Surface

A surface is the outer boundary of an object or space — the outermost layer that separates inside from outside. In mathematics, particularly in geometry (the study of shape) and topology (the study of properties of shape that remain unchanged), a surface is a two-dimensional object: every point on it, when examined closely, appears to resemble a part of a flat plane.

Common examples include a flat plane, the surface of a sphere, the surface of a cylinder, and a torus (a doughnut shape). It is important to distinguish between a surface and volume: a solid ball is three-dimensional, whereas the surface of a ball is two-dimensional.

Mathematical definition

In topology, a surface is a two-dimensional manifold: a space in which each point has a neighborhood resembling a region of a flat plane. Surfaces can be classified by properties such as whether they are compact without boundary, whether they are orientable, and by their genus (number of holes). The surface of a sphere has genus 0; a torus has genus 1.

Curvature and shape

Differential geometry measures the curvature of a surface. Gaussian curvature indicates whether a surface resembles a sphere (positive curvature), a plane (zero curvature), or a saddle surface (negative curvature). A key property revealed here is that intrinsic curvature can be measured from within the surface itself without reference to the surrounding space — an insight that became fundamental to the concept of curved space in modern physics.

The Earth's surface

In geology, the Earth's surface (or topography) is the boundary between the solid crust and the atmosphere or water. It is characterized by features such as elevation and landforms including mountains, valleys, and plains. Geological science uses mathematical models — the geoid and the ellipsoid — to define a reference surface for measuring locations and elevation.