From Pentagons to Pentagrams

The concept of Galois conjugation, as discussed in the book "The Symmetries of Things" by John Horton Conway, Heidi Burgiel, and Chaim Goodman-Strauss, has been applied to polyhedra, specifically the Kepler–Poinsot polyhedra, to reveal a family relationship between them. This operation, which replaces √5 with -√5 in formulas, has been shown to transform regular pentagons into regular pentagrams and vice versa. The golden field, a set of numbers with rational coefficients, equipped with the usual addition, multiplication, subtraction, and division, is central to this transformation.

The application of Galois conjugation to a regular pentagon with vertices in the golden field results in a regular pentagram, and conversely, applying it to a regular pentagram yields a regular pentagon. This transformation preserves the field operations, and when applied twice, returns to the original state, much like complex conjugation. The exterior turning angles of regular pentagons and pentagrams, specifically 72° and 144°, respectively, play a crucial role in this transformation, as their cosines are related through Galois conjugation.

The implications of this discovery are significant, as it reveals a deep connection between geometry and number theory. The use of Galois conjugation to transform polyhedra and polygons has far-reaching potential in mathematics and computer science. For instance, the transformation of a rhombicosidodecahedron with vertex coordinates in the golden field results in pentagrams, while squares and equilateral triangles remain unchanged. This property can be leveraged to create new geometric shapes and patterns, which can be used in various fields, such as architecture, engineering, and computer graphics.