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