Dimensional Baby Steps

[quickfur]

But yes, I do use povray. I have a tool for computing coordinates, which generates a 4D definition file that contains all the information about the vertices, edges, ridges (2D faces), and cells, which are then fed to another program that does the computations for projecting the object into 3D based on a given viewpoint. This projector program also lets me assign textures to various elements of the polytopes, to highlight/hide certain parts of the object for clarity's sake, etc.. Once the projection is done, it writes the 3D model of the result into a povray scene file that can be raytraced from any camera angle (specified externally by a .pov template file).

Aha, OK, so even three programs are needed for these images. I actually thought it would be easier, because POV-Ray is usually so versatile when dealing with all kinds of formulas and functions (at least from what I've seen of the more experienced POV-ers.)

Povray is indeed very versatile in dealing with all kinds of math stuff. I'm not that well-versed with its more advanced features though; I use it mainly because it saves me the hassle of writing my own 3D->2D projector. I'm sure more experienced POV-ers would take a different approach; in fact, I know some on this forum have directly coded 4D->3D projections within povray itself.

I chose to write separate programs for it because povray vectors are limited to 5D (or was it 6D?), but I wanted a general approach that would work with objects of any dimension. Furthermore, I wanted to deal with things like hidden surface culling, free 4D camera placement/orientation, etc., all of which are no doubt possible in povray but slightly inconvenient due to the assumption of a large part of its vector-manipulating functions that one is dealing with 3D vectors. In retrospect, the vector size limitation isn't really that big of a problem, because one could use arrays to represent larger vectors. But one consideration still scores against it, that is, performance. A C++ program specifically crafted to do what I need is significantly faster than using the povray input language, which is interpreted. Performance is an important consideration when dealing with higher-dimensional objects, because of combinatorial explosion: an icosahedron, for example, has only 60-odd elements (12 vertices, 30 edges, 20 faces), but its 4D counterpart, the 600-cell, has 120 vertices, 7200 edges, 1200 faces, and 600 cells. When you go up to even higher dimensions, the number of elements grow exponentially: an n-dimensional cube has 2ⁿ vertices alone, not counting its other elements which also increase very quickly. With such numerous elements, maximal performance is called for, lest one has to endure unnecessarily long waits.

[...] I'm glad you like the idea. Actually I made a mistake above, when describing the idea "in a way that all the voxels of a 3d retina are filling the whole field of view". I rather meant "part of the voxels/voxel array of a 3d retina are filling the whole field of view". I don't know exactly which part that could possibly be (maybe half of the voxel array for representing something like a 180 degree 2d viewing angle). I guess to put the whole voxel array into one image would look too messy.
But I was thinking of placing the camera inside, yes. Maybe experimenting with some fish_eye or ultra_wide_angle camera or something like that. And maybe some colour cues (i.e. different colour for volume with difference light shading for distance within the volume, and different colour for the edges and still different colour (or maybe edge colour but darker) for vertices.

Hmm. I shall have to think more carefully about this. I'm unfamiliar with fish_eye or ultra_wide_angle, but I suppose they are just some ways in which one can render a larger panorama of one's surroundings from a single viewpoint. As for shading, I shall have to consider how best to do it, because obviously the volumes cannot be rendered opaquely; one would perhaps want some kind of fog effect to give a sense of depth when looking into the volume, perhaps? I'm unsure about what works best at this point. Edges and faces I will leave as-is, since our brains do still use them as guideposts in inferring the shapes of 3D volumes.

[gonegahgah]

Following up on the time method - which I think I was thinking too 3D about before - here are some better diagrams to demonstrate dimensional progression:



We start with 0 dimensions. The diagram is a 3D depiction of 0D. We smear it into our 3 dimensions to make it visible.
In the middle frame, in the very middle of that tiny block inside it, the real 0D resides.
Because there are 0 dimensions all the dimension opposites are marked X to show that they don't exist.
The diagram shows several frames which will come into use soon. At the moment we are only using the middle frame.



If we endow our 0D universe with time then although nothing exists it does so describing a line of time.
This is the purpose of the frames - to depict a time series.
In the previous image you can see some red X's and they represented the 0D universe without time. Now we add time and get rid of the red X's.



If we squash and turn those 0D time frames into the single central frame then we will be looking at the line 'face' on - which means by its dot face to a 1Der.
Again we've eliminated the time dimension so some blue X's have appeared to show it no longer exists.
You can see that the green X's have disappeared to show the back-forward dimension is now in use.
Our 1D universe now has depth and nothing else.



Now we endow our 1D universe with time so that the 'deep' line describes a square through time.
The line appears in multiple consecutive frames with a clear frame at its front in time and a clear frame after its end in time with clear frames implied before and after.
The blue X's that appeared for the previous image have disappeared again because the time dimension is back in play.



Again, if we squash and turn the time frames into the central frame then we will be looking at the square 'face' on - which means by one of its edge faces to a 2Der.
Again we've eliminated the time dimension so some green X's have appeared to show it no longer exists.
You can see that yellow X's have disappeared to show that the up-down dimension is now in use.
Our 2D universe now has depth and height and nothing else. You can also see that a 2D observer has now appeared in our 2D universe.



Now we endow our 2D universe with time so that the 'deep' square describes a cube through time.
The square appears in mutiple consecutive frames with a clear frame at its front in time and a clear frame after its end in time with clear frames implied before and after.
The green X's that appeared for the previous image have disappeared again because the time dimension is back in play.
In their place you can see our 2D observer who can now magically see across time all at once.
Because they are looking across time I have shown their line of vision not in line with any of the spatial dimensions.
The time dimension is not in the same line as any of the existing spatial dimensions.



Again, if we squash and turn the time frames into the central frame then we will be looking at the cube 'face' on - which means by one of its square faces to us 3Ders.
Again we've eliminated the time dimension so some gold X's have appeared to show it no longer exists.
You can see that purple X's have disappeared to show that the right-left dimension is now in use.
Our 3D universe now has depth, height and width and nothing else. You can also see that a 3D observer has now appeared in our 3D universe.



