PUZZLING PENTAGONS—HOW MANY WAYS CAN WE COVER A FLAT SURFACE?

Convex polygons, shapes bounded by straight lines in which all of the corners point outward, are the simplest of shapes, but despite their simplicity, there are many unsolved problems concerning polygons. Some polygons have the nice property that they fit together espcially well, so that you can use lots of copies of them to cover a large surface to form what is called a tiling. For some polygons, such as triangles, it is easy to see how to form tilings. For others, such as seven-sided convex polygons, it is impossible to form a tiling. This article discusses the history of a long-standing open question in geometry: which convex pentagons give rise to tilings of the plane? The authors also discuss their contribution to the solution of this problem, which involved developing a computerized algorithm to help them search for a new kind of convex pentagon that can form tilings, and the basic idea of this algorithm is discussed.

mathematics mostly involves learning to use formulas discovered in ancient times, such as the Pythagorean theorem.While that is not true, mathematical discoveries can often be di cult to explain in nontechnical terms, and many times these discoveries do not have immediate real-world applications.Indeed, we (the authors) were unaware of new mathematical discoveries until we were college students, so we want to give you an insider's view on how these discoveries can be made.
Like most new science, new mathematical discoveries are born out of curiosity about how the world works.We believe that developing habits of mind that lead to questioning why things are true and whether those truths can apply to new situations can lead to the discovery of new mathematics.People who want to discover new mathematics must be curious about why and how mathematics works, instead of just accepting facts without question!In that spirit, this article tells the story of our own mathematical discovery.

THE TILING PROBLEM: WHAT SHAPES CAN COVER THE PLANE WITHOUT GAPS OR OVERLAPS?
Tilings are patterns formed from shapes that cover various kinds of   In geometry, the definition of a tiling is an arrangement of shapes that covers an entire flat surface that goes on forever in all directions, called a plane, without gaps or overlaps ( Figures D-F).To understand the science of tilings, mathematicians are interested in the central question: "what shapes can be used to form tilings?" This is called The Tiling Problem.Because the shapes in tilings can be so varied (jagged, curvy, pointy, big, or small), it is impossible to answer this question without simplifying it.
To simplify The Tiling Problem so that we may hope to solve it, we must restrict the conditions of the problem in three ways.First, we require that all the individual tiles in the tilings have the same shape create the tiling be a polygon, which is a shape bounded by straight lines, like a triangle (three sides) or a pentagon (five sides).Last, we required that the polygon be convex, meaning its corners all point outward, as in convex polygons can be used to form monohedral tilings of the plane?" (You can try this for yourself by following the direction in Appendix A).The solution to The Tiling Problem for Convex Polygons has a rich history, spanning over years and involving many people (Table ).Some of the people who contributed to this problem's  solution were not professional mathematicians; they were just curious people who asked "Why?"We hope this encourages you to do so as well!

ISOHEDRAL TILINGS -RECIPES FOR TILINGS
Imagine that you have two copies of the same tiling, one lying on top of the other so that they are perfectly aligned.Now picture sliding the top copy in some direction so that the top copy again perfectly aligns with the bottom copy.Such a motion is called a symmetry of the tiling.

SYMMETRY
A symmetry of a tiling is a way to move a tiling so that the "before picture" and the "after picture" are identical.
Another way to understand the idea of a symmetry of a tiling is to think of a "before picture" and an "after picture"-if you slide a tiling and the before picture and the after picture are exactly the same, then the direction and distance that you slid the tiling is a symmetry of the tiling.To describe the complexity of the symmetries of a monohedral tiling, we can ask whether or not, for any two tiles in that tiling, there is a symmetry of the tiling that moves the first tile to the second tile.If the answer is yes, we say the tiling is isohedral.

ISOHEDRAL
If a monohedral tiling T has the property that for any two tiles T and T in T , there is a symmetry of T that moves T to T , we say T is isohedral.
Isohedral tilings look the same around each tile, so we can understand how the shapes fit together in the whole tiling just by understanding what is happening around the boundary of any one tile.The way tiles fit around each other in an isohedral tiling is described by an incidence symbol, which can be viewed as a sort of recipe for how a shape can tile the plane (see Appendix B to explore the idea of incidence symbols.)

