Mobile game with real 4D tesseract

[globex]

Hey guys,

I've just spend the last year developing a mobile game that uses the a true-to-physics 4D tesseract as the main game piece.

It took us a really long time to figure out how to make this work in a way that would make sense to the average person. I think we found a really clever solution. What do you think?

PS. It's free to download and play up to 3000 points.

http://www.tesseric.com
iOS App Store: https://itunes.apple.com/us/app/tesseric/id621752516?ls=1&mt=8
Google Play Store: https://play.google.com/store/apps/details?id=com.lexansoft.tesseric

[Polyhedron Dude]

This looks interesting, any chance there is a Windows version - I don't have any of the handheld devices.

[globex]

We're considering a Windows release in the near future. The game was designed primarily for a touch interface, so we may need to do quite a bit of tweaking to find a nice way to play with a keyboard/mouse.

[gonegahgah]

Although this probably works as a puzzle game I suspect it isn't truly representative of a spatial 4D world; even one that is fixed to 8 axial directions.
The reason is that I can't see why it is necessary to also move in the 3D space to move into the 4th axis.
Perhaps you can explain globex? Or anyone else here?

[Secret]

So basically speaking, it's a tesseract block resting on a swock (temikaria's term for 3D paper like object) (chorix in Wendy's terms, rind in my terms if it's curved, hypersurface in 4D, cells in quickfur's and wikipedia's terms) and rolling on it in 3 orthogonal direction, and you, the player, looking at the whole scene in a bird's eye view from the w direction

It is truly 4D in the sense that the tesseract itself and its rolling on the swock is 4D (since the rotations causes the color panals (which are actually the cubic cells of the tesseract projected into 3-space) to switch in a manner that only a 4D rotation can bring along (I am not familiar enough with rotational groups to state any further about SO(4)), but in terms of movement, you are restricted to 3D space

It can theoretically be extended to 4D movement but then you will need a way to keep track of the hypersurface on a 2D screen. Globex already use multiple 2D planes to keep track of the location of the tesseract along the z axis so some similar extension (make multiple copies of the playfield with something to tell the players their whereabouts, for example) will be required to show the position of the tesseract along the w axis

[quickfur]

At first I was gonna say that this is a bit disappointing since you can't move around in all 4 directions in 4D space, but on second thoughts, this isn't *too* far from what a gravity-bound Tetronian would experience. They would be confined to the surface of a 4D planet, which at the human scale is basically a 3D hyperplane, and they can roll large tesseracts around while standing on it, without needing to actually move up/down in the 4th direction.

So this game is like the 4D equivalent of a 2D game, where the game world is represented by an overhead view, and movement is restricted to the (hyper)plane. Just like in 2D games, you can add a "fake" 4th direction to it by various tricks, like stairs / ladders that lead to upper/lower floors of a 4D building (basically "portals" that take you between different 3D hyperplanes), etc..

But a "true" 4D game would be like John McIntosh's 4D maze game, where you actually see a projection of a 4D maze and can move in all 4 directions. (It's also extreeemely confusing if you don't keep a fixed vertical orientation, since your brain will just turn into a pretzel trying to figure out what goes where in a full 4D space while being in any one of the 192 orientations possible in 4D. )

[wendy]

It's actually a 3d game, in spheric space. There the tesseract x4o3o3o is a tiling, not a polytope.

In that regard, it's kind of like the usual fare of playing games on a torus or cylinder (KQ3, KQ4).

[gonegahgah]

Another contradiction that I'm guessing exists ties to the use of 8 colours in this game. It would appear that you have to roll to objects in 8 alternate 3D planes.
In a 4x grid in any dimension adding an extra dimension should add only 4 new rows; not eight as this game appears to add.

For example:
- a 1D world of lines would have 4 lines spaces end on end that could be shifted up-down to with our line player.
- a 2D world of squares would have 4x4 square spaces in a grid that could be slid to either up-down or forwards-back.
- a 3D world of cubes would have 4x4x4 cube spaces in a grid that could be rolled to either up-down, forwards-back or sideways.
- a 4D world of tesseracts would have 4x4x4x4 tesseract spaces in a grid that could be rolled to either up-down, forwards-back, sideways 1, or sideways 2.

So in reality there are only 4 rows added in the 4th axis; not 8 as I think this game is providing.

The other error I think is that you can not immediately jump to any other row into the 4th axis; just as you can't in the lower axes dimensions.
For example if you are at the left of the cube you can not immediately jump to the right of the cube.
Instead you have to pass through the second row then the third row before you make it to the opposite side row.

So depicting immediate jumps to all tesseract rows in the 4th axis is wrong I'm guessing?
Instead you must move only to adjacent rows and this doesn't include moving directly between the opposite '4-faces' which are not adjacent to each other?

The game already has movement in 3 axes in both their opposites (ie. left-right, up-down, forward-back). They need only provide movement into the 4th axis in its two directions (ana-kata).
Instead they provide 6 other directions of movement; as if by hyperspacing perhaps; I don't know?

[Secret]

There are 8 colors because the tesseract has 8 cubic cells

Tesseric is a 4D version of a die rolling game, where the "die" here is a tesseract and your mission is to roll so that the die rest on its various colored "3D side" on the "3D floor"

As Wendy had pointed out, it is basically a 3D game in terms of degrees of freedom, since you can only move up/down, left/right or front/back, but as quickfur pointed out the tesseract is rolling on the floor like a 4Der confined by gravity to a 3D floor does

[gonegahgah]

I wanted to distinguish the 4th axis of movement and its absence; though after reading the answers here I noticed how the playfield itself was locked to a 3D space.
It is nice to have the 4th direction of movement as in John McIntosh's 4D maze as highlighted by quickfur. I think that's more the Holy Grail when we talk about 4D games.
Calling "Tesseric" a "true 4D arcade/puzzle" probably exceeds our definition.
The OP here does only refer to the main game piece as "true-to-physics" but the web page isn't as clarified.
The tesseract die rolling notion is still interesting though. Anyone played this?