Now we endow our 3D universe with time so that the 'deep' cube describes a tesseract through time.
The cube appears in mutiple consecutive frames with a clear frame at its front in time and a clear frame after its end in time with clear frames implied before and after.
The gold X's that appeared for the previous image have disappeared again because the time dimension is back in play.
In their place you can see our 3D observer who can now magically see across time all at once.
Because they are looking across time I have shown their line of vision not in line with any of the spatial dimension.
As already mentioned, the time dimension is not in the same line as any of the spatial dimensions.

The thing I hope to make most clearest via these images is that when we add a dimension we aren't doing so from any directions that exist in the lower dimension.
The analogy here is time and time is not a spatial dimension.
So when we see an object across time - to make it a spatial dimension - we do so with a brain that connects all those time frames at once.
So, in each consecutive higher dimension we are looking from a direction that we could not look from before in the lower dimension.

So it is because of this that I apologise that I wasn't able to continue this time method progression here and show us once and for all what a 4Der sees.
At best, all we can do is try to show the best approximation we possibly can. There will possibly always be debate about the best way to do that.
As I say, I'm looking to help contribute to making a way where many more people can get a greater feel for what 4D means.
Hopefully that is achievable and should hopefully make for an enjoyable experience for them as well...

It should be recognised with these diagrams that, just as we view our objects as having a front and back say, our objects in the time frames also begin and end and because of this they end up having a front 'face' in time and a 'back' face in time. So if you look forward in time, from the past, you will see the front time face; and if you look back in time, from the future, you will see the back face. So the time analogy nicely provides us with where the two extra faces are that keep popping up each time we add a new dimension.

In otherwords, the existing 'faces' (at each dimensional level) each trace out a 'face' of the next higher dimension; and the front and back time views become the extra 2 faces.

I'm also just wondering if I can explore the notion of dual rotation using this time frames analogy as well?

It's also interesting to note - I'm not sure whether there is disagreement on this but let me know - that gravity doesn't really kick in until you have 2 dimensions.
With 0 dimensions there is nothing to create gravity.
With 1 dimension gravity is constant in both directions so may as well not exist. Things could not hold together in 1D.
With 2 dimensions gravity would decrease at a linear rate from the surface of a planet.
With 3 dimensions gravity decreases as a squared rate from the surface of a planet.
With 4 dimensions gravity would decrease at a cubed rate from the surface of a planet.

[gonegahgah]

Another interesting thing to note is that the added dimension for a higher dimension has sprung from one of the non-existant dimensions of the lower dimension before it.
Up to growing to 3 dimensions this has been easy for us to see because we can easily notice unused dimensions in anything lower than 3D.

If we could depict all these diagrams as a 4Der could then they would always see an extra pair of 4-faces that were an unused dimension as well.
They could easily depict the up-dimensioning from 3D to 4D and their diagrams of the lower dimensional up-dimensionings would also clearly show this 4th dimension (non-)'face'.

So this is why, for us, we run out after the 3D depiction.
It might be nice to think that the purple pre-non-existance face was always the one expanding into the extra dimension but this is not the case.
Extending on that you might think the 3D cube is also purple on all its inside and that purple becomes the new face.

Although that would be a close analogy, it would be better to think that the time dimension is another colour - say brown or something.
In this case it would be the two new ends of the object in the brown time dimension that expands up to the next dimension's added two 'faces'.

It is convenient to think that the purple (non-)face looks very much like the faces of the next higher dimension but this runs out after 3D because all our available dimensions are used up and we can not longer see any purple. So, although this progression is obvious because of this until 4D, it is not 100% accurate.

[gonegahgah]

The question returns to negotiating higher dimensional worlds.

Step 17. 2Der traveling along a road in 3D:



You can see in these two pictures the comparison of the type of re-orientation typical in most computer games.
You have the sideways movement on the left and the rotation on the right.
Sideways movement is good for avoiding things but it is not by co-incidence that rotation is the preferred method for re-orientation that we favour.

Step 18. Find the road in the first place:



If the 2Der is off the road then they will not know which way to turn to find the road.
They may not even know that the road is there; the same as for any other objects that are off in the 3rd unseen direction.

[quickfur]

[...]It's also interesting to note - I'm not sure whether there is disagreement on this but let me know - that gravity doesn't really kick in until you have 2 dimensions.
With 0 dimensions there is nothing to create gravity.
With 1 dimension gravity is constant in both directions so may as well not exist. Things could not hold together in 1D.
With 2 dimensions gravity would decrease at a linear rate from the surface of a planet.
With 3 dimensions gravity decreases as a squared rate from the surface of a planet.
With 4 dimensions gravity would decrease at a cubed rate from the surface of a planet.

Actually, there can be gravity in 1D too. But it is thoroughly uninteresting, because the lack of a lateral dimension in which momentum could counteract the inward pull of gravity. A 1D universe with gravity will quickly have all matter conglomerate into a single very long line segment. You wouldn't be able to stand on either end of it, since to even lift yourself up will require counteracting the gravity of all other matter in the rest of the universe.

At least 2D is required for gravity to produce meaningful structures -- there must be at least one lateral dimension in which momentum can counteract the force of gravity, so that matter doesn't just all collapse into a single n-dimensional ball.

[gonegahgah]

Actually, there can be gravity in 1D too. But it is thoroughly uninteresting, because the lack of a lateral dimension in which momentum could counteract the inward pull of gravity. A 1D universe with gravity will quickly have all matter conglomerate into a single very long line segment. You wouldn't be able to stand on either end of it, since to even lift yourself up will require counteracting the gravity of all other matter in the rest of the universe.

Ah, yes, I see my error, thanks QuickFur.
4D: gravity decreases with distance at a cubed rate.
3D: gravity decreases with distance at a squared rate.
2D: gravity decreases with distance at a constant rate.
1D: gravity remains constant with distance.

So 1Der falls with the same acceleration no matter where they are.
A 2Der falls with a linearly increasing acceleration.
A 3Der falls with an acceleration that goes up in a hyperbolic fashion.
A 4Der falls with an acceleration that increases more at a cubed sort of rate.

And the rest, that you mention about momentum or lack of, produces the results that you said.

