Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. For example, a line has a dimension of one because only one coordinate is needed to specify a point on it, like the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two because two coordinates are needed to specify a point on it, such as both latitude and longitude.

The inside of a cube, a cylinder, or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different, but in modern physics, they are combined into a four-dimensional space known as spacetime. This helps describe complex theories like electromagnetism and general relativity.

The concept of dimension is not limited to physical objects. High-dimensional spaces often appear in mathematics and the sciences. These can be Euclidean spaces or more general parameter spaces and configuration spaces, which are abstract spaces independent of the physical space. Understanding dimensions helps us describe and study many aspects of the universe, from the smallest particles to the largest structures.

In mathematics

The dimension of an object in mathematics is the number of coordinates needed to describe any point on it. For example, a line has a dimension of one because only one number is needed to show where a point is on the line. A plane, like a flat surface, has a dimension of two because two numbers are needed to show a point’s location.

Dimensions help us understand how objects are shaped and how they fit into space. A point has zero dimensions, a line has one dimension, a plane has two dimensions, and objects like a cube have three dimensions. Some special shapes, like a tesseract, have four dimensions.

In physics

Classical physics theories describe three physical dimensions: up/down, left/right, and forward/backward. Any movement in other directions can be described using just these three. For example, moving diagonally combines moving up and forward at the same time.

Time is often called the "fourth dimension" because it measures change in the universe. Unlike the three spatial dimensions, we can only move forward in time, not backward. Physics theories like special relativity and general relativity treat space and time together as spacetime. Some theories, like superstring theory, suggest there may be more than four dimensions, but we have not found evidence for these extra dimensions yet.

In computer graphics and spatial data

Main article: Geometric primitive

Many digital systems, like illustration software, computer-aided design, and geographic information systems, work with shapes using special building blocks called geometric primitives. These blocks match the dimensions of space:

• Point (0-dimensional), a single spot in a Cartesian coordinate system.
• Line or Polyline (1-dimensional), a list of points that the computer connects to make straight or curved lines.
• Polygon (2-dimensional), a closed line that marks the edge of an area, helping the computer know what’s inside and outside.
• Surface (3-dimensional), often shown as connected flat pieces called a polyhedron, to mark the outside and inside of a 3D shape.

Sometimes, these systems show real-world things in simpler shapes. For example, a city might be shown as a point, or a road as a line, to make maps easier to understand and store. This works well as long as we remember these shapes are just representations, not the actual things.

More dimensions

Dimension can also refer to different concepts in mechanics, physics and chemistry, and statistics. Examples include exterior dimension, Hurst exponent, isoperimetric dimension, metric dimension, order dimension, and q-dimension, which includes fractal and correlation dimensions. These ideas help scientists and mathematicians describe and understand complex systems and patterns in various fields.

List of topics by dimension

Here is a list of topics grouped by how many dimensions they have. In math and science, a dimension is a way to describe space. For example, a line is one-dimensional because you only need one number to describe a point on it. A flat surface, like a piece of paper, is two-dimensional because you need two numbers to describe a point.

The list includes topics from zero dimensions, like a single point, up to very high dimensions used in advanced theories like string theory. Each dimension level has different shapes and concepts that help us understand space in new ways.