It should however be possible to make a simple 4D 4x4x4x4 tesseract space visually accessible.
Rather than exactly superimposing the four rows of cubes, if we use a small but distinguishable angle of rotation, this should allow the 4th axis rows to be visually distinguishable from each other.
You could simply use four colours but rotation allows distinguishable objects to be superimposed and simultaneously seeable as being in a difference 4th axis row.

I have a feeling that the above approach would make it too easy to move around a 4D space and making a similar puzzle game in a 4x4x4x4 space would not be as likely?
Would we lose the sense of 360° of sideways in the above approach as per the 192 orientations mentioned by you quickfur?

Perhaps a snake game could be made using rotation to distinguish which 4th axis row we and objects are in?
The more squares you add the more visually confusing it is going to be as you add in an extra number of fine rotations to cover the additional rows; which I guess could be part of the playing challenge?

[quickfur]

[...]
It should however be possible to make a simple 4D 4x4x4x4 tesseract space visually accessible.
Rather than exactly superimposing the four rows of cubes, if we use a small but distinguishable angle of rotation, this should allow the 4th axis rows to be visually distinguishable from each other.
You could simply use four colours but rotation allows distinguishable objects to be superimposed and simultaneously seeable as being in a difference 4th axis row.

If your goal is to make a tesseractic grid visible, there are plenty of ways to do it. I even wrote a 4D game some years ago that does this in ASCII: you just represent it as a 4x4 grid of 4x4 grids (sorta like 4D tic-tac-toe, if you will). You can play 4D board games this way, and you can make an RPG out of moving pieces on the board.

I have a feeling that the above approach would make it too easy to move around a 4D space and making a similar puzzle game in a 4x4x4x4 space would not be as likely?
Would we lose the sense of 360° of sideways in the above approach as per the 192 orientations mentioned by you quickfur?
[...]

Well, if you're making a grid-based game, then orientation wouldn't be an issue.

I did add gravity to a variant version of my 4D game, which makes it interesting as you can only walk to a particular position if there's a solid tile underneath you, otherwise you'd fall down to the next floor. But as long as the ground beneath you is solid, you can move around on a full 3D floor freely, which is quite interesting since it allows a lot of things to "hide in plain sight" -- passages leading off a 3x3x3 room that you miss because you're not used to thinking about the floor itself being a 3D (hyper)surface. One-square-wide passages can also twist in amazing ways, even when gravity is present: I made it so that you can climb up a 1-square high tile if the top of the tile is vacant; so you could climb up and down "stairs" made of 1 tile blocks even though you'll fall through empty space. So a passage can have an extremely convoluted twisting as it spirals upwards, winding around all 4 directions, and you can have gaps on either side so if you're not careful you'd just fall back down to the ground floor again. You can also experience, first-hand, how a road built on the 3D floor can have another road encircling it, yet the two never intersect. Or how you can walk around a river without ever needing to cross it. Or how a 2D wall is not enough to block movement "around" it.

Of course, the grid-based view doesn't really give you a good feel for how a 4Der would see these things, but at least it does give you a good idea of how they're actually possible in 4D space in the first place. My dream is to one day write a 4D game that represents what you see as a 4D -> 3D projection (the same way a native 4Der would see things), so that you can almost "experience 4D first-hand". Sorta like McIntosh's 4D maze game, but with proper surfaces instead of just a tangle of lines. But that's probably years, if not decades, away.

P.S. Also, if you introduce gravity, then you won't have to deal with the 192 orientations, just 24.

[gonegahgah]

If your goal is to make a tesseractic grid visible, there are plenty of ways to do it. I even wrote a 4D game some years ago that does this in ASCII: you just represent it as a 4x4 grid of 4x4 grids (sorta like 4D tic-tac-toe, if you will). You can play 4D board games this way, and you can make an RPG out of moving pieces on the board.

Certainly is one way to do it for a grid situation. It highlights the situation of depicting 4D where it can take a lot of space to do so even for a grid scenario.

I did add gravity to a variant version of my 4D game, which makes it interesting as you can only walk to a particular position if there's a solid tile underneath you, otherwise you'd fall down to the next floor. But as long as the ground beneath you is solid, you can move around on a full 3D floor freely, which is quite interesting since it allows a lot of things to "hide in plain sight" -- passages leading off a 3x3x3 room that you miss because you're not used to thinking about the floor itself being a 3D (hyper)surface. One-square-wide passages can also twist in amazing ways, even when gravity is present: I made it so that you can climb up a 1-square high tile if the top of the tile is vacant; so you could climb up and down "stairs" made of 1 tile blocks even though you'll fall through empty space. So a passage can have an extremely convoluted twisting as it spirals upwards, winding around all 4 directions, and you can have gaps on either side so if you're not careful you'd just fall back down to the ground floor again. You can also experience, first-hand, how a road built on the 3D floor can have another road encircling it, yet the two never intersect. Or how you can walk around a river without ever needing to cross it. Or how a 2D wall is not enough to block movement "around" it.

Cool.

Of course, the grid-based view doesn't really give you a good feel for how a 4Der would see these things, but at least it does give you a good idea of how they're actually possible in 4D space in the first place. My dream is to one day write a 4D game that represents what you see as a 4D -> 3D projection (the same way a native 4Der would see things), so that you can almost "experience 4D first-hand". Sorta like McIntosh's 4D maze game, but with proper surfaces instead of just a tangle of lines. But that's probably years, if not decades, away.

I would like to see that as well. I would also like to have a game one day that incorporates 4th axis visual 'rotation' to help convey a greater sense of 4th dimensional bulk and position.
That should hopefully make it easier to manoeuvre around I'm hoping.

P.S. Also, if you introduce gravity, then you won't have to deal with the 192 orientations, just 24.

I was wondering about that. Gravity does simplify things for depicting 4D.
[Sadly we seem to have to fake it and use 3D gravity - as per other discussions - otherwise possibly all we might need to depict is an amorphous squash of nothing useful!]

