konstruerbar

Konstruerbar refers to the concept of something being capable of being constructed or built. In mathematics, particularly in geometry, it denotes a figure or object that can be constructed using only a limited set of tools, typically an unmarked straightedge and a compass. This concept is fundamental to classical geometry and has been studied for centuries, famously leading to the unsolvability of certain classical construction problems such as squaring the circle, trisecting an angle, and duplicating the cube using only these tools.

The rules of construction with straightedge and compass are precise. A straightedge allows the drawing of a line segment between two existing points or the extension of an existing line segment. A compass allows the drawing of a circle with a given center and radius, or the transfer of a distance. Any point that can be obtained through a finite sequence of these operations from a given set of initial points and lines is considered constructible.

The theory of constructible numbers, which are numbers that can be constructed using straightedge and compass, is closely related to field extensions in abstract algebra. A number is constructible if and only if it belongs to a field extension of the rational numbers that can be obtained by a sequence of quadratic extensions. This mathematical framework explains why certain geometric problems are impossible to solve with the restricted tools. The term "konstruerbar" thus carries a specific technical meaning within this mathematical context, highlighting the limitations and possibilities of geometric construction.