[gonegahgah]

Step 19. A straight 4D road:



Here is a fairly good rendition of a straight stretch of 4D road.
A 4Der wouldn't see the rotation; to them these frames would all be side by side in the 4th direction.
But it is the best way to show to us their extra 359deg of sideways that we don't have.

I could have rendered every angle of sideways but it would end up looking like a cylinder. This would have prevented us from seeing something important.
That is that the 4Der sees every angle as being a flat road.
To them the road is flat in the forwards and each sideways angle just as our roads are flat forward and our one pair of sideways direction.

I'll go more on about this soon and show it a bit more clearly and also show how this naturally progresses up through the 2D road -> 3D road -> 4D road.

[quickfur]

[...]
1D: gravity remains constant with distance.

So 1Der falls with the same acceleration no matter where they are.
[...]

Hmm, you're right! I didn't realize that earlier.

And since we're at it, I thought we should make explicit the assumption at work here: we're assuming that the reason gravity (in 3D) has an inverse square law (as opposed to a reciprocal law or an inverse cube law) is because the flux emanating from the source of the gravity is spread thin over the surface area of a sphere of radius equal to the distance from the source. This wikipedia page explains it best: Inverse-square law.

In 3D, the surface area of the sphere increases in proportion to r², and so we have an inverse square law.

In 2D, the surface area of a circle increases in proportion to r, so gravity in 2D would diminish according to a reciprocal law (1/r).

In 1D, the force would not diminish, because the surface area of the line segment never increases. So in 1D, no matter how far apart two objects are, they would feel the full strength of each other's gravity. Thus, the collapse into a single very long line segment would happen very fast indeed.

[gonegahgah]

And since we're at it, I thought we should make explicit the assumption at work here: we're assuming that the reason gravity (in 3D) has an inverse square law (as opposed to a reciprocal law or an inverse cube law) is because the flux emanating from the source of the gravity is spread thin over the surface area of a sphere of radius equal to the distance from the source.

Cool, thanks, it is useful to elucidate that for everyone reading here.

Step 20. Hilly roads:



Here is a picture on the left of a road going over hills in a 2D vertical plane. On the right is a picture of a road going over and among hills which 3D allows.
I'm not going to attempt to depict a road going over, among and around hills in 4D at this stage.
Any descriptions of such are thing are welcome here... cheers!

Before I even try to think about that myself, I'll deal with just straight roads as a first scenario in my next post.

[gonegahgah]

Step 21. 4D Roading:



Curiosity got the better of me. I thought I'd try to do a little depiction of a road that is 4D for a 3Der to follow.
What I have done is show, by shadowy roads, what will rotate into the 3Ders view if they rotate their head towards the ana or kata dimensions.
You can see that their immediate road ends very shortly after it begins in our 3D pane and there is unimpeded nature land after this.
To stay on the road you would not walk forwards but would follow the road in the kata direction by turning that way.

The nice thing with this approach is that the flat road that you are standing on doesn't change. It does but it doesn't appear to do so.
As you rotate into the kata direction you feet move through the road to another frame that is rotated kata-wards.
Your feet stay on the road the entire time even as you do this.

What does happen is that as you turn kata-wards, or ana-wards, bits of road that weren't available to you in the previous frame become available to you to then turn left or right or go straight ahead onto.

The depiction I have done has a very tight spiral into the kata-wards direction. There is a slight hill kata-wards+forward and after that a slight valley kata-wards+forward.
The road doesn't quite make it full turn back to our plane which perhaps I could have shown to let you see how this would look. Ah, well...
The actual turn kata-wards is rather tight and probably wouldn't ever be quite like that except for horsebends.
At no point have I made the road do an S-turn in the kata-wards direction but this could have been done too.

With more sane turning degrees it should appear even better hopefully.
I'm also hoping that with correct shadowing and shading that this becomes a bit more obvious still.
Sadly a road doesn't itself offer any obvious signs of up and down which will help with orientation. I'll try to show this later on for other objects.
You could add street posts and things like that.

In provided a reductionist view I've heeded what QuickFur said about overloading our eyes and confusing our interpretation.
I could show the road as a tube but this would just confuse the viewer.
I could also try to rotate the entire landscape but again this would confuse the viewer.
It is best to only 'project' a few items in that way. I could have, for example, projected the sun through the full 360 degree of ana-kata rotation so that it formed a donut starting in the sky and passing through the ground and underneath back to the sky.
It's all about helping us to negotiate the 4D world with greater ease and developing understanding.

I'm sure it would be fun just standing in one spot and rotating ana- or kata-wards as the land transforms before you eyes while, in the process, giving sense to the roads strange meanderings.

[gonegahgah]

In the depiction, that I made in the previous image of a road that is made to exist in 4D, I have rotated the other sideways directions around the axis of the depth.
What I am more inclined to want to do is rotate the other sideways directions up around (and down and back over your head) in front of you. I think that would experience better.
Can you imagine doing an overhead loop to loop to get around mountain while all the time seemingly staying on the ground. How cool would that be.

Another thing I consider out of this, showing that even attempts have their own worth, is that just as a road doesn't tend to veer suddenly to the right; changing to a different angle of sideways-ness is not usually a sudden occurrance. So turns will always tend to be more gradual. You don't have to go suddenly from 90deg sideways to 200deg sideways.

So using the rotation of forward up for objects (and road) laying at a kata-wards angle and forward down for objects laying at a ana-wards angle would probably be easier to follow...

[quickfur]

[...] I'm not going to attempt to depict a road going over, among and around hills in 4D at this stage.
Any descriptions of such are thing are welcome here... cheers! [...]

The easiest way to understand 4D roads is by using a map.

A map is basically an orthogonal projection of the landscape onto a hyperplane placed some distance above the ground. Such a projection collapses any height differences to 0, so that only the connectivity of the roads is shown, not any variations in their elevation. We may optionally connect select points of the same elevation -- i.e., draw contour lines -- to show how elevation varies across the map, but this is not mandatory. Features of interest may be specially marked on the map -- for example, we can represent houses or landmarks with some symbol or other such markings to indicate where they are along the road(s).