[quickfur]

[...] P.S. Also, if you introduce gravity, then you won't have to deal with the 192 orientations, just 24.

I was wondering about that. Gravity does simplify things for depicting 4D.
[Sadly we seem to have to fake it and use 3D gravity - as per other discussions - otherwise possibly all we might need to depict is an amorphous squash of nothing useful!]

3D gravity? Why?

There's nothing wrong with using 4D gravity on a planet that exists by fiat. A game doesn't have to explain everything, like how the planet can have a stable orbit. In fact, we don't even need to invoke planets. Just say the ground is an infinite 3D hyperplane that you're standing on, and you're good to go. Unless you're making a game about 4D astronomy, this is more than good enough. As for how there can be sunrise/sunset with a (hyper)planar world, see "doesn't have to explain everything".

Worrying about 4D "realism" in a game intended to help you understand 4D via dimensional analogy (which is basically what this is all about, ultimately -- since there's no guarantee that a truly "native" 4D physics will be anything even remotely comprehensible to us --), is a bit like starving to death because you're too busy worrying about the exact chemical composition of your lunch. I say just draw the usual simple 3D->4D analogies, and leave it at that. It's not as though that doesn't already introduce enough "strange" features that players will find hard to understand. One prime example that comes to mind is how, in 3D, if you look through a doorway at the side of a corridor, you only see (parts of) the ceiling, the floor, and the far wall. But in 4D, you actually see *three* of the corridor's 4 walls (or parts of them thereof), two of which are in an orientation that most people would find rather foreign and counterintuitive, and would have trouble interpreting.

[gonegahgah]

In 4D we depict corridors as 4 walls - which are made of square prisms; well square prism faces; our existing walls are rectangular prisms with rectangle faces - to define a corridor, along with a floor, ceiling, entrance and exit.
Generally a corridor is depicted as empty and walls as solid.
So in particular 3D views I imagine that you will see just walls and no actual corridor?

Twisting our view 'opens' the corridor which, as per our corridors, is uni-directional dimensionally. ie you travel from one end to the other end.
4D rectangular prism corridors benefit from allowing doors to exit not just left or right but any of 4 sideways directions.

If you have circular walled corridors then you could have a door that is continuous around the 360°?
Such a door would open with no problems; wouldn't it?
Problem with square prism faced corridors is they tend to suggest rooms off to left-right-ana-kata whereas you can have rooms off in any of the 360° of sideways directions. There is just so much more space.
That limitation is similar to us thinking like a 2Der and having a corridor up the centre of the house with doors only at the ends; and no side doors...

Even if we had square prism walled corridors the doors would not need to be out of one wall but could be out of one, two, three or all walls at once and still open no problems? Is that correct?

[quickfur]

In 4D we depict corridors as 4 walls - which are made of square prisms; well square prism faces; our existing walls are rectangular prisms with rectangle faces - to define a corridor, along with a floor, ceiling, entrance and exit.
Generally a corridor is depicted as empty and walls as solid.
So in particular 3D views I imagine that you will see just walls and no actual corridor?

Here's the thing about sight in general: it's impossible to see empty space. What we imagine as empty space is actually a construct of the mind, not something actually in the visual images that our eyes see. Think about a 3Der looking down a corridor that, say, branches off to the right into a perpendicular corridor. We unconsciously understand that there's an empty space in the wall where the side corridor begins, where one may pass through. But look again at what you see from a 2Der's POV. Where's that empty space in the wall? The 2Der sees no such thing. Instead of an empty space where the side corridor is, he just sees a pattern of polygons embedded in the image of the right wall: a triangular extension of the ceiling's tetragon (the ceiling of the side corridor), a square (the leftmost part of the corridor's far wall), and a triangular extension of the floor's tetragon (the floor of the side corridor). What empty space? Where? There is no empty space at all!

The empty space exists only in 3D, you see. Your brain has to be able to interpret the image in a 3D sense before you can understand how there can possibly be an empty space in the right wall, and how that triangle-square-triangle configuration represents, as a whole, the start of a side corridor.

It is analogous when you look down a 4D corridor. The corridor itself appears not unlike a tesseract projection, with 6 frustum-shaped volumes that converge on the "inner cube" (representing the far wall of the corridor, or just a point if the corridor is too long for you to see the end). The top and bottom frustums are images of the ceiling and floor, respectively, and the other 4 are the corridor's side walls. The side corridor appears as a configuration of polyhedral cells: 4 parallelopiped-shaped volumes surrounding a cube that's embedded in one of the side wall frustums. But how does this configuration of cells even represent the empty space that opens out to a side-corridor? From our 3D-centric POV, there is no empty space anywhere, even just in the main corridor, let alone that complicated-looking arrangement of polyhedra on the side. For all we know, that could represent some complicated object stuck on the wall; how are we to know that it's supposed to represent empty space opening out to a side corridor? What empty space, where?

Now, if we apply dimensional analogy, then we can understand that the cube is the image of the beginning of the far wall of a side corridor, and that the top and bottom parallelopipeds are the images of the ceiling and floor of the side corridor. But what about the 2 remaining parallelopipeds? What are those? It requires a bit of effort to analyse the situation to realize that actually, they are two other walls of the side corridor. But wait... how can we possibly see three walls of the side-corridor's 4 walls? From our 3D experience, the only wall of the side corridor that's visible when seen from that angle is the far wall, because that's the only wall that's facing us. But now in 4D, there are three walls that are facing us?! How can that be? No amount of dimensional analogy will help you with this one: this phenomenon only occurs in 4D (and above). It's something totally foreign to our 3D sensibilities. It's hard enough already to perceive empty space where we don't see any, and now we have to grapple with interpreting things that have no analogue in 3D? Yep, that's why I said that we don't even need to think about exotic things like modified 4D physics that allows orbits to work; even the "simple" generalizations of 3D phenomena to 4D already introduces enough new concepts that the prospective 4D game players will have to come to terms with.

