
And none of them intersect. Every single 360° of sideways is independent in space from its adjacent forward or backward slice.

I think this helps to give us a greater picture of the size of 4D. As I've mentioned, where we can fit 16 people the 4Ders can fit 64 people.
And you can start to understand this more clearly when you appreciate that, for every single one of our sideways that I've shown in the top diagram, the 4Der has the whole complete circle of space in the second diagram. That's for every single slice! No overlap!

I should really have put arrows going left and right, in the first diagram, to show that they extend further but that is okay; everyone knows this anyway.
In the second diagram I have shown that ana-kata-left-right extend further outwards as well.
I've also shown some arrows going around in a circle to show that there is no preferred direction for ana, kata, left, or right.
They would all be considered equally equivalent in all directions to a 4Der so I'm not sure how they would even think to distinguish them.

That's why I argue that there is no reason for a 4Der to have four legs. They don't have 4 preferred directions of sideways. They have a whole 360° of sideways.

Anyhow, I'll depict exactly the same as this, but with rotated slices, next...

[gonegahgah]

The following is a picture depicting our 3D Space in pieces (pie slices. Hey it even combines nicely ie. pie[sli]ces. Maybe there is something historical in that?)
Anyhow the following is a picture of our 3D Space in rotated slices; instead of the sequential slices in the previous post.



Still we have the 2Der world that is directly forwards (and backwards) and this time we rotate their slice to encompass our full 3D world.

Again to convert this to a 4D world you would imagine that each slice is replaced with a full 360° of space.
This time they all have a common centre of intersection; whereas for sequential slices their is no overlap or intersection.
I used the term pieces because I like to think of the slices as flat pie pieces; to be more accurate.

This diagram is a good representation of how I said I would depict 3D to a 2Der; but the slices, off into 3D, would be rotated up into the 2Der's available sky view rather then being lost somewhere off in 3D. So, as well as representing 3D, the above diagram also represents the initial co-ordinates, for drawing 3D using the rotation method, for the 2Der.

Although, this represents how we could work out the co-ordinates to draw 3D using the rotation method for the 2Der, it does give us the pre-final clue to how we would work out the co-ordinates to draw 4D, for us 3Ders, using the rotation method for one of the two orientation options I've mentioned.
One of the diagrams, in the previous post, gives us the pre-final clue to how we would work out the co-ordinates to draw 4D for the other orientation method I mentioned.
I'll show the two clues again:


You might be able to work out how from these two diagrams yourselves. They give me the clues, I needed, on how to depict the two different 4D orientation options for us.
Feel welcome to add anything if you see this too. Either way I will draw pictures to show what I mean soon.

[gonegahgah]

The right diagram needs the least modification to show how it can be used to depict the 'locate' view option.



All I have done is substitute ana for left and kata for right to show that the rotation now occurs into those two directions for locating objects off in 4D, in place of the left and right that was needed for the 2Der to locate objects off in 3D; because we can already see those directions. Now the extra circle of space per slice, that replaces each shown, and unshown slice, is representative of each slices left-right + front-back views instead as per:



I've left in the arrows pointing out because they still extend further outwards but I've taken away the arrows around the space because we definitely have a forward and a left and right directions. Each slice in the first diagram is replaced with a unique copy of this circle of space to represent each particular rotation into 4D space for us 3Der's for the purpose of locating objects more easily that are off in the 4th dimension. With this rotation method, when you turn into the 4th dimension, your forward turns too to that new direction. Left and right also change to maintain the left-right alignment to yourself. Again, the extra space represented by the circle for each slice in the first diagram, intersects at the centre; because the rotation method is being utilised.

I'll work on depicting the 'fine tune' view model using the other image but that will take a little time to construct. I can also depict this 'locate' view with the other image as well so I'll draw that too to show you how that looks...

[quickfur]

[...]
In the second diagram I have shown that ana-kata-left-right extend further outwards as well.
I've also shown some arrows going around in a circle to show that there is no preferred direction for ana, kata, left, or right.
They would all be considered equally equivalent in all directions to a 4Der so I'm not sure how they would even think to distinguish them.

That's why I argue that there is no reason for a 4Der to have four legs. They don't have 4 preferred directions of sideways. They have a whole 360° of sideways.
[...]

They will distinguish between the lateral directions based on their own anatomy. Just like in 3D, we distinguish between front/back and sideways because our anatomy has a preferred "front" direction, which just so happens to divide the remaining across-space (to use wendy's term) into left and right. If we were starfish, for example, we'd think of the world in terms of 5 lateral directions instead of front/back and left/right.

