Ebenenanzahl

Ebenenanzahl is a German term that refers to the number of planes involved in a geometric configuration or counting problem. In three-dimensional Euclidean space, a plane is a flat surface extending infinitely in all directions, defined by a linear equation ax + by + cz + d = 0. The Ebenenanzahl describes how many such planes participate in a given setup and influences how space is partitioned and how complex the arrangement becomes.

A central question in this context is how many regions space can be divided into by a given number of planes. If n planes are positioned in general position (no two parallel, no three meeting in a common line, etc.), the maximum number of regions is R3(n) = (n^3 + 5n + 6)/6, which is equivalent to the sum of binomial coefficients 1 + C(n,1) + C(n,2) + C(n,3). For example, 0 planes yield 1 region, 1 plane yields 2, 2 planes yield 4, 3 planes yield 8, and 4 planes yield 15 regions.

In real configurations, the actual number of regions may be smaller if planes are parallel, coincident, or share intersection lines. The Ebenenanzahl therefore serves as a key parameter in problems of combinatorial geometry, computational geometry, and related fields, affecting algorithms for spatial partitioning, visibility, and 3D modeling.

Related concepts include arrangements of hyperplanes, geometric partitioning, and planar versus spatial subdivision.