Sorry .... I knew I should have put some graphics in my post. But my laptop is 5 floors down in the carpark, and I was using my girlfriends laptop. No graphics program ... lol.
First, just a clear up .... the unit-circle I describe as 2-dimensional, is the universe within the flat circle. The challenge is to describe a spatial (not temporal .. time does not enter it for this purpose) location by the minimum number of co-ordinates. Imagine throwing a dart into the circle, and then mathematically describing its location.
So if you draw an imaginary polar axis thru the circle, you can now draw a line from the centre to the impact point of the dart, by measuring the length of the line, and its angle from the polar axis .... you can recreate the exact location of the point... ?? Is this clear ... please stop me here if I confused the issue ??
So ... extending it to 3-dimensional, the universe becomes a sphere, instead of a circle. But the universe is within the sphere (not the flat outer surface of the sphere)
So to answer your second question first ... lol. I am picturing all co-ordinates as being within the sphere. (and within the circle for 2D)
Well ... 0-dimensional space would require, in order to locate a point spatially, either negative-1-measurement, or an infinite number of measurements ... lol. I can't conceive of 0-dimensional space.
If 1-dimensional space has to have a point described within it then you need zero measurements .... ?
If the Universe was 1-dimensional, then in order to travel from London to New York you would need zero measurements .... because London, New York and the rest of the Universe are all within the point. You are already in both places, so no co-ordinate (or map) is required.
Hope I'm not confusing you more ??
If the Universe was 2-dimensional (the circle) then you would need the distance from the centre of the circle to London (if you were in New York) and the angle from the polar axis. In other words the minimum you would require was a single measure of length (dimensional space ?)
If the universe was 3-dimensional (lol ... it is) then you would not only need the 2-dimensional measure, but also the measure within the volume (sphere) otherwise you might end up exactly where London is axially, but a 100 klms below it ? or above it ?
And so on, adding a coordinate and an angle describes an extra dimension.
If I have made this explanation worse, let me know ... and I will do the graphics and repost ... a pic paints a thousand words ... lol
Wick ... your first link worked, but the other 2 just took me to the top of this page.
I am still reading your first link, will get back to it.
cool bananas ... greg


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