So the number and arrangement of limbs is inherently tied to our perception of the world around us. They are inter-dependent.

To use a slightly less common example: so far we've tended to imagine 2Ders as some kind of flat being with a front and a back. However, this may not be the ideal arrangement at all. For one thing, the 2Der would be unable to face the other direction, so he could only walk forward, else he wouldn't be able to see where he's going! Also, he'd be unable to talk to another 2Der face-to-face unless the other 2Der was facing the opposite direction. And 2Ders who face, say, left, will be unable to ever face right, and those who face right will be unable to ever face left (unless they stood on their heads -- a rather uncomfortable proposition).

So an alternative arrangement is for the 2Der to have eyes on both sides. The left eye will always see what's on the left, and the right eye will always see what's on the right, so one eye will be most useful depending on which direction the 2Der is moving. Such a 2Der will have no sense of "front" or "back", because for them, there is only left or right. Their left/right is the equivalent of the "traditional" 2Der's front/back, except there is no preferred direction. With the traditional 2Der, front is the preferred direction of movement, but for the "bilateral 2Der", neither is preferred. Accordingly, their world-view will be drastically different. They will think of everything in the world as 2-sided. If we were to try to teach them 3D, their difficulty would not lie in the perception of sideways, but in the perception of front/back! For such 2Ders, they may find it easier to adopt the POV of the 3D starfish, who has not just two lateral directions but 5, and where these 5 lateral directions are interchangeable via a 3D rotation.

Of course, we humans tend to think of the traditional 2Der instead of the bilateral 2Der, because that's closer to our own worldview. So we also tend to think of 4Ders in the same vein: as having front/back and sideways. More exotic 4D body plans would be so far removed from our everyday experience that we would have a hard time understanding how they perceive their world, let alone draw useful dimensional analogies. But even with the "traditional 4Der" with front/back and sides, there's one drastic difference between us 3D humans' sides and a 4Der's sides: while our left/right are detached, the 4Der's sides form a continuous circle. Where for us left/right are clearly delineated as two separate laterals, for the 4Der they have a single continuous lateral. How they demarcate this continuous lateral will depend very much on what lateral limbs they have.

For example, they could have 5 arms and 5 legs around their laterals, in which case their perception of sideways will not be unlike that of a starfish with 5 arms. No one arm would be singled out as belonging to a special class, unlike our left/right which are clearly distinct, even though they are symmetric in their own way. For a starfish, there are no left arms or right arms, there is only one kind of arm, and any arm can take the place of any other arm by a simple rotation. Thus it is for the 5-armed 4Der: they would not have a distinction between arms, since they could rotate any one arm into any other just by a simple rotation. And unlike starfish, who do not have a singled out front/back direction, the 5-armed 4Der will maintain front/back as a special pair of directions in addition to having 5 equivalent arms.

[quickfur]

Continuing on the line of body plans, I chose 5 arms for the 4Der just for the fun analogy with starfish. But just as starfish in 3D also vary in number of arms, so it is also possible to have 4Ders with other numbers of arms. The choice of 4 arms may be biased to our own quadrilateral worldview (caused by our own human anatomy). It is easy to imagine 3-armed 4Ders, where the arms are arranged in a triangle around the continuous lateral side. It is also possible to have 2-armed 4Ders, in which case their perception will be very close to ours save for possessing an extra pair of lateral directions in which there are no arms.

But before we go further about other 4D body plans, we may consider some hypothetical (or not-so-hypothetical maybe?) 3D cases. We've already talked about starfish, which certainly would have quite a different worldview from us, having no preferred front/back directions but 5 (or in some cases 6 or 7) more-or-less equivalent lateral directions. But we may also imagine some perhaps less-practical cases which may be helpful mental aids to understand the 4D case: such as a 3Der with a cylindrical body with perfect circular symmetry, with 1 leg (or tail stub) and no arms. A vertical worm, if you like. Suppose further that this being has a face that's perfectly symmetrical, so that it can rotate around its axis without any preferred orientation. Its neck is such that it can point in any of the 360deg of lateral directions, and since its face is symmetrical, it would perceive all of the lateral directions as equal. Left/right, front/back would be meaningless to such a being. The only special direction is up/down, so its perception of the world may well be in terms of what angle something makes with the vertical, rather than the frontal direction. So it would perceive things as lying at various points in concentric circles centered on the vertical. If such a being were to imagine 4D, then it would not think of 4Ders as having front/back at all, but rather as having a spherindrical body, in which things lie at various points in concentric spheres centered on the vertical. Such 4Ders, being also armless, would be different only in the sense that they have more lateral directions in which they can point (by turning their neck), but none of these directions would be special in any way. Rather than merely a 360deg of sideways, these beings would have a whole sphere of sideways.

