Klitzing wrote:.[...] Physics there is arguable, as that one has to be proved by experiment. Not so geometry (as far as no calculation errors come in). There are fascinating objects to be found e.g. within a 4D spatial space.
("Spatial" here means: all coordinates are equivalent, i.e. "space-like". There is not one being specialised, by using imaginary units there. - Such non-spatial geometries e.g. are used within special and general relativity, with 3 common space coordinates and one time coordinate. - Not to be confused with!)
E.g. polytopes are not only possible within 2D (polygons) and 3D (polyhedra), but likewise in any other space. E.g. within 4D they meanwhile are called polychora.
[...]
Prashantkrishnan wrote:What is the nature of a temporal dimension?
I can imagine a polytope which has a temporal dimension, as it determines a duration of time. But how about the existence of two temporal dimensions? Is such a thing possible? Since we are moving towards the future at an almost constant rate, and cannot change the direction, we find the concept difficult. How would it be for a one dimensional living being, who, for some reason, is moving in a direction at a constant speed and cannot go the opposite direction due to some mysterious force? Is that in any way similar to our concept of time?
quickfur wrote:So dealing with two temporal dimensions just produces something that is probably incomprehensible to us, because we have no concept of what "time moving forward" means when there's a 2D area in which events can progress. The closest I got to two temporal dimensions is the (very) crude analogy of the evolution of a story's plot (in "internal time") over (external) time. That is, your first draft of the story has the plot going forward in a certain way, so you can say at t1=0, time in your story's universe progresses from t0=0..n in a certain way. But as you revise your story, the new version of the plot now progresses from t0=0..n in a different way. So if you collect all of your story's drafts together, you can lay out the different versions of the plotline across a second time dimension, that is, the external time ("real world time"), which is perpendicular to internal time ("in-story time"). Then you can consider a perpendicular cross-section of these different versions of the storyline, and look at, for example, how a particular character changed from its original conception to its final version, at the same point in the story's time. This then represents the progression of the second time dimension: where time in the story's universe has stopped, but the characters and scenery still change because now you're moving from the original version of this event to newer revisions of the same event in your story. Obviously, the characters in your story are oblivious to this perpendicular motion, since they live only in story-time. But conceivably, one could write a story where a character acquires a 90° rotation from in-story time to revision-time, wherein he travels to a "parallel storyline" where characters, scenes, and objects have changed while story-time stays still, and then he can rotate 90° back into in-story time and then proceed to experience the rest of the story in a new version of the plotline. (But as I said, this is just a very crude analogy of 2D time -- I haven't been able to get any further than this 'cos the analogy breaks down.)
Klitzing wrote:I do like that 2-dimensional time flat in the way quickfur explained it. The internal time dimension of a storyboard versus the external time dimension of the author. Providing thus the possibility to plot the change of scenes within the lifetime of the author.
Dreams, I'd think, also are kind of non-linear paths within that flat. They start within a specific storyboard but shortly after they leave the long-term intrinsic causual necessities by deviating from that direction to other spaces of possibility.
So in grand total that 5-dimensional space with 2 imaginary time dimensions could be considered as the realm of possibilities (of stories).
--- rk
wendy wrote:Of course, in the real world, all of this is a triffle, because physicists consider something called 'phase-space'. One considers a gas, like air, and then for every molecule allocate six dimensions. That's right, you have something like 1.2*10^24 dimensions per gram of air. For 10 grams, that's 12*10^24 dimensions. Now in that space, the whole of air, in all of its glorious motion and adetation, is one point. Kind of makes the leech lattice looks like a peice of string.
anderscolingustafson wrote:If there were two time dimensions there could be one in which entropy changes and one in which entropy remains constant.
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