2D

A map of a 2D road is thus very simple. Since there is no lateral dimension, it consists of a single straight line, perhaps with a mark at one point to indicate your current location, and a mark at another point to indicate your destination. In between, there may be other marks to indicate intermediate stops, or features of interest, etc.. All in all, quite a simple affair: just a single line with various points labelled. If we wanted to indicate elevation, we could specially mark points along this line to indicate elevation of some multiple of a chosen step (say we mark points on the road where the elevation is exactly a multiple of 10 feet). Thus, we obtain a series of "contour points" along the line, and we can tell at a glance how the road rises or falls as it traverses the terrain.

In 2D, walking on the road is a simple affair: there is only forwards or backwards, so you just keep going forward and eventually you'll reach your destination.

3D

A map of a 3D road is more interesting. We are already familiar with such maps, so I'll only point out some salient points: the map is drawn on a 2D surface, and roads are shown as lines that are interconnected. We may optionally mark our starting location as a point somewhere on this 2D surface, and maybe also a destination at another point. Given such a map, we can then trace the series of roads that we may take to get from our starting point to our destination. Here, we see a significant departure from the 2D case: there may be multiple paths from one point to another, due to the presence of a lateral dimension. Furthermore, there is now the possibility of multiple roads: in the 2D case, there is only enough room to fit a single road over the ground. Where there are no roads, we can have other terrain features, like rivers. Rivers may intersect roads, in which case a bridge is needed to avoid having to take a dip in the water.

Path-finding becomes a significant concern here: unlike the 2D case, walking forwards along the same road does not guarantee you'll end up at your destination. You may need to turn onto a different road -- a series of roads, in fact -- to get there. Roads also may wind sideways, to the left or to the right. (Remember that we're disregarding elevation here, since we're projecting the 3D terrain onto a 2D map.)

Elevation may be optionally shown on the map by noting points of equal elevation, and connecting them with curves. Thus, we obtain contour lines, that give us an idea of how elevation varies as we walk across the roads from starting point to destination. Unlike the 2D case, where contours are indicated by points along the single road, in 3D we have contour lines that may or may not intersect with any road. A hill with roads winding around it, for example, will be shown as a series of concentric roughly-circular contour lines, with the top of the hill situated in the innermost contour.

4D

In a 4D world, a map would have to be drawn on a 3D surface. As with the 3D case, roads show up as a network of lines within this 3D surface, and we may mark two points as our starting point and destination, respectively. Like the 3D case, there is ample room to fit multiple roads, and some non-road terrain features between the roads too. In fact, there is a lot more room on the map to fit things. For example, unlike the 3D case, rivers need not intersect any road at all, because the roads can simply wind "around" the river, thereby evading the need for bridges. In fact, it's possible to have two disjoint road systems that fit within each other, thereby covering the same geographical region yet have no interconnections between them at all.

Unlike the 2D case, where all roads are straight, and unlike the 3D case, where roads can only wind to the left or to the right, here in 4D roads have two degrees of freedom with which to wind. They can form spiralling or other such complex curves. A road can spiral around a river and follow the river's course upstream or downstream, for example, all without ever intersecting the river. Path-finding becomes a much harder problem in 4D. In 3D, if you get lost, you can in theory just walk in the general direction of a road and you're bound to find the road. In 4D, you can easily miss the road, since lines (roads) do not divide the 3D surface of the terrain, so there's no guarantee your path will intersect the road. Furthermore, even if you stay on the road, there's no guarantee you'll arrive at your destination, since it's possible for multiple disjoint road systems to cover the same geographical region. If you're on the wrong road system, you'll never find your destination, even if the roads you're taking are circling around very close to it!

Elevation can be optionally indicated by noting the points of equal elevation, and connecting them. It takes more than mere contour lines to do this, though: you'll need to draw 2D surfaces to indicate regions of equal elevation. So we would have contour surfaces on our map. A 4D hill, then, would appear as a series of roughly spherical contour surfaces nested within each other, with the highest part of the hill represented as the region contained by the innermost contour. To determine the changes in elevation along a road, then, we simply look at how it intersects these contour surfaces as it crosses the terrain.

[quickfur]

Iin retrospect, it seems that we have missed another method of 4D visualization: using contours. Just as contour lines on a 2D map shows how the surface of the ground in 3D varies in elevation, so we may use contour surfaces on a 3D map to show how the surface of the ground in 4D varies in elevation.

For things that are mostly flat, like the ground, contours are possibly the best way of visualizing 4D, since it is not prone to illusions or other deficiencies in the other methods of visualization. However, it may not work so well with more complex objects that aren't mostly-flat.

[gonegahgah]

In retrospect, it seems that we have missed another method of 4D visualization: using contours. Just as contour lines on a 2D map shows how the surface of the ground in 3D varies in elevation, so we may use contour surfaces on a 3D map to show how the surface of the ground in 4D varies in elevation.
For things that are mostly flat, like the ground, contours are possibly the best way of visualizing 4D, since it is not prone to illusions or other deficiencies in the other methods of visualization. However, it may not work so well with more complex objects that aren't mostly-flat.

Yes, absolutely. Our thoughts must have been almost linked together QuickFur because I started working on a 4D contour map last night.

I was thinking that like our simpler roads it would make sense for a 4D road to follow the path of least resistance and/or effort where possible. Of course we have the means to move mountains these days or tunnel through them but you can find many instances of roads that follow the landscape in a meandering fashion still.

It is interesting to note, in all higher dimensions, that generally where you stand will follow a generally graduated pattern. If you are standing on the side of a hill then you will have a downside, an upside and a graduation of variations between curving down and decreasingly curving down until going up in the 359deg of remaining angle. The same goes for any point in 4D; though you may have further up curved and down curved lines somewhere in the extra 359deg of sideways provided.

But, the same goes for them as for us. Anywhere you have a road is going to be, or fashioned to be, generally level and flat in all sideways directions whether it is left-right for us or 360deg of sideways for the 4Der. So this, and the graduated change in landscape, creates some rules for us that should make it easier to develop a 4D topology. It would even hopefully be possible and interesting for other intrepid people at some point to create something of a 4D science for such things as geology from these things. Would certainly be interesting to see and read.