Twisting our view 'opens' the corridor which, as per our corridors, is uni-directional dimensionally. ie you travel from one end to the other end.
4D rectangular prism corridors benefit from allowing doors to exit not just left or right but any of 4 sideways directions.

The possibilities are far more exciting than that. Corridors don't have to be rectangular... you can have hexagonal corridors (or any polygon, for that matter) which allows 6 doors around a single point in the corridor at a time, yet the corridor can still stretch forwards and backwards without interruption.

You can even have a cylindrical corridor, as you point out below, (or more accurately, a cubindrical corridor) in which case doors can be placed in any of the 360° of lateral directions.

Even better yet, there's at least another kind of cylindrical (cubindrical) corridor, in which the orientation of the cylinder is sideways. In this case, the corridor will have two cylindrical walls and a kind of rounded 3-manifold surface that wraps around the corridor from ceiling to floor, covering the space of where the other two side walls would be. Then you could either have doors in one of the cylindrical walls, or in the wrap-around surface. This kind of corridor has a curved floor, though, so it's sorta like a tunnel in 3D... but whereas in 3D a tunnel only has rounded walls, the 4D tunnel has two flat walls in the shape of cylinders. Weird, huh?

If you have circular walled corridors then you could have a door that is continuous around the 360°?
Such a door would open with no problems; wouldn't it?

Of course. And speaking of doors, here's another interesting thing: in 3D, doors have 1D hinges: that is, the door rotates around a line (the rotational axis) when you swing it open/shut. In 4D, doors have 2D hinges: the door rotates around a plane when you swing it open/shut. So whereas 3D doors are rectangular in area, 4D doors are cubical in volume (or cuboidal). Merely rectangular doors wouldn't be doors in 4D, since they wouldn't stop you from just walking "around" them!

Problem with square prism faced corridors is they tend to suggest rooms off to left-right-ana-kata whereas you can have rooms off in any of the 360° of sideways directions. There is just so much more space.

On the other hand, though, they do allow more consistent use of space, since if the doors are too close to each other, that constrains the size of the rooms behind them. And while cylindrical corridors are cool, the side rooms will have to be odd-shaped in order to tile the floor space, otherwise there'll be gaps between the rooms that will be wasted space.

That limitation is similar to us thinking like a 2Der and having a corridor up the centre of the house with doors only at the ends; and no side doors...

Even if we had square prism walled corridors the doors would not need to be out of one wall but could be out of one, two, three or all walls at once and still open no problems? Is that correct?

Of course. You could have an octagonal corridor with 8 doors around a single point. There's plenty of space for them to open (though if they are to open all at the same time, some of them would have to open inwards instead). Or any n-gonal corridor, for that matter. It's just that certain values of n works better, because it allows you to subdivide the floor space of the side rooms more efficiently. Like n=4 (cubical tiling of floor space), n=6 (hexagonal prism tiling of floor space), n=8 (octagonal-square prism tiling, the square prism parts can be treated as alcoves leading to an octagonal-prism room deeper inside). Triangular corridors (n=3) also work, because it can fit into the truncated-hexagonal tiling of floor space (so the triangular corridor is surrounded by 3 dodecagonal-prism rooms -- the corridor will be narrow but the rooms spacious). Most other values of n (including n=infinity: cylindrical corridor) leads to difficulty in evenly subdividing the floor space for the side rooms, so they don't really work that well in practice, even if they are cool in theory.

Another way to arrange rooms is to use a spiralling arrangement of doors, so that the doors are all right next to each other, but spiral around the corridor as you go down. The spiralling allows the side rooms to access more lateral space, so they can still have plenty of space; it's just that they will have a "skewed" orientation relative to the direction of the corridor. Not that this has ever stopped modern architects in our 3D world from designing buildings where rooms have unusual angles.

Boring ol' rectangular corridors (n=4) are still the most practical, though, because cuboidal subdivisions of floor space are the easiest to maximize the usable space for. The other "cool" shapes are cool in theory, but they're an annoyance when you're trying to maximize usable floor space. (If you've ever lived in a room that isn't rectangular but has odd angles, you'll know how annoying it is to figure out how to arrange the furniture to even fit, let alone maximize usable space!)

[gonegahgah]

Just thinking about it - and following this precept of dividing rooms into tesseracts (or their equivalent rectangular prism) - I believe that they would extend our principle one step further.
In 2D there is no need for halls at all as they would be wasted space.
In 3D we have halls because these can lead off to the left and right side rooms.
In 4D we have halls which then branch into offshoot halls which then lead off to their up to 4 side rooms so as to allow better compactification and use of all the additional space available?
Thoughts?

[quickfur]

Just thinking about it - and following this precept of dividing rooms into tesseracts (or their equivalent rectangular prism) - I believe that they would extend our principle one step further.
In 2D there is no need for halls at all as they would be wasted space.
In 3D we have halls because these can lead off to the left and right side rooms.
In 4D we have halls which then branch into offshoot halls which then lead off to their up to 4 side rooms so as to allow better compactification and use of all the additional space available?
Thoughts?

Well, in 2D there can be no halls (assuming the circular planet + gravity model, of course -- with Edwin Abbott's flatland model you can have halls).

In 3D, actually, we already have halls that branch into other halls -- you see this in large buildings and places like hospitals where different corridors lead to different subsections which have their own corridors. Of course, you can't take that too far, since it quickly turns into a labyrinth and becomes more confusing than useful. If you visit those old castles, though, some of them are arranged like that, and some even have secret passages.

Speaking of which, in 4D there would be so much more room for unsuspected secret passages. One interesting thing about 4D is that you can have two (or more) completely disjoint corridor networks intertwining each other on the same floor (just like you can have disjoint yet intertwining road systems). So you can have an entire secret passage network that runs through the building just as extensively as the "public" corridor network, and nobody would know any better!

