Konstruierbare

Konstruierbare is a term used in mathematics, specifically in Euclidean geometry, to describe a geometric object that can be constructed using only a compass and straightedge. These tools are idealized geometric instruments with specific limitations: a compass can draw circles and arcs of a given radius, and a straightedge can draw a line through two given points. The construction must be achieved through a finite sequence of operations using only these tools.

Many fundamental geometric figures are constructible. For instance, constructing an equilateral triangle given a line segment, bisecting an angle, or constructing a perpendicular bisector of a line segment are all classic compass and straightedge constructions. However, not all geometric problems are solvable with these limited tools. The trisection of an arbitrary angle, the squaring of a circle (constructing a square with the same area as a given circle), and the doubling of a cube (constructing a cube with twice the volume of a given cube) are famously proven to be impossible using only compass and straightedge. These impossibilities are generally demonstrated through algebraic proofs, showing that the lengths required for such constructions are not expressible as roots of polynomials with rational coefficients, which is a condition for constructibility. The concept of constructibility is deeply intertwined with field theory and Galois theory, which provide a rigorous framework for understanding which geometric problems are solvable.