Euclid’s point has no part yet builds every figure
In geometry a point idealises exact position with no size—a zero-dimensional primitive. Euclid called it “that which has no part,” fixed by axioms rather than smaller pieces. Analytic geometry later tags points with coordinates; modern talk treats spaces as point sets.
Because a point is a primitive notion, it is pinned down not by definition but by axioms it must obey, such as the rule that exactly one straight line passes through two distinct points. On paper, a compass tip, scriber or pen marks a small dot or pinprick to stand in for it. Two curves or three surfaces can meet at a single point, called a vertex or corner, and an isolated point of a set is one with a neighbourhood containing no other member of that set.
In the plane a point becomes an ordered pair, with x conventionally horizontal and y vertical, and the idea extends naturally to triples in space. Many Euclidean objects are infinite collections of points: a line in n dimensions is the set of points whose coordinates satisfy one linear equation, and a segment containing a single point is called degenerate. Euclid’s postulate that any two points can be joined by a straight line let him build almost every figure known in his day, though his treatment of points was incomplete and sometimes assumed unstated facts.
Mathematics has several inequivalent definitions of dimension, and a point scores zero in all the common ones. A vector space containing only the zero vector has no linearly independent subset. Every open cover of a single point can be refined to one open set, giving covering dimension zero, and a single ball of arbitrarily small radius covers it, giving Hausdorff dimension zero.
Some frameworks drop points altogether: noncommutative geometry and pointless topology describe spaces through algebras of functions or sets instead. Physics goes the other way, idealising electrons as points carrying charge, and the Dirac delta function, zero everywhere except at the origin yet integrating to one, models such a concentrated mass or charge.
Source: Point (geometry)