You're right that 4Ders would tend to have contours that look like the outlines of a 3D potato rather than our 2D potato outlines. That was a good point that I hadn't thought of. They would have 3 dimensional 'outlines' (or should that be 'outplanes') of height whereas we have 2 dimensional outlines of height. I think that would be the best way to map 4D in computer terms too. Then, when we translate it to the rotational model, it will use the 4D map but would present just the current 3D slice of that map to not confuse the viewer. Shadowy features in the landscape would ultimately give the indication that there are other angles of sideways to turn into and also some idea of the topology.

I agree with you that this is another - and important - new 4D representation method. Well done. It's good to see this flow of ideas...

[gonegahgah]

Step 22. A contour map of a normal 3D landscape:



Step 23. How the 2Der would slice up the above 3D landscape with rotation:



I thought that it would again be useful to go back to our 2Der to work out what it is that they see in a rotated slice fashion.
The thing that rotating provides is that the 2Der is always on the ground and the ground immediately below them doesn't change drastically.
As things move away from the 2Der the change can be greater but you still see a fairly consistent changing pattern due to the general formation of landscapes.

Things like cliffs would make for sudden changes but they tend to be the exception.

The 2Der can get a better sense of the lay of the land by rotating; then they could by moving directly sideways.
It should be the same for the 3Der when it comes to providing access to the extra 359deg of sideways.
The 3Der doesn't need to picture the 3D lay of the land so they can already see more but they do need help in picturing the extra dimension in which the land would lay.
Rotating is probably the easiest way for them to do this. The other bonus is that you can get a greater picture of the bulk as well by taking in a full rotation.

[gonegahgah]

Step 24. Shaping towards a 4D contour map for a 3Der:



The 3Der has a slightly different situation to the 2Der in that they can see a view before them that spans from left to right; which the 2Der can't see.
However, the 3Der still can't see the 4D landscape.
In the same manner, that we would rotate the 3D landscape into the 2Der's view, the 4D landscape can be rotated into the view of the 3Der.

The above picture only shows the one landscape view, shown as a contour map, that the 3Der is currently in.
The other landscapes would need to progress, in a flowing manner consistent with the seen lay, through the full 360deg of sideways.

Interesting things to note using this approach are the that the lay of the line of site directly before the 3Der changes little even while rotating through the 4th dimension.
Primarily the front lay of the land would change little.
Areas away from this change to a greater degree due to changing land formations off in the 4th direction.

Hopefully this idea is okay with everyone here? Are there any concerns about it?

[quickfur]

Step 24. Shaping towards a 4D contour map for a 3Der:



The 3Der has a slightly different situation to the 2Der in that they can see a view before them that spans from left to right; which the 2Der can't see.
However, the 3Der still can't see the 4D landscape.
In the same manner, that we would rotate the 3D landscape into the 2Der's view, the 4D landscape can be rotated into the view of the 3Der.

Here I'd like to point out that the different "slices" as shown above should not be confused with rotation involving the vertical direction; the circle of 360° angles lie in the horizontal hyperplane perpendicular to the "up" direction.

[...] Interesting things to note using this approach are the that the lay of the line of site directly before the 3Der changes little even while rotating through the 4th dimension.
Primarily the front lay of the land would change little.
Areas away from this change to a greater degree due to changing land formations off in the 4th direction. [...]

This is interesting indeed. I'm trying to reconcile this with my own method (4D->3D projection) to understand what's happening here, and it appears that your approach is equivalent to taking a series of 2D slices of the 3D projection image, each slice of which intersects with the vertical axis, so that together they cover a cylindrical section of the projection image. That's a rather clever way of what amounts to showing what's in the 3D projected image.

[gonegahgah]

Here I'd like to point out that the different "slices" as shown above should not be confused with rotation involving the vertical direction; the circle of 360° angles lie in the horizontal hyperplane perpendicular to the "up" direction.

Exactly. Thanks QuickFur. Here's a diagram showing how for the 2Der we would create the view for them; just to give us a comparison again to our situation.

Step 25. Vertically rotated slices to give our 2Der a mental impression of the left-right lay of our world:



On the left it shows our 3D landscape and the 2Der but the 2Der can only see the vertical slice directly in front of them. The right picture shows how, using rotation, we would demonstrate the lay of the land to the 2Der. They certainly wouldn't see all that at once as it would just confuse them. They would only see the 0deg to start with. As they turn right into our 3D world they progress through the rotated slices; or that is how they develop an impression of our world. The land undulates before them from the 0deg slice through the intermediate slices until that shown at 20deg and so on through greater degrees of right turn. In this way they get to see the bulk of our world in a fairly understandable way for them. The ground under their feet stays pretty much the same and at the same height; but the land before them undulates as they rotate their 2D eyes right (or left) into our other 359deg of land. The reason I show the rotation from above is 1) that is the only direction the 2Der can think of other then forward and that way is already occupied with land which just confuses their mind if they try to superimpose them (they just can't get the notion of where our sideways is) and 2) that is where I wish to show shadowy objects so that they know how far to turn left or right. They know how far to turn left or right because they see the object tilted - depending upon how rotated to left or right it is - and it is not usual to see a house, for example, tilted at 45deg towards us. That would tell them that they have to turn 45deg to the right to bring the house into their plane.

This is interesting indeed. I'm trying to reconcile this with my own method (4D->3D projection) to understand what's happening here, and it appears that your approach is equivalent to taking a series of 2D slices of the 3D projection image, each slice of which intersects with the vertical axis, so that together they cover a cylindrical section of the projection image. That's a rather clever way of what amounts to showing what's in the 3D projected image.

Thanks QuickFur and correct, that is pretty much the technique of the method. I was starting to think this morning about how a tesseract would be represented using this rotated method. I'll draw some diagrams to show what I came up with, tomorrow hopefully...

[Hugh]

Here I'd like to point out that the different "slices" as shown above should not be confused with rotation involving the vertical direction; the circle of 360° angles lie in the horizontal hyperplane perpendicular to the "up" direction.

I'd like to thank gonegahgah and quickfur for their continuing efforts in this thread, it is fascinating trying to understand it.

When I see the picture of a 2Der with a 1D line in front of it, I think of a 3Der being able to move around the vertical 1D line 360 degrees and see the exact same 1D line from any of those directions.

