Point (geometry)

In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. Points are important because they are the building blocks of shapes and spaces. Just like tiny seeds can grow into big trees, points can come together to form lines, surfaces, and even higher-dimensional objects.

As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and higher-dimensional objects consist. In classical Euclidean geometry, a point is a primitive notion, defined simply as "that which has no part." This means a point has no length, width, or height—it is just a location.

Points help us describe the world around us. When we draw or measure, we often use tools like a compass or a pen to mark points. These points can also be found where two lines or curves meet, called a vertex or corner. With the help of numbers called coordinates, we can pinpoint exactly where a point is on a map or a graph.

Points in Euclidean geometry

Points are some of the most basic ideas in Euclidean geometry. Euclid, the ancient Greek mathematician, described a point as "that which has no part"—meaning it has no size or shape.

In a flat, two-dimensional space called the Euclidean plane, we show a point using two numbers written as (x, y). The first number, x, usually shows how far to the right or left the point is, while the second number, y, shows how far up or down it is. When we add a third number, z, we can describe points in three-dimensional space, adding depth to our flat plane. This way, points help us describe exact locations in space.

Dimension of a point

Main article: Dimension (vector space)

Main article: Lebesgue covering dimension

Main article: Hausdorff dimension

In mathematics, a point is considered to have no dimension. This means it has no length, width, or height — it is just a single position in space.

There are different ways to measure the dimension of objects, and in all of these ways, a point is always seen as zero-dimensional. This is because a point does not contain any smaller parts or directions, unlike lines, surfaces, or solids which have more dimensions.