A Spatial Analog of the Compass Rose Constructed Using Galois Fields

This paper proposes a spatial analog of the compass rose, constructed using finite fields and the discrete logarithm operation. The basic idea is to match the geometric elements of regular and semiregular polyhedra with elements of Galois fields (GF), which allows for the introduction of discrete spherical coordinates defined in algebraic form. The icosahedron is considered as a basic example. It is shown that using the icosahedron faces and the GF(41) field results in a 20-directed spatial structure that can be interpreted through discrete analogs of polar and azimuthal coordinates. Next, a variant based on the icosahedron edges and the GF(31) field is investigated, in which the number of directions increases to 30 while maintaining the regularity of the construction. A further generalization to the case of a truncated icosahedron, associated with the GF(181) field, is also considered, demonstrating the possibility of increasing the angular resolution without abandoning the algebraic organization of the set of directions. The obtained results demonstrate that the spatial rose of compass points can be represented as a finite system of directions with an explicit internal structure, convenient for coding, enumeration, and algorithmic processing. The proposed approach is of interest for problems of discrete description of rotations, construction of finite coordinate systems, and development of sectoral control algorithms, including those applicable to UAVs and their groups. The proposed formalism may also be considered as a sector-level coding layer for command-and-control architectures in which it is sufficient to identify a spatial sector rather than reconstruct full continuous coordinates.