Gömb

Gömb is the Hungarian term for the three-dimensional geometric shape commonly known in English as a sphere or ball. In geometry, the word is typically used to denote a solid ball—a set of points whose distance from a fixed center is at most a given radius. The hollow surface of such a shape is usually referred to as a gömbfelület or gömbhéj. In everyday usage, gömb can denote either the solid or the surface, depending on context.

Definition and basic forms: Let c be a center in Euclidean space and r > 0. The solid gömb B(c, r) is the set of all points x with Euclidean distance ||x − c|| ≤ r. The boundary of B(c, r) is the gömbfelület S(c, r) = { x : ||x − c|| = r }. In three dimensions this yields a perfectly symmetric, convex object.

Formulas and properties: The volume of a solid gömb with radius r is V = 4/3 π r^3, and its surface area is A = 4 π r^2. A gömb with radius 1 is called a unit gömb; its volume is 4/3 π and its surface area is 4 π. The gömb is a central object in geometry and is used to model many real-world shapes and processes due to its symmetry.

Higher dimensions: In n-dimensional space the n-ball B^n(r) has volume V_n(r) = π^{n/2} r^n / Γ(n/2 + 1), and its boundary S^{n−1}(r) has area S_{n−1}(r) = n π^{n/2} r^{n−1} / Γ(n/2 + 1). These formulas generalize the basic 3D case.

Applications: Gömbs appear in physics, computer graphics, architecture, and various branches of mathematics as a fundamental model of round objects and isotropic spaces.