In this blog post, we’ll explore the basic concepts and various types of tessellation, the history and research behind the convex tiling problem, and their applications in everyday life, art, and engineering.

Puzzles and Tessellation

Everyone has probably had the experience of moving pieces back and forth while playing the Seven-Piece Puzzle with friends to complete the shape just a little faster, or spending late nights putting together a 100- or 200-piece jigsaw puzzle. The fun and challenge these puzzles offer can be explained mathematically by a single concept called “tessellation.”
Tessellation refers to the method of arranging multiple shapes or tiles so that they do not overlap, thereby covering a plane without any gaps. Although it may seem like a simple game, it is one of the important fields of mathematics that is actively researched today in various disciplines, including geometry, combinatorics, and computer science.

Types of Tessellation

Tessellations are generally classified into three main categories.
The first is regular tessellation. This refers to a tessellation that covers a plane without gaps using only one type of regular polygon. The only regular polygons capable of forming a regular tessellation are the equilateral triangle, the square, and the regular hexagon. This is because the sum of the interior angles at a single vertex must be exactly 360 degrees. For example, since the measure of one interior angle of an equilateral triangle is 60 degrees, six such angles add up to 360 degrees, allowing the plane to be filled without gaps (360 ÷ 60 = 6). In contrast, since one interior angle of a regular pentagon is 108 degrees, 360 cannot be divided exactly, so a regular tessellation cannot be formed using only regular pentagons.
The second type is a semiregular tessellation (also known as an Archimedean tessellation). This refers to a tessellation composed of two or more types of regular polygons, where the types of polygons meeting at each vertex and their arrangement order are identical. For example, a case where one vertex is shared by one equilateral triangle and two regular dodecagons is denoted as (3,12,12). The possible forms of semiregular tessellations are limited, and there are a total of eight known cases to date. Representative examples include (3,12,12), (4,6,12), and (4,8,8).
Third is the irregular tessellation. This refers to a tessellation that uses two or more types of regular polygons but does not follow a consistent arrangement rule for the polygons meeting at each vertex. In other words, it is a form where different arrangements appear at each vertex or where different patterns coexist in different regions.
There is also a concept in tessellation called a dual tessellation. This method involves taking the center of each polygon in an existing tessellation as a new vertex and connecting the centers of adjacent polygons to create a new pattern.
A prime example is the Cairo tiling. This pattern became famous for its widespread use in the sidewalk pavers of Cairo, Egypt, and features a regular repetition of non-concave pentagonal tiles. Cairo tiling can be easily found in the flooring of buildings, in plazas, and in the paving designs of public facilities; it is known as a prime example of tessellation that demonstrates both geometric beauty and efficient use of space.

The Convex Polygon Tiling Problem: An Extension of Tessellation

While the tessellations examined so far have primarily focused on regular polygons, extending this concept to more general shapes gives rise to even more intriguing mathematical problems. A prime example is the convex tiling problem, which classifies shapes that can cover a plane without gaps using convex polygons.
According to current research, all triangles and all convex quadrilaterals can cover the plane without gaps, regardless of their shape or size.
Conversely, it has been proven that no convex polygon with seven or more sides can tessellate the plane. Furthermore, convex hexagons can cover the plane only if they satisfy certain conditions, and such hexagons are currently classified into three basic types.
Conversely, the convex pentagon remained a long-standing puzzle that captivated mathematicians. Determining which convex pentagons can tessellate the plane—and exactly how many distinct types exist—was a flagship problem that attracted over a century of research.
Reinhardt was the first to systematically study this problem, discovering five distinct types of convex pentagons in 1918. Subsequently, as various researchers discovered new types one after another, the number of known types gradually increased. In 2015, the problem garnered significant attention once again when Casey Mann and his research team discovered the 15th type—the first new discovery in about 30 years.
Even after Reinhardt’s work, the problem remained unsolved. Researchers continued to discover new types, and at one point, a total of 14 types were known. Then, in 2015, when Casey Mann and his colleagues discovered the 15th type, the problem once again captured the attention of the global mathematical community.
This was followed by large-scale computer-aided verification and mathematical proofs, and it has now been proven that there are exactly 15 types of convex pentagons capable of tiling the plane with a single shape. In 2017, Michaël Rao demonstrated through a rigorous computer-aided proof that no further new types exist. This problem has long been known as a prime example of a “problem that can be explained simply enough for a child to understand but is extremely difficult to solve,” and is now regarded as one of the most significant solved problems in the history of mathematics.
Each discovered convex pentagon must satisfy specific relationships between its side lengths and interior angles in order to cover a plane without gaps. Therefore, analyzing and classifying these different geometric conditions is at the core of the research. Recently, with advances in automated computer-aided proofs and computational geometry techniques, active research is being conducted to explore and verify various tiling problems, and this research is expanding to more complex spatial tiling problems.

Applications of Tessellation

Tessellation is, surprisingly, widely used in various fields around us. Its scope of application is extremely broad, ranging from art and architecture to computer graphics, materials science, life sciences, and structural design.
The Dutch printmaker M. C. Escher created unique tessellation works by using geometric transformations and optical illusions to arrange the shapes of creatures such as birds, fish, and lizards in repetitive patterns. He freely employed symmetry transformations—such as translation, rotation, reflection, and glide reflection—to artistically transform regular tessellations. Escher’s works are still regarded today as a prime example of the intersection of mathematics and art and are frequently used in mathematics education.
Tessellation is also a crucial technique in the field of computer graphics. By subdividing the relatively simple polygons that make up a 3D model into even smaller polygons, surfaces can be rendered much more smoothly and naturally. In real-time rendering, in particular, tessellation techniques that subdivide only the necessary areas in detail are used to achieve high-quality graphics while maintaining performance. Furthermore, when combined with displacement mapping, it enables more realistic representations of surfaces such as skin, rocks, terrain, and architecture in movies and games. These techniques are also widely used in game engines and 3D graphics software.
Tessellation also plays an important role in life sciences and materials science. Repetitive structures observed in nature—such as crystal structures, honeycombs, turtle shell patterns, certain plant tissues, and viral capsids—are formed by the regular arrangement of a consistent basic unit. The concept of tessellation serves as a highly useful tool in the process of analyzing and understanding these structures. Furthermore, the principles of tessellation are applied in various engineering fields, including the design of new materials, metamaterials, crystal structure analysis, and the optimization of architectural structures.
Physicist Richard Feynman once said that to understand nature, one must learn the language nature uses. Tessellation can be considered a prime example of a mathematical language that helps us understand complex planar structures through the regular combination of simple shapes. Originating from simple puzzle games, tessellation is now widely used not only in pure mathematics but also in art, architecture, computer science, materials engineering, and the natural sciences, serving as a vital bridge connecting different disciplines.