HOW WE FOUND A NEW TYPE OF PENTAGON
Led by our curiosity, we began to look for common features of the types of tilings known at the time (types -in Figure ).We made two important observations, which generated even more questions!First, types -can produce isohedral tilings.In types -, if you cluster together two or three pentagons to act like a single tile, then that cluster produces an isohedral tiling; some types form tilings by clusters of two pentagons, as in Figures G, L, and others form tilings by clusters of three pentagons, as in Figures J, N.This observation guided us to consider how we might discover new convex pentagons that similarly tile the plane in isohedral clusters.
Our second observation was, along the boundaries of the clusters in tilings of types -, we saw that the middle part of the edges of some pentagons met the corners of other pentagons, as illustrated by the red points in Figure N.We call these points flat nodes.We

FLAT NODE
A flat node in a k-block transitive tiling by pentagons is a point on the boundary of a k-block where the corner of one pentagon meets a point in the middle of an edge of another pentagon.

K-BLOCK TRANSITIVE
k-block transitive tilings are those in which identical clusters of k pentagons, acting as a single tile, form an isohedral tiling.
wondered how many flat nodes there might be in each cluster, and we proved that in tilings involving clusters of two pentagons, there can be at most two flat nodes per cluster.Similarly, in tilings involving clusters kids.frontiersin.org

Mann and McLoud-Mann
Puzzling Pentagons The types of convex pentagons that can tessellate the plane.(A) Type : of three pentagons, there are at most three flat nodes.This was crucial to understanding all the possibilities.Using these observations, we developed a computer algorithm to search for all possible pentagons that can produce these clustered tilings in which the sizes of the clusters are up to four.Type (Figure O) was discovered through this computerized search.The algorithm works as follows.First, flat nodes are placed on the boundary of a template cluster of three pentagons.Second, an isohedral type was

Mann and McLoud-Mann
Puzzling Pentagons chosen for the cluster (out of possible types) and the cluster was labeled according to the "recipe" for that isohedral type.Third, copies of the clusters were placed around a central copy of the cluster, according to the isohedral recipe.Fourth, the corners and sides of the individual pentagons of the clusters were labeled (there are many ways to do this, and the computer checks them all).Finally, from the resulting figure, equations describing the sides and angles of the pentagons were generated.The steps of this process are illustrated in Figure .To justify this was a new type, we verified that we could make a pentagon with the specific angles and side lengths, and that such a pentagon did not satisfy any of the equations that define Types -(check for yourself!).
After our discovery of Type , French mathematician Micheal Rao heard the news and decided to investigate the problem.In doing so, he proved that the pentagon we discovered completed the classification, so types -form the complete classification of pentagons that tile the plane ([ ] ), settling the longstanding Tiling Problem for Convex M. Rao posted an article on the online repository ArXiv, but the article has not yet been verified through a process of peer review. Polygons!SUMMARY have probably seen tiled kitchen floors and patios, or maybe more elaborate examples such as M.C.Escher's artwork.Look around in your daily life, and you will likely see many tilings, as in FiguresA-C.Tilings appear in the natural world, such as in a bee's honeycomb, the cracked mud of a dried lake bottom, and a gira e's fur.They are even used to understand how the atoms of crystals fit together.Mathematicians have worked for many years to understand and classify various kinds of tilings.

Figure
Figure

Figure
Figure Real-world tilings (A-C) and tilings of the plane (D, E). (A) Gira e with tiling fur pattern.(B) Dried mud forms a tiling.(C) A mosaic pattern at the Alhambra.(D) A tiling by squares of sizes.(E) A monohedral tiling.(F) A tiling by nonconvex polygons.
Figure E. -sided polygons are called triangles; -sided TRIANGLES -sided polygons are called triangles.polygons are quadrilaterals; -sided polygons are pentagons; -sided QUADRILATERALS -sided polygons are quadrilaterals.PENTAGONS -sided polygons are pentagons.polygons are called hexagons; in general, polygons with n sides are HEXAGONS -sided polygons are called hexagons.called n-gons.n-GONS In general, polygons with n sides are called n-gons.When we use these limitations, our simplified version of The Tiling Problem, called The Tiling Problem for Convex Polygons, asks: "Which

Figure
Figure

FigureA
Figure A -block cluster that generates an isohedral tiling.The red and blue labels and arrows the incidence symbol [a + b+ c + d + e + f + ; a + + d − c − b + f + ]. understand the incidence symbol, see Appendix B.