You do have a point, though, that branching corridor networks would likely be more extensive in 4D, mainly because there is so much floor space to cover! Imagine if you have a single floor in the shape of a cube, where you can fit n*n*n rooms. Even for n=2, that's already 8 rooms, and for n=3, it's already 27 rooms. For larger floor spaces, the number of rooms increases pretty fast, so you have to somehow design the corridors to be able to access all these rooms in the most efficient manner. Note that most of these rooms would be internal rooms (no windows), so it could be advantageous to introduce a courtyard in the middle, so that more rooms can have windows. In 3D, a courtyard design allows you to have just a single corridor encircling the floor, with rooms on either side, but in 4D, this is still not enough. You'd need at least 3 main corridors (think of it as latitude/longitude lines at 0°, 90°, 180°, and 270° divisions) to make all rooms reasonably accessible. Then you'll need to branch off into smaller corridors to reach the rooms at the corners of the cube. So just the sheer number of rooms would be quite a good reason to have a branching corridor network, if nothing else.

On the flip side, because of the fast increase in the number of rooms as floor space size increases, it also means large buildings are very expensive to build: think of the amount of materials you'll need to make a 3x3x3x1 single-storey building, which already has 27 rooms -- and don't forget every wall you put in requires x^3 units of wall materials, since each wall must cover a 3D area in order to be a wall (i.e. divide space). That's a lot of materials required just to make a single wall! And think of the number of walls you'll need to build: remember that each cube-shaped room requires 6 walls (compared to the 4 in 3D). So all in all, it's a pretty steep ramp up from 3D. Because of this, I speculated some time ago that 4D buildings will likely lean towards the small side, since the cost of large buildings would increase too quickly as size increases. Due to the large amount of extra floor space you get just by increasing each dimension of a room a little, it would seem more economically viable to have larger, but fewer, rooms, than to have lots of small rooms (which requires too many walls, each of which adds a huge dent in your building materials budget).

[gonegahgah]

This also would tend to reveal a trend in building heights as well.
With there being so much more available floor space and more floor space immediately surrounding you in 4D there would be a tendency towards less height in buildings in 4D.
Even books would tend to be smaller as you can fit more content into the smaller space.
So you may tend more towards having a single book shelf instead of shelves. So you don't need as much vertical space to store all our stuff in 4D.
Multistory houses would be extremely rare and high scrapers would not scrape very high compared to ours.

I wonder how we would read with their being no preferred left or right? And how would the text flow on a page? Maybe from the centre out in a spiral?
Or just from some corner font space to another corner then onto a line (rectangular prism) under this until the directional edge is encountered and then behind this, then behind that, until the page is full?
For us it would most closely resemble a cube full of 3D characters that look nothing like ours filling up the entire cube.
I guess they could also add more alignments then our left (eg 'p') and right (eg 'q') alignments against the alignment of line of writing as the letters must still traverse in a line even in 4D.
Instead of left to right I imagine they would have words meaning 'writing from direction' and 'writing towards direction'.

[quickfur]

This also would tend to reveal a trend in building heights as well.
With there being so much more available floor space and more floor space immediately surrounding you in 4D there would be a tendency towards less height in buildings in 4D.

Yep. Not to mention that the higher dimension of each floor would mean that the weight of upper storeys would increase very quickly, making a need for a much larger number of load-bearing walls. That would also increase the cost of tall buildings.

Even books would tend to be smaller as you can fit more content into the smaller space.
So you may tend more towards having a single book shelf instead of shelves. So you don't need as much vertical space to store all our stuff in 4D.
Multistory houses would be extremely rare and high scrapers would not scrape very high compared to ours.

Yeah, a single 4D bookshelf would be equivalent to a whole series of bookshelves in 3D, because with a 3D front surface, you'd be able to access so many more books just from a single shelf! In 3D, you can fit k books on a single shelf, so if you have say 6 shelves per bookshelf, you can fit 6k books in it. In 4D, a shelf of comparable dimensions would fit k^2 books, so even if there are only 3-4 shelves per bookshelf, that's 4k^2 books. So if k=20, the 3D bookshelf would hold 120 books, but the 4D bookshelf could hold 1600 books!

I wonder how we would read with their being no preferred left or right? And how would the text flow on a page? Maybe from the centre out in a spiral?
Or just from some corner font space to another corner then onto a line (rectangular prism) under this until the directional edge is encountered and then behind this, then behind that, until the page is full?
For us it would most closely resemble a cube full of 3D characters that look nothing like ours filling up the entire cube.
I guess they could also add more alignments then our left (eg 'p') and right (eg 'q') alignments against the alignment of line of writing as the letters must still traverse in a line even in 4D.
Instead of left to right I imagine they would have words meaning 'writing from direction' and 'writing towards direction'.

I still haven't reached a conclusion on what would be the most likely form of 4D writing. But even in 3D, if you study historical writings, you'd find that all sorts of things have already been tried, like spiralling writing, bidirectional writing (boustrephedon: you write from left-to-right, then at the edge of the paper you reverse the direction and write right-to-left, then when you get back to the starting edge, you reverse the direction once more, etc. -- in some forms of this you actually write mirror-image glyphs when going the reverse direction). Eventually the consistency of a single direction with line wrapping seemed to win over the other creative ways of writing layout, so in modern writing, you find a fixed direction (left-to-right, right-to-left, or top-to-bottom) with a fixed wrapping direction (line by line top down, or column by column right-to-left or left-to-right).

Now in 4D, at least as far as road signs are concerned, I'm thinking that a vertical-based writing (rather than horizontal) is probably preferable, because of the 360° rotational freedom in the "across-space". If the writing were horizontal, then drivers approaching the sign would see it in any of the 360° possible orientations, which makes reading the sign rather difficult (it would have the same level of difficulty as us reading rotated writing in 3D, like upside-down writing or slanted writing). But if the writing were vertically-based, then even if the glyphs are rotated you could at least always read top-to-bottom, which reduces the difficulty of quickly scanning the sign. Otherwise it would divert too much attention from the driver trying to process arbitrarily-rotated writing on the sign, and would lead to more accidents.

