Cycles and spheres and center points.

 Exactly—and that’s an important observation.

Both a circle in 2D and a sphere in 3D are defined in terms of a point that does not belong to the set of points being described (except in a degenerate case).

Why the center is not in the set

• A circle is the set of all points at distance \(r > 0\) from \((a,b)\).
  
• A sphere is the set of all points at distance \(r > 0\) from \((a,b,c)\).
  

But the distance from the center to itself is 0, not \(r\).
So the center fails the defining condition and is therefore not included.

Exception: the degenerate case

If \(r = 0\):

• The “circle” or “sphere” collapses to a single point.
  
• The center is the entire set.
  

Outside of this trivial case, the center is excluded by definition.

Conceptual takeaway

This highlights something subtle but important in mathematics:

• Objects are often defined relationally, not by membership.
  
• The center acts as a reference point, not an element of the set.
  
• Circles and spheres are boundaries, not filled regions.
  

If you instead include all points at distance less than or equal to \(r\), then you get a disk (2D) or a ball (3D)—and those sets do include the center.

If you want, we can go further into how this distinction matters in topology, analysis, or definitions of dimension.