Generalizing BBP-type constructions to other regular polygons (e.g., octagon/√2)

Determine whether geometric and algebraic properties analogous to those of the regular pentagon that underlie the golden ratio φ can be identified for other regular polygons—such as the regular octagon associated with √2—and used to construct BBP-type series formulas for mathematical constants in the corresponding algebraic bases.

Background

The paper proves a BBP-type formula for π² in the golden ratio base using a geometric identity tied to the regular pentagon and φ, and demonstrates a Machin-like identity for ζ(3). The method’s success relies on special geometric and algebraic properties of φ within the field of fifth roots of unity.

In the concluding discussion, the author highlights that it seems unlikely the same approach extends straightforwardly to other irrational algebraic numbers lacking comparable geometric structures, but explicitly raises the possibility of analogous phenomena for other regular polygons, such as the octagon (linked to √2). The author labels this possibility as a challenging open question.