pentagonales

Pentagonales is a term used in geometry and number theory to refer to pentagonal-related objects. In practice, it is used to describe pentagonal polygons, pentagonal numbers, and tilings involving pentagons.

Pentagonal polygons: A pentagon is a five-sided polygon. A regular pentagon has five equal sides and five equal interior angles of 108 degrees. Its diagonals intersect to form a pentagram, and the ratio of a diagonal to a side equals the golden ratio φ ≈ 1.618. The area of a regular pentagon with side length s is A = (1/4) √(5(5+2√5)) s^2.

Pentagonal numbers: The nth pentagonal number is P(n) = n(3n−1)/2 for n ≥ 1, producing the sequence 1, 5, 12, 22, 35, … These are figurate numbers that can be arranged in a pentagonal pattern and appear in various number-theoretic contexts.

Pentagonal tilings: Not all pentagons tile the plane, but certain convex pentagons do. The plane can be tiled by pentagons of specific shapes, and researchers have identified multiple families of convex pentagons that tile the plane, with at least 15 distinct families documented in classification efforts.

Overall, pentagonales encompasses a set of related concepts tied to the five-sided polygon, its geometric properties, and its number-theoretic and tiling manifestations.