I'm making a 3D Game of Life, now lets forget that I mentioned Game of Life and get to the problem. When I made my 2D version last year I had the edges of my array connect to become a 3D torus. Now I'll be starting with a 3D array and I can't really visualize the connection. Right now I have a hollow torus, with the inside face of the torus and the outside face of the torus needing to connect. So How does one visualize that? Its a 4D torus right? But that means nothing to me because I can't conceive of even a 4D Cube. The closest I've come is to think that the width of the objects in the array becomes infinitely small, so that the distance between the innner face and outer face approaches 0 and they touch. Does that work or do I have it all wrong?
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Visualizing 4-D objects.
- Thread starter dorion
- Start date
There is no "easy" way to do this.
There are essentially two methods: Either you look at "slices" which in this case would be 3D objects. This works well if you can "pretend" that one dimension is just a parameter.
The second, more complicated, way would be to look at the shadow of the 4D object which again is a 3D object´. It should be easy to find pictures showing the shadow of a hypercube; the most famous being Dali's "Jesus on the hypercube" (or whatever it is called).
Fort most people the first method is probably the easiest.
There are essentially two methods: Either you look at "slices" which in this case would be 3D objects. This works well if you can "pretend" that one dimension is just a parameter.
The second, more complicated, way would be to look at the shadow of the 4D object which again is a 3D object´. It should be easy to find pictures showing the shadow of a hypercube; the most famous being Dali's "Jesus on the hypercube" (or whatever it is called).
Fort most people the first method is probably the easiest.
wwswimming
Banned
might the 4th dimension be multiple 3D torus'es (tori ?) super-imposed on top of each other ?
Hi,
Yes - this is possible, and No, it is not easy. When I was at Cambridge I knew a couple of good Mathematicans (I'm a bad Mathmatician) who could see objects in higher dimensional space. They had a great deal of difficulty communicating what they could see though. The ability was real - they coiuld find valid (and usually very elegant) proofs of propositions in N-dimesnional space which baffled the rest of us.
This one may be a bit easier....
Yes. If dorion is connecting up the edges (faces) of his space they way I think he intends, then one view of the 4D object must be as a sucession of 3D toruses stacked up one on top of the other.
However, the ends must be connected, so the stack would actually be bent into a circle, to create a torus of toruses.
In a continuous 3D space that view would create what appears to be a single torus (though with a hollow inside). That's just the 3D shadow (thanks f95toli) It's not like that - in 4D the toruses in the stack do not intersect to form a solid surface, so trying to visualize it as a small number of discrete toruses in 4D rather than a larger number, or a continuous surface, may be easier.
There is an important pont here - I assume the space is discrete, not continuous. By this I mean it exists as a set of cells whose co-ordinates are integer values. Now think of the game in 3D as a torus displayed as a wireframe. Shrink the cells to points, so that they exist only where the wireframes intersect. Now take n of those wireframe toruses and arrange them to form a ring as I describe above. The toruses no longer intersect, and you could add in the 4th dimesnion as a new set of wireframes connecting the equivalent points in the ring of toruses. Provided you see the cells as points - the place where the wiredrames intersect - there should be no problems with this view.
It's intersting to note that this view must be incorrect in the sense that it lacks symmetry wrt to the dimensions that is present in the underlying formulation - i.e a cube whose opposite faces are joined. The same problem exists withthe 3D torus for the 2D game. I assume you started with a square surface and joined two opposing edges to form a cylinfer, then bent the sylinder into a torus, thus joining the other two edges which had become circles in the first transformation.
Since the choice of which pair of edged to connect is arbitrary, the view is to some extent (probably irrelevant) defective. Hmmmmm
Peter
Yes - this is possible, and No, it is not easy. When I was at Cambridge I knew a couple of good Mathematicans (I'm a bad Mathmatician) who could see objects in higher dimensional space. They had a great deal of difficulty communicating what they could see though. The ability was real - they coiuld find valid (and usually very elegant) proofs of propositions in N-dimesnional space which baffled the rest of us.
This one may be a bit easier....
Yes. If dorion is connecting up the edges (faces) of his space they way I think he intends, then one view of the 4D object must be as a sucession of 3D toruses stacked up one on top of the other.
However, the ends must be connected, so the stack would actually be bent into a circle, to create a torus of toruses.
In a continuous 3D space that view would create what appears to be a single torus (though with a hollow inside). That's just the 3D shadow (thanks f95toli) It's not like that - in 4D the toruses in the stack do not intersect to form a solid surface, so trying to visualize it as a small number of discrete toruses in 4D rather than a larger number, or a continuous surface, may be easier.
There is an important pont here - I assume the space is discrete, not continuous. By this I mean it exists as a set of cells whose co-ordinates are integer values. Now think of the game in 3D as a torus displayed as a wireframe. Shrink the cells to points, so that they exist only where the wireframes intersect. Now take n of those wireframe toruses and arrange them to form a ring as I describe above. The toruses no longer intersect, and you could add in the 4th dimesnion as a new set of wireframes connecting the equivalent points in the ring of toruses. Provided you see the cells as points - the place where the wiredrames intersect - there should be no problems with this view.
It's intersting to note that this view must be incorrect in the sense that it lacks symmetry wrt to the dimensions that is present in the underlying formulation - i.e a cube whose opposite faces are joined. The same problem exists withthe 3D torus for the 2D game. I assume you started with a square surface and joined two opposing edges to form a cylinfer, then bent the sylinder into a torus, thus joining the other two edges which had become circles in the first transformation.
Since the choice of which pair of edged to connect is arbitrary, the view is to some extent (probably irrelevant) defective. Hmmmmm
Peter
silverpig
Lifer
Difficult to visualize? Yes.
Difficult to implement? Not really.
Oh, and your first 2d game that you turned into a torus is still a 2d shape (as your game is limited to the surface of the torus). You just need 3 dimensions in order to visualize it. Think of it like a piece of paper (neglecting thickness). You can write an x,y grid on the paper while it's flat, and then crumple it up, or bend it into a tube, and you'll still have x,y ordered pairs describing every point on its surface.
I think I know what you did in your original case, and that is you just made it so an automaton on the right boundary was influenced by the automatons on the left boundary and vice-versa, as well as making the same condition for the top/bottom boundaries correct? Well if you wanted to do the same thing in 3d, and started with a cube, then you just map x(max) goes to x(min), y(max) goes to y(min) and z(max) goes to z(min).
Difficult to implement? Not really.
Oh, and your first 2d game that you turned into a torus is still a 2d shape (as your game is limited to the surface of the torus). You just need 3 dimensions in order to visualize it. Think of it like a piece of paper (neglecting thickness). You can write an x,y grid on the paper while it's flat, and then crumple it up, or bend it into a tube, and you'll still have x,y ordered pairs describing every point on its surface.
I think I know what you did in your original case, and that is you just made it so an automaton on the right boundary was influenced by the automatons on the left boundary and vice-versa, as well as making the same condition for the top/bottom boundaries correct? Well if you wanted to do the same thing in 3d, and started with a cube, then you just map x(max) goes to x(min), y(max) goes to y(min) and z(max) goes to z(min).
imported_inspire
Senior member
You might want to ask a Topologist.
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