Is it correct that a 3Der with a 2D plane of vision in front of it, could be similarly encircled by a 4Der being able to move around the vertical and see the exact same 2D plane of vision from any of those directions as well?

[quickfur]

[...] When I see the picture of a 2Der with a 1D line in front of it, I think of a 3Der being able to move around the vertical 1D line 360 degrees and see the exact same 1D line from any of those directions.

Is it correct that a 3Der with a 2D plane of vision in front of it, could be similarly encircled by a 4Der being able to move around the vertical and see the exact same 2D plane of vision from any of those directions as well?

Yes, that's correct. In 4D, rotation happens not around an axis, but a plane. (We like to think of rotation in 2D as happening around an axis, but actually that axis doesn't exist in 2D space; it's protruding into our 3D space. Rotation in 2D happens around a point.) So given any 2D plane, one can go around it in 4D in much the same way as we can walk around a pole (i.e. vertical line) in 3D.

[Hugh]

[...] When I see the picture of a 2Der with a 1D line in front of it, I think of a 3Der being able to move around the vertical 1D line 360 degrees and see the exact same 1D line from any of those directions.

Is it correct that a 3Der with a 2D plane of vision in front of it, could be similarly encircled by a 4Der being able to move around the vertical and see the exact same 2D plane of vision from any of those directions as well?

Yes, that's correct. In 4D, rotation happens not around an axis, but a plane. (We like to think of rotation in 2D as happening around an axis, but actually that axis doesn't exist in 2D space; it's protruding into our 3D space. Rotation in 2D happens around a point.) So given any 2D plane, one can go around it in 4D in much the same way as we can walk around a pole (i.e. vertical line) in 3D.

Okay, thanks quickfur! That's great to hear!

[gonegahgah]

Yes, that's correct. In 4D, rotation happens not around an axis, but a plane. (We like to think of rotation in 2D as happening around an axis, but actually that axis doesn't exist in 2D space; it's protruding into our 3D space. Rotation in 2D happens around a point.) So given any 2D plane, one can go around it in 4D in much the same way as we can walk around a pole (i.e. vertical line) in 3D.

That's quite interesting; didn't think of that. Food for thought... Thanks QuickFur & Hugh.

Step 26. Our 2Der in line with the side of a house rotating across face of the house:



Of course to examine how we would see a tesseract we grab the assistance of our 2Der again and observe what happens when he scans across a simple house of ours.
You can see that he sees pretty much a series of lines that could be deep squares to the 2Der. By the end of the scan they are actually seeing the corner but it still gives them an idea.

We must always remain aware that our 2Der can not think where sideways is so although it forms a nice picture for us; to the 2Der they are simply seeing a series of lines before their eyes. They must infer that it goes somewhere perpendicular but they can not produce the total image in their mind. If I had stacked the lines on top of each other this would have given us a better understanding of how difficult it is for the 2Der to understand sideways. To illustrate:



Though, to be fair, the 2Der does at least get to see a gentle morphing while scanning across the house. They see the grey wall as they turn, then the door suddenly pops into vision for a little while, before popping out of vision by grey wall again. The grey walls themselves pop into being. You see grass and blue sky, then the walls pop up, and as you continue rotating, the walls at some time are replaced by grass and blue sky again. In fact, it is a little like our time viewers that I depicted just recently. Objects pop into existence and then out of existence from their beginning 'edge' and at their ending 'edge'. And, like the magic time viewers, our enabled 2Der can scan backwards and forwards across these frames. Though, unlike our magic time viewer, our enabled 2Der can still only see one frame at a time.

The interesting thing for this is that the 2Der is seeing the square from the same POV as us but each viewed line is unconnected. They see multiple single lines whereas we see a continuous spread of non-discernible lines; or a square. Our perception is of a whole; their perception is of a sequence of discrete lines. Or more accurately, they see a line of view with lines that morph before them in length or suddenly appear - while they turn - and then suddenly disappear - after turning a bit further. This is the difference between a 2Der, under these circumstances, and us.

This also has consquences when it comes to upscaling our view to 4D. I'm certain, with a few more pictures, that this will help us to better understand how 4D 'flat' (read 3D) objects still only have - what can be considered to be - two main views to a 4Der - just as we see a square as having two main views and a 2Der sees a line as having just two main views. I'm fairly certain this will become clear with the next few pictures and will then also help us to understand the nature of 4D roads, etc...

[gonegahgah]

Step 26b. 2Der looking at 3D house via corner:



Here is a 2Der's view before a 2D computer screen that rotates them through our sideways dimensions. They see the grass and sky and as they rotate they pass through: grass -> grey wall -> darker grey wall -> door -> back to just wall -> and finally back to grass on the other side of the building. Again their view is very similar to what we would see except it is unconnected except as a morph; unlike what we see which is connected as a whole.
It is the extra-connectedness that gives an object is appearance of 'bulk' at each level of higher dimension.

[gonegahgah]

Today we are going to go where no 2Der has gone before.

Here is a picture of a 2Der attempting to look at a 3D building:



Normally the 2Der is limited to seeing by the following means:



Our evil scientist is going to dissect a few 2Ders (he'll recruit some volunteers for a secret lifetime mission and drag them off into our 3rd dimension never to return).
He removes several 2Der eyes and optic nerves and combines them into this monstrosity:



Each column of eye seeing cells lies in a different plane slightly rotated to the previous planes.
The mad scientist wires these into a poor 2Ders brain who is just developing sight.
Although our 2Der is still stuck in their 2D plane they grow up to be able to see across a section of our 3D plane.
Which is a bit confusing to them as they can't touch these things they see except directly vertically in front of them.



The scary thing is, though this is very science fiction, something like this could possibly be simulated one day.
One day we may be able to create a computer simulation of a 4D spatial environment and wire a human up to it.
The human brain has a high degree of plasticity so it wouldn't surprise me if our brain could accept this stimuli and accept it as real.

We learn direction by associating movement and sensation with vision (or sense if we are blind). This is how we conceive left and right when our 2Der counterpart can't.
If, in a computer simulation, we could seem to move with 360deg of sideways freedom, as well as forward-back and up-down, then we may very well believe it to be real.
Our brain might be equipped to fully deal with whatever environment it encountered.