But for books and other writings that don't have the same requirements as road signs, many possibilities are still open. It wouldn't matter if a book had horizontal writing, because you could simply just rotate the book to get it into the most comfortable reading orientation, so this becomes a non-issue. Also, while it seems advantageous to restrict road signs to a single vertical column, in a book it wouldn't matter if it were packed into a 3D cubic area, since someone reading the book would be in no hurry and can take the time to orient the book correctly and find the next line no matter how it's wrapped.

[gonegahgah]

I do suspect that because they have so many more combinations; and because of the directional confusion you've clarified; that they would have little need to use similar looking characters in different orientations like 'b' and 'd'.
Probably in 4D you would get to the stage of quickly recognising a character no matter what its sideways orientation. I wonder if they would develop preferred orientations for say cursive writing? Printing might be a bit more relaxed?
Even though they might not need to, upside versions would still be easily distinguishable like 'b' and 'p'.
It would be whimsically nice to be able to see the evolution of 4D characters. I can imagine oriental glyphs that have strokes in all directions of their little box space.

[gonegahgah]

And then we get to hands. I have more thoughts on this but am a bit time squeezed so comments are very welcome.
I'm thinking that 2D hands have two fingers basically for gripping, our 3D hands have four fingers and an opposable thumb.
This makes our 3D hands useful for spear throwing.
If a 2Der were holding our spear poking sideways through their world it would not be very stable and easily wobble.
I figure it would be the same if we held a 4D spear that we could not stop it from wobbling around in the extra dimension when it pokes the extra sideways through our dimension.

If the 2Der held our spear forwards from themselves they would hold it different to us and be clasping its end whereas we wrap our hand around the side.
Again we would probably hold a 4D spear by the edge whereas a 4Der could get a good grip of their spear.

I also figure that, in terms that we can conceive, that the 4Ders hands, when flat, would be like a cup (when the away from palm dimension is discarded for simplication).
They would tend to have fingers that are placed side by side in a circle (as best we can conceive; again).
Their thumbs, because they still need opposable digits, would mainly sit opposite the fingers. I figure they would have at least two thumbs.

I'm giving my imaginary 4Ders 7 fingers and 2 thumbs for a total of 9 digits on each hand and a total of 27 digits altogether across three hands.
Should make their multiplication tables interesting... Thank goodness for all the extra brain cell combinations, and also their extra glyph combination possibilities to uniquely pictorise 27 separate numbers!

[quickfur]

And then we get to hands. I have more thoughts on this but am a bit time squeezed so comments are very welcome.
I'm thinking that 2D hands have two fingers basically for gripping, our 3D hands have four fingers and an opposable thumb.
This makes our 3D hands useful for spear throwing.

The fact that we have 4 fingers is probably not a consequence of the 3D-ness of our space. Many other animals have other numbers of fingers, but all animals that can effectively grasp objects (necessary for handling tools, etc.), have opposable thumbs (or its equivalent). I'd say at least 2 fingers + 1 thumb is necessary in 3D; the fingers+thumb is needed to maintain grip on the object, and the 2 fingers, lying on an axis perpendicular to the axis of fingers+thumb, is necessary to maintain the orientation of the object. It's possible to grip an object with just 1 finger and 1 thumb, but it will be very difficult to maintain its orientation, since it can easily rotate in the axis perpendicular to the finger+thumb plane (the "wobble" that you described).

So we see that effective grasping of object requires at least a finger + an opposable thumb (to fix an object to 1 plane, i.e., 2 dimensions), plus at least one more finger (to fix orientation in the remaining dimension). Of course, having more than 3 digits is useful for other reasons, like being able to manipulate protrusions from the object while holding it, like pushing a button on a TV controller without needing to put it down somewhere first, etc.. So while 2 fingers + 1 thumb is the minimum, it seems advantageous to have more than 2 fingers.

[...] I also figure that, in terms that we can conceive, that the 4Ders hands, when flat, would be like a cup (when the away from palm dimension is discarded for simplication).
They would tend to have fingers that are placed side by side in a circle (as best we can conceive; again).
Their thumbs, because they still need opposable digits, would mainly sit opposite the fingers. I figure they would have at least two thumbs.

I tend to agree. You need at least 3 digits (finger + thumb + another kind of thumb not parallel to the first thumb) in order to fix orientation in a 3D hyperplane (by wrapping around a spherical cross-section of a 4D spear, say). If there are less than 3 digits, you'd only be able to wrap around the equator of the spherical cross-section, which means it can easily slip from your grip. You need to wrap around it in at least 3 different dimensions in order to hold it in place. Then you need at least 1 other finger to maintain the orientation of the spear in the remaining dimension.

So I'd say the absolute minimum number of digits in 4D is 4, with at least 2 different kinds of oppositions (if both thumbs lie along the same axis as the fingers, then you only effectively have 2 dimensions covered, which cannot maintain grip), plus at least 1 more digit for maintaining orientation in the remaining dimension. But it's probably useful to have more than just 2 fingers + 2 kinds of opposable thumbs, so that you can, say, push buttons on a controller without needing to put it down first. The exact number, though, is anybody's guess, for lack of any direct evidence.

I'm giving my imaginary 4Ders 7 fingers and 2 thumbs for a total of 9 digits on each hand and a total of 27 digits altogether across three hands.
Should make their multiplication tables interesting... Thank goodness for all the extra brain cell combinations, and also their extra glyph combination possibilities to uniquely pictorise 27 separate numbers!

That's nothing. Even here on 3D Earth, ancient civilizations have used base 60 counting systems before. In fact, the very common base 10 (decimal) system in widespread use today is actually rather suboptimal in many ways, one of the most glaring of which is the inability to exactly represent fractions involving multiples of 3 (1/3 has an infinite digit expansion). Base 60 is superior because 60 is highly composite, so a large number of fractions with small denominators can be evenly divided (so 1/2, 1/3, 1/4, 1/5, 1/6, all have terminating digit expansions). This makes arithmetic much nicer than in base 10. Of course, memorizing 60 different digits is no fun, so as a compromise, base 12 shares in some of the nice properties of base 60 while still being relatively tractable to common people.

