Vinkelens

Vinkelens is a term sometimes used in geometry education to refer to a constructed family of angles based at a common vertex. It is not a standard mathematical object in formal geometry, but appears in problem sets and expository writings to illustrate how angles relate under rotation and relabeling of rays.

Definition and structure: A vinkelens consists of a fixed vertex V and a finite or infinite set of rays {ri} emanating from V. Each ray is assigned an angle θi relative to a chosen reference ray r0. The collection {θi} describes the arrangement of the rays around V, with θ0 typically defined as 0. Angles are considered modulo 2π, reflecting wrap-around at full rotation.

Operations and properties: Rotating the entire vinkelens by an angle φ adds φ to every θi. The angular separations between successive rays are given by Δi = θi+1 − θi (mod 2π). The sum of all successive separations around V is 2π for a full circle. Vinkelens can be used to illustrate angle addition, modular arithmetic, and circular order.

Applications and relationship to standard angles: In teaching, vinkelens helps students visualize how changing a single ray affects multiple angles and how angle measures behave under rotation. When the reference ray is fixed and all rays are treated as ordinary angles, a vinkelens description reduces to a conventional angle configuration. It also connects to concepts from group theory, notably the action of the rotation group SO(2) on the set of rays.

Examples and notes: A simple vinkelens with rays r0, r1, r2 at θ0 = 0, θ1 = π/4, θ2 = π/2 yields angular separations of π/4 and π/4, forming equal gaps around the vertex.