The only obstacle would the increased processing required which would probably be our current visual processing requirements multiplied by the squared root of our current requirements. Maybe? Or maybe it would even be more? I'm not sure...

[gonegahgah]

There is one thing that I have been trying to reconcile in my mind and that has been how many eyes are needed by our 4Der friend.
This has been a puzzle in my mind early in the peace and now, for the last couple of days, from discussions I have seen here.
The stumbling block has always been that it seems unavoidable that a 2Der needs to have two eyes (one above the other) to see distance just as we do.

It is something that I had been planning to explore through future diagrams - that I have planned for other purposes.
Something occurred to me tonight that maybe extends upon the debate.

First off I have been pondering if the 4Der needs an extra cue because of the 360deg of sideways. That thought wasn't enough to help me think any further by itself.
I was then deliberating on some other facet and, as an example of that facet, I was reminded of how much we struggle to determine how far things are away or how big they are in space or in blizzards.
It would seem that we do rely on being able to see the horizon or other objects to help us with determining some of the information that we need.

So where a 2Der needs two eyes to pinpoint things, I wonder if having three eyes wouldn't help us to pinpoint things a bit better then we can?
If we had a third eye above our two other eyes; would this eliminate the size illusions we experience - due to our reliance on information provided by the horizon?

This actually came into my thoughts aside a sudden realisation that when we look forward there can be a series of focus lines in front of our eyes that are the same distance from both our eyes for their lengths. These lines form a circle before and over us vertically. I'll draw some pictures later if it seems a good idea.

When the 2Der has two eyes there are no two points that are identically defined by the same dual focus of their eyes.
This is not the case with us - and we must rely on up and down to determine where also they lay in relation to us. This doesn't seem to be as acute as our left-right dual focus.
As I said, our eyes usually rely on relative environmental information to help sure up this bit of information.
If we had three eyes we might have no such problem.

The situation grows more complex for the 4Der who, if I understand correctly, actually has a square of points before their eye that are all equal distant from their eyes.
As they can see all that square space at the same distance, I wonder if it is possible, without other cues, for the 4Der's eyes to discern which sideways things are without having an extra eye - along with the assistance of their horizon. I wonder if a 4Der in space would be even more confused and if having four eyes would be an ideal number for them in space.

So now I'm wondering if for space it would actually be better if our various dimensional people actually had:
2D: two eyes
3D: three eyes
4D: four eyes?

I still get this feeling that a 4Der will need three eyes to more successfully cope with the extra 359deg of sideways... but I am not convinced yet either; either way.
As I said it is something I hope to show conclusively - either as true or false - in upcoming pictures.

Does this add anything to the debate at this stage?

[wendy]

Using the model that the eyes form a simplex in the across-space, then the Nd person would have N-1 eyes.

The across-space corresponds to all-space, less height and forward. We have two eyes, which lie in the same line, perpendicular to height + forward.

Only one eye is needed to see things. Many 'prey' animals have an eye on each side to search for preditors.

Two eyes are needed to create a stereo effect. However, this is most intense in the same 2-space that sets height/forward.

Three eyes in 4d would give better triangulation of the across-space, but ye have also to consider what happens when the eyes give confilcting messages.

[gonegahgah]

Cool, thanks Wendy. I thought about the aspect of triple focus, if that relates, but I think that should be okay because we already rely on both our eyes to work together.
When they don't it makes it difficult for us to focus on a point.
We build up this ability to dual focus as a growing experience from fairly young. I imagine that a 4Der would develop the triple focus through the same process.
It's an interesting process in itself where we move our hands randomly at first and start to realise that contact with things happens at a particular point in relation to our vision. It takes a little while for controlled vision to kick in when we are born via this process of trial and error.
As you mention, any mix up of one or two eyes would certainly make it more difficult for them to focus on a particular point.

[quickfur]

[...]

[...]

Aha! I think I can see where this is going. You're thinking of a simulated scenario (say, some kind of virtual world simulation on the computer) where the user only gets to see a 3D scene, but this 3D scene is a slice of the 4D world? And the user gets an extra "hyperspatial movement" dial that lets him rotate himself through 4D, so that he can "scan" the 4D scene left-to-right (or rather, ana-to-kata) as it were?

I like this idea, actually. I think it's a very good way to introduce a native 4D environment to someone new to the concept.

[...]

[...]
The scary thing is, though this is very science fiction, something like this could possibly be simulated one day.
One day we may be able to create a computer simulation of a 4D spatial environment and wire a human up to it.
The human brain has a high degree of plasticity so it wouldn't surprise me if our brain could accept this stimuli and accept it as real.
[...]

AFAIK, there are two schools of thought on this. One is that our visual apparatus is hard-wired to work with 3D, and no matter what we do, we'll never be able to perceive anything else; the other is that the brain just receives a bunch of signals from the optic nerves, and over time, through an innate instinct to rationalize the input, spontaneously develops a 3D interpretation of these signals.

There is some evidence for both: in relation to the former is the way certain structures in our optic nerves seem to be inherent to how stereopsy (the perception of depth from binocular disparity) works. This seems very specific to 3D vision. On the other hand, it has been proven that people who are given lenses that inverts the image their eyes see (so that the image that forms in the retina is "right side up" instead of the normal "upside-down") eventually learn to "flip" the images subconsciously, and have no problems with sight/motion after the initial period of fumbling.

I personally lean somewhat towards the second camp, which is why my interest got really piqued when I chanced upon a report some years ago that an experiment was done where the subjects were put in an immersive VR environment where they eventually learned to navigate in 4D space and manipulate 4D objects (assembling 4D blocks to form certain target shapes). However, I could never find any details on this experiment -- what exactly the VR environment was, how the environment was presented to the participants, how they interacted with the simulated 4D environment, how complex the maneuvers were, etc.. But it does seem to show that our brain is much more flexible than those of the first camp appear to think. Given the right kind of stimulus, we may very well be able to wire ourselves to a "native" 4D environment (i.e., simulated 3D retina) and thereby see 4D "natively".

[quickfur]

On the subject of eyes: I've thought about this many times in the past, but never really could settle on any single answer as to the optimal number of eyes in 4D.

