Two Dimensional Coordinate Plane with Distance as Third Dimension

Click For Summary
SUMMARY

The discussion focuses on visualizing a two-dimensional finite plane, specifically a square, and extending it into a three-dimensional space by incorporating distance as a third dimension. The resulting shape is identified as a rectangular solid with a pyramid removed from one end. When considering a circular finite plane, the shape transforms into a cylinder. The conversation explores the implications of distance measurements, suggesting that the height of the 3D object may vary based on the maximal distance from each point, leading to complex geometrical forms such as scalloped boxes and conical shapes.

PREREQUISITES
  • Understanding of basic geometry and three-dimensional shapes
  • Familiarity with coordinate systems and plotting
  • Knowledge of distance metrics in Euclidean space
  • Concept of rotational geometry and its applications
NEXT STEPS
  • Explore the properties of three-dimensional shapes derived from two-dimensional figures
  • Study the mathematical principles of distance in Euclidean geometry
  • Investigate the implications of rotating shapes around axes in 3D space
  • Learn about the significance of state spaces in mathematical modeling
USEFUL FOR

Mathematicians, geometry enthusiasts, educators, and students interested in advanced geometric concepts and spatial reasoning.

crastinus
Messages
78
Reaction score
9
TL;DR
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.
 
Mathematics news on Phys.org
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.
 

Similar threads

High School Moving along one axis in a four-dimensional plane
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Expressing any given point on plane with one unique number
  • · Replies 4 ·
Replies
4
Views
2K
High School Invariant under rotation: Banal, obvious, or noteworthy?
  • · Replies 2 ·
Replies
2
Views
2K
High School Reprise: Calculate the distance between two points without using a coordinate system
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad What is the role of geometry and dimensional operations?
  • · Replies 3 ·
Replies
3
Views
2K
High School Cartesian Space vs. Euclidean Space
  • · Replies 10 ·
Replies
10
Views
2K
High School Proof Pertaining to Possible Paths Through 3 Dimensions
  • · Replies 18 ·
Replies
18
Views
3K
Map of a 4D planet
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Three equations of planes, dimension should be 1?
  • · Replies 10 ·
Replies
10
Views
2K
Graduate Alternative topology of our 3D space
  • · Replies 1 ·
Replies
1
Views
2K