I Two Dimensional Coordinate Plane with Distance as Third Dimension

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The discussion explores the concept of adding a third dimension to a two-dimensional coordinate plane, specifically focusing on how distances between points create a three-dimensional shape. When starting with a square, the resulting shape is described as a rectangular solid with a pyramid removed from one end. If the finite plane is a circle, the shape becomes a cylinder, potentially with a cone-shaped cut-out. The conversation also touches on whether the height of points in this 3D space corresponds to their maximum distances from other points or remains constant. The implications of this "state space" of position and distance are questioned, suggesting potential significance in understanding spatial relationships.
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What shape will this 3-D object have?
Imagine we draw a two dimensional finite plane with coordinate axes; for simplicity, let's make it a square. Now, suppose we add a third dimension that represents the possible distances between any two points on the square. Now we have a three dimensional space. What shape will that space have?

I've worked it out some myself, but I don't think I quite understand how to do it in the best way. Obviously, the resulting shape is some kind of rectangular solid. What I get when I think about this is a rectangular solid with a pyramid removed from one end.

How would it be different if we made our finite plane a circle? Then the resulting 3-D object would certainly be a cylinder of some kind.

Thanks.
 
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Possible distances between any points range from 0 to sqrt(2) times the length of your square, and from 0 to the diameter of your circle. Does that mean the object just has a constant height equal to that maximal distance? Or does every point get a height according to its maximal distance to other points in the object (making a pyramid out of the square and a cylinder with a cone-shaped cut-out out of the circle)? Or something else?
 
If you first consider a one dimensional case along the x-axis then each x value would be its distance from zero. Plotting the x,dist on on an xy plot would give the line y=x

Extending to your 2D case is equivalent to rotating the y=x about the y-axis giving a cone. Considering you have a square then youll get scalloped box in 3D where the scallops are parabolic from the definition of cutting a cone with a plane parallel with its central axis along the y direction. The planes are the sides of the square which become the sides of the box when extended in 3D.

Did i say that right?
 
Thanks for the responses! I just keep thinking about it.

This is a sort simple "state space" with just position and distance. I wonder if there is any significance to it.
 

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