Obviously, 1 eye is enough if we rely only on foreshortening to perceive distance, but this is fraught with illusions, a prime example being the moon, which is too far away for our binocular vision to perceive its true distance, and hence the illusion that it appears smaller when high in the sky but bigger when close to the horizon -- actually, the circular image of the moon that falls on our eye has exactly the same radius in each case, but in the latter case the presence of familiar objects such as trees or buildings recalibrate our brain's perception of its size (i.e., the brain can't figure out just how much foreshortening has occurred and changes its guesstimate when presented with objects of known size nearby).

Two eyes will give us stereoscopic vision, but it is unhelpful in the directions perpendicular to two lines of sight (e.g. up/down for us). It is instructive to note, though, that binocular disparity occurs with two eyes regardless of dimension (two points plus target object = triangle, and you can triangulate distance in any dimension with just 3 points), so stereopsy is possible with only two eyes. The question, though, is whether the disparity is sufficient to give reliable depth information in all directions.

What about animals like flies, that have compound eyes? Do we know how the multitude of images help it determine distance accurately?

[gonegahgah]

Aha! I think I can see where this is going. You're thinking of a simulated scenario (say, some kind of virtual world simulation on the computer) where the user only gets to see a 3D scene, but this 3D scene is a slice of the 4D world? And the user gets an extra "hyperspatial movement" dial that lets him rotate himself through 4D, so that he can "scan" the 4D scene left-to-right (or rather, ana-to-kata) as it were?

Exactly QuickFur but also with the addition of extra cues that help the user to feel that they are in a 4D environment and where they are. These include shadowy images of buildings and landmarks that would otherwise be hidden in the 4th dimension. Adding tilt with rotation in the sky, or underground, to familiar objects would give the user the direction these objects are in 4D. Care would have to be taken with overcrowding especially initially and for awhile into the game. Perhaps, as the player grows more familiar with 4D, it could be part of the increasing challenge, as they progress, to have some crowding as one type of challenge? I've already thought of some scenarios that would provide for the player to find themselves in the 4D world in an interesting way. Initially they would find themselves in a 3D maze of corridors where they appear to be trapped with no apparent way out...

I'm also suspecting that we will be able to see that familiar objects, such as a tesseract, have more than one cube 'face' (or 4-face) and get some notion of how the 4-faces relate and move together through this approach hopefully; though I have to be careful of overcrowding. I'm hoping so and am keen myself to see if this is going to eventuate.

I like this idea, actually. I think it's a very good way to introduce a native 4D environment to someone new to the concept.

Thanks QuickFur. I'm hoping that it will help us to interact with it in a meaningful way and allow more people to get more used to and proficient with the environment.

On the other hand, it has been proven that people who are given lenses that inverts the image their eyes see (so that the image that forms in the retina is "right side up" instead of the normal "upside-down") eventually learn to "flip" the images subconsciously, and have no problems with sight/motion after the initial period of fumbling.

Cool, that sounds a lot less drastic than what I was proposing . Wow, I hadn't thought of that more practical approach. That's awesome to hear.

I personally lean somewhat towards the second camp, which is why my interest got really piqued when I chanced upon a report some years ago that an experiment was done where the subjects were put in an immersive VR environment where they eventually learned to navigate in 4D space and manipulate 4D objects (assembling 4D blocks to form certain target shapes). However, I could never find any details on this experiment -- what exactly the VR environment was, how the environment was presented to the participants, how they interacted with the simulated 4D environment, how complex the maneuvers were, etc.. But it does seem to show that our brain is much more flexible than those of the first camp appear to think. Given the right kind of stimulus, we may very well be able to wire ourselves to a "native" 4D environment (i.e., simulated 3D retina) and thereby see 4D "natively".

I lean towards the second camp more too. I wonder if they were able to do that; sounds cool. I wonder what model they would have used too.

Just before I read this I was working on the following picture as well:

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



This shows how 3Der hooked up to a box that is receiving multiple pictures from a great number of rooms. You can see just some of the rooms. The middle rooms have a cube on the table, and then some rooms appear that have an empty table, and then no table at all; until you hit the last room into the 4th dimension. The reason that there are empty tables is because the tesseract only has a certain length into the 4th direction equal to its lengths in our 3 directions. Just like our table has space to either side of the cube; so would the tesseract only occupy so much space of tables into the 4th direction.

The number of rooms needed would probably have to be equal to the number of seeing cells in our eye multiplied by the square root of that number. This should hopefully give a number equal to the number of eye cells needed to give us a 4D eye.
You might need to multiply this again by a further rotational factor to allow us to move through the 4th direction as well and allow different views of the cube from different 4th dimensional angles. You would also need to multiply this again by another formula to allow for close-up viewing.

Either way, that is a lot of rooms. Maybe some redundancy would help to reduce that number but it's still probably a lot.
All the images from the different rooms, as necessary (or they turn off), are fed to the apparatus on the table and these are then sent directly to the 3Der's brain to process.

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

What it does hopefully show us is that we would no longer see individual face squares of a cube; just as we don't see individual lines when looking at a square.
Our brain would cement all the squares together, of each forward available face, and these would form a cube in our brain; but one where we can see the same corresponding faces from each square. No matter how we turned the tesseract we would always be compounding the same faces together.
This is what gives a cube one of its two 4-faces in 4D; that is the compounding of all these square faces that all face the same outward way.

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

I'd have to think about how to fix this. At the moment I'm not sure how to fix that problem with this multi-room approach yet.

I think the problem is us. We can still only look at the cube (even across rooms) across our 3 available directions and we need to be able to look at it also across all 4 available directions. Ah, I think that's it. I think we need an extra eye and we need to place it in a series of parallel rooms that are looking at the cube from a different aspect. I'll have to think some more about that though...

I'm not sure if that relates to the debate on the need for three eyes yet - though I had already been pondering about where the 4Der's third eye would lay if they had three eyes. If two of their eyes crossed our 3D plane, the third eye would not; and would be somewhere off, at the same distance, in the extra 359 deg or sideways; most likely forming an equilateral triangle arrangement of eyes. Maybe, when their 3 eyes lay in our 3D plane they only see a single cube; just like us with our two eyes. More stuff to think about...