Is Point a Circle with Zero Radius?

Today, I came across a statement that said that a point is a circle with zero radius. But is it so? Let me give you a to-the-point answer.

The definition of a circle incorporates the principles of circumference and area, which designates it as a two-dimensional body, whereas a point is a zero-dimensional object, as you would understand in the following paragraph.

If somebody invited you to demonstrate a circle of zero radius, you may perhaps put the contention that it was a point. But then, you may possibly, in the same way, make the same point a sphere of 0 radius, or a square of side measuring 0 units, or even a line segment of 0 length. The point is (no pun intended!); a point is just a point. It is no further a circle than a sphere or a line segment. A point is an abstract concept that can't be drawn. It is purely only an element in space. It has a position, but it does not take up any space. It is a 0-dimensional object. It lacks length, width, and height.

                   

Furthermore, if the radius is zero, then it is not really a circle, but might be termed as a degenerate circle – which comes under the ambit of inverse geometry. Nonetheless, it would still not be right to characterize a point as a ‘regular’ circle because other related definitions would cease to validate. Consider, for example, the case of tangents to a circle which strictly postulate that the radius has to be greater than zero.

Mathematics is as deep as Romeo’s love for Juliet. Such a pointless (again, no pun intended!) definition of a point would only produce ambiguity. Think about someone who is delved into topology. A theorem like "no circle is contractible" would definitely turn some heads in his office if the question of “a circle of radius 0" is raised.

Taking about inverse geometry and degeneracy, let me further elucidate that a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class. Degenerate cases are exceptional cases where changes to the object’s dimension usually occur. For instance, a triangle is a 2-D object, but a degenerate triangle is contained in a line, and its dimension is therefore 1.

Hence, it is quite evident that, technically, a point ought not to be termed as a circle with 0 radius. Having said that, a loose definition of a point as a 0-radius circle can be accepted if the scope of the work under consideration is fairly limited and precise.