But if you're talking about superior intellects, then you might want to consider the so-called "phinary", which uses the irrational Golden Ratio as a base. This base has the peculiar property that certain irrational numbers have terminating digit expansions. In base 10, 1/2 = 0.5, which is terminating, while 1/3 = 0.3333..., which is non-terminating; but all irrational numbers have non-terminating digit expansions. In phinary, however, certain irrational numbers have terminating digit expansions. In particular, certain numbers involving √5 have finite digit expansions, which makes it extremely nice for exact arithmetic of quantities involving the golden ratio (y'know, pentagons and stuff, which we all know and love thanks to 4D objects like the 120-cell and 600-cell). Furthermore, since the golden ratio phi is approximately 1.61803, you only need to use 2 digits (a 3rd digit can be useful for intermediate calculations, but the final number can always be expressed using only 2 different digits). (The disadvantage of phinary, OTOH, is that while whole numbers have terminating digit expansions, they also look like fractional numbers -- e.g., 5 = 1000.1001_φ, and all simple fractions like 1/2, 1/3, etc., are all non-terminating -- the latter isn't a big problem, though, since 1/3 in base 10 is non-terminating anyway, and we never have any problem with that.)

(Some of you, upon reading about phinary, may wonder about using pi (3.14159...) as a base. However, the problem with pi is that it is transcendental, meaning that it cannot be directly expressed as some algebraic combination of rational numbers with +, -, *, /, n'th-root extraction, exponentiation, etc.. This has the consequence that in base pi, all integers have non-terminating digit expansions, and worse yet, they have non-repeating digit expansions. The only numbers with "nice" representation are numbers directly involving pi -- which makes it nice for representing angles in radians, I suppose, but kinda puts a damper on trying to perform everyday arithmetic involving, y'know, integers. )

[wendy]

I use all sorts of these bases, 2.618, 2.414 and 3.732. Just dig out the pennies and run the stone board.

sqrt 5 is 10.1 in base phi, but f.f in phi2.

[gonegahgah]

Cool. I was also thinking about simple rubbish bins and cups last night. This is where the 360° of sideways comes into its fore.
A simple evenly round bin and simple round plastic cup then become easy to envisage in 4D.

The equivalent can be seen moving from the 2D to the 3D world.
Their simple cup (which admittedly has some rigidity issues) is an out angled plastic front and back with plastic bottom.
To transform from their cup to ours you simply rotate the 2D cup into our 3D space around its vertical centre.

The same is done when moving from our simple cup (& simple bin) to a 4Ders cup.
You simply rotate this around the centre into the 360° of sideways space until you end up back at the beginning (or 3D space).
This process then represents a completely self-contained 4D cup (& bin).

The advantage of this analogy is that it helps me to understand that more bulk exists in the outside rotation as opposed to the no bulk being added to the object by the rotating axis itself.
ie. More bulk is on the outside of the object than towards the centre.
Just as for the 2D cup being transformed into a 3D cup, more circular diameter is added by the outside than the inside.

This helps give me a better understanding of the 4D world.
Of course this is just mathematics.
A tesseract still directly stretches a finite distance into 4D space when we observe it as a cube and what we see is merely an edge.
But, where 'overlap rotational projection' is used - instead of conventional space 'overlap displacement &/or shrinkage' projection - to represent 4D space, then movement through into 4D sees less distortion at the centre of rotation.
Which is useful for orientation and object recognition purposes.
And the centre of rotation does not add to the mass or exist beside itself in rotational projection and is just different 4D orientations of the same 'planar' axis.
A 4D 'planar' axis has the added dimension but its size remains 0d₁ x 0d₂ x 0d₃ x Ld₄; which is important to understand.
Just as our 3D 'linear' axis has the dimensions 0d₁ x 0d₂ x Ld₃.

I also realise that another advantage of rotational projection is that the 4Der's fingers can not 'exist' inside our 3D cup - as it can with 'spatial displacement' projection.
Instead they will always rest outside our cup space even if they aren't in our current 3D space.
With 'displacement projection' the 4Der's fingers can appear to exist inside our solid objects whereas of course they can only really hold them by the two opposite faces of 'flat' objects.

The same goes for a tesseract using rotational projection.
The 4Der's fingers will lay majorly outside the 3D space representing the tesseract using 'rotational projection' (excepting the corners where some intrusion will occur).
With 'displacement projection' the 4Der's fingers can be anywhere inside the 3D space representing the tesseract in order to make contact with all 8 faces and myriad edges of the tesseract.

There are probably pros and cons to both approaches to projection? What are the pitfalls of 'rotational projection'?
I guess one is that you need to rotate the object to gain a fuller sense of it mentally; but that is probably a good thing?

[gonegahgah]

Is it okay for me to refer to the various projection methods as: Displacement Projection, Shrinkage Projection, and Rotated Projection?
We've also been presented with projection involving both displacement and shrinkage of course.
Are these acceptable names, or are there other preferred names for these?

The three methods of projection as I understand it are:
Displacement - Moving the 3D space of each step into the 4th axis slightly away from each other and depicting them all in our 3D canvas; rather than leaving them directly overlapping.
Shrinkage - Shrinking or enlarging the 3D space of each step into the 4th axis and again depicting them all in our 3D canvas; leaving them directly overlapping at a common point.
Rotated - Utilising the vertical space to project into the 4th axis with each angle into the 4th axis corresponding with an angle into the air around a 0x0x0x∞ plane of rotation; leaving them directly overlapping but rotated.

I think generally for displacement and shrinkage methods we have at least the one (or two) end face(s) fully represented in our 3D space (as we can easily perceive them) with the other six (or how ever many) faces distorted?
In rotation we can generally align one end face to be as we more easily perceive it but the rest rotates off into the sky to correspond with their angle off into the 4th bulk of 4D space.

Are there any other projection techniques for 4D (besides combining displacement and shrinkage of course)?
Are there better ways to term these?

