Point (geometry)

In geometry, a point is an abstract idealization of an exact position. It is considered a zero-dimensional object because it has no size, width, or depth. While we often draw points as small dots, these are merely physical representations. A true mathematical point is an indivisible element that serves as a building block for space. Points comprise one-dimensional curves, two-dimensional surfaces, and even higher-dimensional objects.

Points can be identified through several different mathematical mechanisms. One common way is through the intersection of geometric shapes. For example, a vertex or corner is a point formed where two curves meet or three surfaces intersect. In analytic geometry, points are defined using numerical coordinates. In a two-dimensional Euclidean plane, a point is an ordered pair of numbers. The first number represents the horizontal position, while the second represents the vertical position.

This coordinate system can be expanded to describe more complex spaces. In three-dimensional Euclidean space, a point is represented by an ordered triplet. This triplet includes a third number to represent depth. Mathematicians can generalize this even further using an ordered tuplet of terms. The number of terms in the tuplet corresponds to the dimension of the space. In modern mathematics, a collection of these points is often treated as a point set.

Historical foundations for these ideas began with the Greek mathematician Euclid. He defined a point as "that which has no part." In classical Euclidean geometry, the point is a primitive notion. This means it is not defined by other concepts but by axioms. Axioms are formal properties that points must satisfy. One such axiom states that exactly one straight line passes through two distinct points.

Euclid's work had lasting consequences for the development of geometric concepts. However, his original postulates were not always complete or definitive. He sometimes assumed facts that did not follow directly from his axioms. For instance, he occasionally assumed the ordering of points on a line. Modern expansions of Euclidean geometry have since worked to remove these assumptions. Today, we can define a line as an infinite set of points.

Mathematics offers several ways to define the dimension of a point. In all common definitions, a point is zero-dimensional. In the context of vector space dimension, a space consisting of a single point is the zero vector. This vector is not linearly independent because a non-trivial linear combination makes it zero. In topological dimension, a point is zero-dimensional because every open cover has a refinement of a single open set.

Another method is the Hausdorff dimension, which uses metric spaces. A point has a Hausdorff dimension of zero because it can be covered by a single ball of any tiny radius. Even though points are fundamental, some mathematical systems exist without them. Noncommutative geometry and pointless topology are examples of these systems. In these "pointless" spaces, the structure is defined by algebras rather than sets of points.

In physics, points are often used to model real-world properties. Scientists may treat a point as having a non-zero mass or charge. This is common in classical electromagnetism when modeling electrons. The theoretical physicist Paul Dirac introduced the Dirac delta function to assist with these models. This function acts as an infinitely high, infinitely thin spike at the origin. It represents an idealized point mass or point charge where the total area under the spike is one.