So you see, when it comes to body plans, many things are possible; the question really is what we're after. If we're after 4Ders that most closely resemble ourselves (for ease of dimensional analogy, say), then we will probably choose a body plan that reflect the biases of our own body, such as having a quadrilateral arrangement of some sort, having a preferred front/back with laterals, etc.. If we're after realism in terms of what is most likely to naturally occur in 4D, then we may end up with rather exotic body plans -- most likely very different from the former case. The fun thing about 4D, though, is that the geometry itself is so rich that even when we stick to the front/back bias, there's still ample room for semi-exotic arrangements, like the 5-armed 4Der, or 3-armed 4Der, or a variation of the 4-limbed 4Der that I thought of a while ago, where there are a pair of arms on opposite vertices of a square, and wings on the other pair of vertices -- this allows both flight and complex manipulation of objects with arms.

[gonegahgah]

Absolutely. Very interesting additional ideas QuickFur.

The interesting thing, which I have to sheepishly admit I only just realised earlier today triggered by your added topic QuickFur, is that a 4Der can have any number of legs and none of them need be in front of or behind any other leg. They could have, for example, ten legs all at the front and each those legs would be equally beside its two neighbour legs. So they could still be 'fourmo-erectus' even with those additional legs. Their animals could still have rear and front legs of course.

Probably there will always be a 'go to' direction - the direction that we are currently heading towards - so probably evolution will provide higher creatures with a front.
It would be certainly interesting to see what 4D environmental effects would have on evolution.

I think, as for us, the trend would still eventually be towards minimalisation so that creatures could focus less energy on all aspects and more energy on higher cognitive ability but who knows how much greater differentiation the extra brain space, provided by the extra dimension, would allow? There is the problem of volume vs surface area to contend with. I wonder if that would affect cell size in the 4D world. In 3D a cell can get too big to the point where the surface area hasn't increased at the same rate as the volume and the cell can't pass enough things through its cell wall at enough of a rate to sustain itself. Smaller is usually easier for that reason.

What is the math situation, for the 4Der, when it comes to their 4-surface rate of exchange versus 4-volume space needing to be serviced by that 4-surface?

[gonegahgah]

Here's the first diagram. I wasn't exactly sure how to depict it. Should I show our left-right that we would see as the angle of ana-kata or should I show that to us it will still look like left and right to us. I went with the later:



It was also easier to draw
Just remember that the left-right is not rotating through our 3D space but is rotating through the 4th direction hence the changing kata value at the bottom right.
The left will always fall to our left and the right will always fall to our right.

At the moment I'm calling this the interact view and the other one the locate view.
The key thing with this view is that anything in front of you stays directly in front of you even as you rotate into the ana or kata directions.
This allows you to more finely interrogate objects that you see.
Basically it just follows the principle of rotating about a 2D-plane which is the most recognised form of rotation into 4D.

You can still move about in 4D by turning left and right and moving forward combined with rotating the left-right view into the ana-kata directions; so that you then move towards them when you turn left and right and move forward.

Now, as I've alluded to previously, you could somewhat simultaneously depict the things hidden off in the 4th direction by taking the other 359° of alternate rotated slices that can't be seen in the current 3D plane and rotating them up into the sky instead of sideways, added with making them colour shadowed and keeping the tilt that this automaticaly adds. This would give us a good sense of where things are located I am thinking and guessing. Hopefully I will add some example pictures to show some simple examples soon.

I'll add an animation of the other navigation option soon.
I would like to ask for considerations on what to name the two different navigation options mentioned.
As I've said, I've called one the 'interact' view and the other the 'locate' view. Any better suggestions?

[Hugh]

...The left will always fall to our left and the right will always fall to our right.

...The key thing with this view is that anything in front of you stays directly in front of you even as you rotate into the ana or kata directions.

That's a fascinating picture gonegahgah, thanks!

I just thought I'd let you know I responded to this post in the VRI thread if you're interested.