Can a Pentagon Tessellate? | The Math of Tilings

Exploring how shapes fit together to cover a surface without gaps or overlaps, known as tessellation, offers a fascinating look into geometry. This concept applies directly to many real-world designs, from floor tiles to intricate architectural patterns. The question of whether a pentagon can tessellate often comes up because, unlike triangles, squares, or hexagons, pentagons present a unique challenge in tiling.

Understanding Tessellation Fundamentals

Tessellation, or tiling, involves arranging one or more geometric shapes (tiles) on a flat surface without any gaps or overlaps. A fundamental characteristic of a tessellation is that it must completely cover the plane. This geometric principle is central to understanding how different polygons behave when arranged together.

For a regular polygon to tessellate the plane on its own, its interior angle must be an exact divisor of 360 degrees. This condition ensures that multiple copies of the polygon can meet at a single vertex without creating gaps or overlapping. Regular polygons have all sides equal in length and all interior angles equal.

• An equilateral triangle has interior angles of 60 degrees (60 6 = 360).
• A square has interior angles of 90 degrees (90 4 = 360).
• A regular hexagon has interior angles of 120 degrees (120 3 = 360).

These three regular polygons are the only ones that can form a regular tessellation, where only one type of regular polygon is used and all vertices are identical.

The Unique Challenge of Pentagons

A regular pentagon has five equal sides and five equal interior angles. Each interior angle of a regular pentagon measures 108 degrees. When we attempt to place regular pentagons around a point, we find that 108 degrees is not a divisor of 360 degrees. Three regular pentagons meeting at a vertex would sum to 324 degrees (3 108), leaving a 36-degree gap. Four regular pentagons would sum to 432 degrees, causing an overlap.

This mathematical property means that a regular pentagon cannot tessellate the plane on its own. The challenge then shifts to irregular pentagons, which do not have all sides or angles equal. These irregular shapes introduce complexity, as their varied angles and side lengths might allow for arrangements that fill the plane.

Pentagons can be classified as either convex or non-convex. A convex polygon has all its interior angles less than 180 degrees, meaning all vertices point outwards. A non-convex (or concave) polygon has at least one interior angle greater than 180 degrees, causing at least one vertex to point inwards.

Convex Pentagons That Tessellate

The search for convex pentagons that tessellate is a notable chapter in mathematics, spanning over a century. Unlike regular polygons, there is no simple formula to determine which irregular convex pentagons can tile the plane. The discovery of these tiling pentagons has largely been through systematic exploration and, in some cases, serendipitous findings.

The first five types of convex pentagonal tilings were discovered by Karl Reinhardt in 1918 as part of his doctoral thesis. These discoveries established that while regular pentagons fail, certain irregular convex pentagons can indeed tessellate.

Decades later, in 1968, Richard Kershner discovered three more types, bringing the total to eight. This expanded the understanding of the conditions under which these complex shapes could tile. The search continued, driven by the mathematical curiosity about whether more types existed.

A significant breakthrough came from Marjorie Rice, a homemaker with a passion for mathematics, who discovered four new types between 1976 and 1977. Her work, conducted outside traditional academic settings, demonstrated the power of dedicated exploration. Rice’s method involved systematically drawing and testing variations of pentagons.

Rolf Stein added another type in 1985, bringing the count to 14. For many years, mathematicians believed that the list of known convex tiling pentagons was complete. However, in 2015, Casey Mann, Jennifer McLoud, and David Von Derau from the University of Washington Bothell announced the discovery of a 15th type of convex pentagon that tessellates. This discovery reignited interest and research in the field.

The comprehensive proof that only 15 types of convex pentagons can tessellate the plane was provided by Michaël Rao in 2017. His rigorous computational analysis confirmed that no other types exist, solidifying the known set of these unique tiling shapes. This proof closed a long-standing problem in geometry.

Characteristics of Tiling Convex Pentagons

The 15 types of convex pentagons that tessellate possess specific geometric properties that enable them to tile. These properties often involve precise relationships between their side lengths and interior angles. For instance, some types require certain angles to be multiples of others, or specific side lengths to be equal.