[Klitzing]

First of all, projection generally uses some D-1 dimensional manifold and mapps all points of D space onto that manifold. But your avisaged 3D perception of 4D space rather describes different slicing methods.

Your "displacement projection" uses a complete affine hyperplane (flat, infinite in any except 1 direction), and displaces that one along the missing direction. Your "shrinkage projection" OTOH in fact uses (finite parts) of spherical surfaces (then re-mapped into flat space), which section hyperspace, the radius of which shrinks, and therefore too the representation in lowspace gets correspondingly scaled. You could unite both views within projective geometry, i.e. by considering some projection center: in the first case that one would be somewhere at (the space of) infinity, while in the second one it is within finite reach. Despite of that usage of "projective geometry", "projection centre" etc. both perceptions in fact are pure sectionings, thereby producing sequences (in fact: a continuity) of neighbouring sections. Your third case "rotation projection" neither uses cartesian, nor spherical coordinates, it rather uses cylindrical ones: you first define some special 1D axis and all the sectioning spaces then would be half-hyperplanes containing that axis, and being "displaced" according to an axial rotation around that chosen axis.

You surely could use cylindrical coordinates in a different sense too: you might consider (finite portions of) the cylinder surface, which slice all space in displacements of increasing radius. Thus you would result in a perception of 4-space which has correct length values in one direction (parallel to the chosen axis), but becomes scaled in the radial direction.

--- rk

[quickfur]

Agree with Klitzing, the word 'projection' has a specific sense in mathematics, it means to map an n-dimensional object to (n-1)-dimensions by drawing straight lines from the object onto some lower-dimensional manifold (the projection surface, or "screen" if you will), where the intersection of the lines with the projection surface produces the resulting image. Your cylindrical approach, though, isn't really a projection, since it's not based on tracing lines from the object to a surface, but rather it's a way of slicing 4D space as a way to explore it from a purely 3D viewpoint.

[gonegahgah]

Cool, using combinations even with rotational projection! And other forms of rotation too.
I should also apologise too because the other forms of projection are already well established, especially through QuickFur's and other's work, but rotational projection is still only theoretical.
My descriptions were meant to be more exploratory than explanatory as I try to gain further levels of understanding. At this stage it is interesting rather than definitive.

[gonegahgah]

Agree with Klitzing, the word 'projection' has a specific sense in mathematics, it means to map an n-dimensional object to (n-1)-dimensions by drawing straight lines from the object onto some lower-dimensional manifold (the projection surface, or "screen" if you will), where the intersection of the lines with the projection surface produces the resulting image. Your cylindrical approach, though, isn't really a projection, since it's not based on tracing lines from the object to a surface, but rather it's a way of slicing 4D space as a way to explore it from a purely 3D viewpoint.

Hi quickfur, what I'm thinking of is still meant to plot the points and trace the lines between. So it would be projection still wouldn't it? You still see the object in its fullness.
Well, fullness in the sense that it is still hard to portray 'bulk'. I'm hoping that movement will make the 'bulk' more tangible to observers.

[quickfur]

Agree with Klitzing, the word 'projection' has a specific sense in mathematics, it means to map an n-dimensional object to (n-1)-dimensions by drawing straight lines from the object onto some lower-dimensional manifold (the projection surface, or "screen" if you will), where the intersection of the lines with the projection surface produces the resulting image. Your cylindrical approach, though, isn't really a projection, since it's not based on tracing lines from the object to a surface, but rather it's a way of slicing 4D space as a way to explore it from a purely 3D viewpoint.

Hi quickfur, what I'm thinking of is still meant to plot the points and trace the lines between. So it would be projection still wouldn't it? You still see the object in its fullness.
Well, fullness in the sense that it is still hard to portray 'bulk'. I'm hoping that movement will make the 'bulk' more tangible to observers.

We 3Ders see 3D only as 2D projections, yet we have no problem with 3D bulk (i.e. volume) being tangible.

OTOH, a big part of that may be due to stereoscopic vision: based on the discrepancy between left/right images, our brain reconciles it by "inventing" the 3rd dimension, thus we can see curvature and thereby infer 3D volume. I'd imagine a native 4Der would also have some kind of stereoscopic vision, which would allow 4D curvature to be perceived, and thereby 4D bulk.

Unfortunately, stereopsis is something that happens instinctively in the brain, and our brains have been wired for only 3D stereopsis (I'm not sure if this is innate, or if it's developed by exposure to 3D as a baby -- I'm not sure it's something you can experiment on though!). Presenting two slightly different 4D projections to us doesn't magically make us see 4D curvature.

Note also that even in 3D, with stereoscopic vision, sometimes illusions happen, and we often instinctively move sideways slightly or look around an object in order to resolve the ambiguities. (Magic shows use these illusions to great advantage, by making the illusion of something impossible happening due.) One would imagine that in 4D, one would have a similar instinctive response in order to ascertain the general shape of an ambiguous object. Perhaps allowing free-form movement in an interactive setting may also help us resolve ambiguous 4D projections too.

One experiment that may or may not yield insight into this issue, is to write a 3D simulator where the display is only 1D, using some kind of 3D->2D mapping that tries to preserve as much information as possible. How easy/hard would it be to "see 3D" in this setting?

A slightly less-extreme experiment is to do normal 3D->2D projection, but render the 2D projection from a sideways view (i.e., stretch the pixels into line segments, then render the resulting flat rectangular cuboid as seen from one side). This would be roughly equivalent to us looking at 4D->3D projections "from the side" rather than face-on, and trying to figure out what's actually in 4D. (This is "slightly cheating" because it's not a true 2D view: a slanting line, for example, will still be rendered as a 3D rectangle slanting into the 3rd direction, thus showing up as a trapezoid, so polygons would appear as polygonal prisms seen from the side, and we can probably tell what its shape is by looking at the slanting boundaries, whereas a true 2Der would see only a 1-pixel wide projection. But the idea is to get a feel for how well we can visualize 3D space from a sideways POV of the projected image instead of a frontal POV, since we're in the same situation with 4D->3D projections, where we can't see the "frontal" view.)