Tilings with diamond, star and pineapple shapes based on the geometry of the regular pentagon

Abstract

This paper presents a gallery of tessellations created by combining the following equilateral shapes: diamonds and stars, which can be further assembled into a pineapple shape. These three shapes can be decomposed into simpler shapes: an isosceles triangle and a rectangle. The triangle is formed by a subdivision of the regular pentagon into five equal sections, so that each have a base of length 𝒂�� and legs of length 𝒃��. The rectangle is created by having the same 𝒂�� and 𝒃�� for its sides. The diamond is formed by two such triangles and one rectangle, while the star is formed by a radial arrangement of five triangles back into the pentagon onto which five more triangles are added (elevated). Using these two shapes, we can tile the Euclidean plane without overlaps and gaps in different ways, including the pentagonal matrix. They can be further assembled into a "pineapple" shape, which can also tile the plane arranged in different ways, using only one shape (tile). We present several examples that include: periodic, non-periodic, rotational, radial and free-form tessellations. These shapes, in addition to their visual attractiveness and decorativeness which can be used in design, also hide the connection with patterns that can be found in nature, similar to Turing patterns.