Many of these tiling pentagons do not form edge-to-edge tessellations, meaning that the sides of adjacent tiles do not always align perfectly. Instead, a vertex of one tile might meet the middle of a side of another. The tiling patterns often involve combinations of translation, rotation, and reflection to fit the pieces together.

Key Discoveries in Convex Pentagon Tilings

Year	Discoverer(s)	Number of Types Added
1918	Karl Reinhardt	5
1968	Richard Kershner	3
1976-1977	Marjorie Rice	4
1985	Rolf Stein	1
2015	Mann, McLoud, Von Derau	1

Non-Convex Pentagons and Their Tilings

Non-convex, or concave, pentagons offer greater flexibility for tessellation due to their reflex angles (angles greater than 180 degrees). These inward-pointing vertices can often fit into the outward-pointing angles of other tiles, facilitating complex arrangements. The conditions for non-convex pentagonal tilings are less constrained than for convex ones.

While the focus of historical research often centered on convex shapes, non-convex pentagons can also form various tessellations. A prototile is a basic shape that can be used to tile a plane. Many non-convex pentagons can serve as prototiles, creating intricate and often visually striking patterns. The presence of reflex angles allows for more varied ways for the tiles to interlock.

Aperiodic Tilings and Pentagons

Beyond periodic tessellations, where a pattern repeats indefinitely, there are aperiodic tilings. These tilings cover the plane without any repeating unit. Penrose tilings, for example, are famous aperiodic tilings that often exhibit five-fold rotational symmetry, a characteristic associated with pentagons. While Penrose tilings typically use rhombuses or kites, they demonstrate how shapes with pentagonal symmetry can contribute to complex tiling structures. The individual tiles in a Penrose tiling are not single pentagons, but the overall patterns can display pentagonal features.

The Mathematical Conditions for Pentagon Tessellation

For any polygon, including a pentagon, to tessellate the plane, the sum of the angles around any vertex where tiles meet must be exactly 360 degrees. This is the fundamental local condition for tiling. Additionally, the side lengths must be compatible, allowing the edges of adjacent tiles to fit precisely without gaps or overlaps.

The ability of a pentagon to tessellate often relies on specific combinations of its angles and side lengths. For instance, some tiling pentagons have pairs of equal angles or equal side lengths that facilitate their arrangement. The operations of translation, rotation, and reflection are key to arranging these shapes. A single pentagon might need to be rotated by a specific angle or reflected across an axis to fit alongside its neighbors.

The geometry of these tiling pentagons is not random. It follows precise mathematical rules. For example, some tiling pentagons have angles that are related by simple ratios, such as one angle being twice another. These relationships are crucial for ensuring that the total angle around any vertex sums to 360 degrees, even when different angles of the pentagon meet at that point.

Conditions for Polygon Tessellation

Condition	Description
Angle Sum at Vertex	The sum of interior angles meeting at any vertex must be 360 degrees.
Side Compatibility	Corresponding side lengths of adjacent tiles must match exactly.
Plane Coverage	The tiles must completely cover the plane without gaps or overlaps.

Real-World Applications and Learning Connections

The study of tessellations, particularly those involving complex shapes like pentagons, extends beyond abstract mathematics. These geometric principles find application in various fields. In art and architecture, tessellations are fundamental to creating intricate patterns, such as those found in Islamic geometric art or the visual illusions of M.C. Escher. Exploring these patterns helps develop spatial reasoning and an appreciation for mathematical beauty.

In material science, the arrangement of atoms and molecules often forms tessellation-like structures. Quasicrystals, discovered in the 1980s, exhibit five-fold symmetry similar to aperiodic tilings, which was previously thought impossible for crystalline structures. Understanding how shapes pack together is also relevant to the study of materials like graphene, where hexagonal arrangements of carbon atoms create strong, lightweight structures. You can learn more about these fascinating structures through resources like Khan Academy.

The exploration of tiling pentagons serves as an excellent example of ongoing mathematical discovery and problem-solving. It shows that even in seemingly well-understood areas of geometry, new findings can emerge. This pursuit encourages analytical thinking and perseverance, valuable skills for any learner. The process of identifying and proving the existence of these tiling shapes demonstrates the iterative nature of scientific inquiry. Further insights into geometric principles are often available from institutions like the National Science Foundation.