Pentagon Tiling: The Shape That Refuses to Tile the Plane on Its Own

Why 108 degrees is the whole problem

Only three regular polygons tile the plane on their own with no gaps and no overlaps: the equilateral triangle (60°), the square (90°) and the regular hexagon (120°) — each because their interior angle divides 360° evenly, so a whole number of copies fits exactly around every vertex (6, 4 and 3 respectively). A regular pentagon's interior angle is 108°, and 360/108 = 10/3 is not an integer — three pentagons around a point leave a 36° gap, and a fourth would overlap. Regular pentagons alone can never tile the plane, full stop, no matter how cleverly you arrange them. That single failed division is the reason pentagon tiling is interesting at all: every pentagon that does tile the plane has to be irregular, with angles and side lengths engineered specifically to close the gaps that a regular pentagon leaves open.

Fifteen known families, found over a century

Mathematicians classify convex pentagons that can tile the plane into named families, each defined by a specific set of angle and side-length constraints (not a single fixed pentagon, but a continuous family satisfying those constraints). Karl Reinhardt found the first five types in 1918. Types accumulated slowly over the following decades — Richard Kershner found three more in 1968, Marjorie Rice, an amateur mathematician with no formal training beyond high-school geometry, discovered four entirely new types between 1976 and 1977 working from a Martin Gardner column, and Rolf Stein and later Casey Mann's team added more. In 2015, Mann, McLoud-Mann and Von Derau used a systematic computer search to find a fifteenth type — and in 2017 Michaël Rao proved, via an exhaustive computer-checked case analysis, that these fifteen types are complete: no convex pentagon outside them can tile the plane.

Type 1: the simplest working recipe

The Type 1 family is the easiest to build by hand: it requires two of the interior angles to sum to 360°, for instance angle A + angle B = 360°, in a pentagon shaped so that a pair of copies interlock into a hexagon-like unit that itself tiles the plane by simple translation. Its defining constraint is purely on angles — several of the fifteen types additionally constrain specific side lengths to be equal, which is what makes some families finite in shape and others (like Type 1) genuinely continuous, with a whole two-parameter family of distinct pentagon shapes all satisfying the same tiling rule.

Type 1 angle constraint:      A + B = 360 deg
tiling unit: two mirror-image (or rotated) copies of the pentagon
             interlock to form a strip; strips stack to tile the plane

The Cairo tiling: pentagons that look hexagonal

The best-known pentagon tiling in the real world is the Cairo pentagonal tiling, named for its use on Cairo street pavements, built from a Type 4 pentagon with two right angles. Each pentagon has a distinctive elongated, near-hexagonal silhouette, and four of them meet at every vertex in a pinwheel arrangement — rotated 90° from their neighbours rather than mirrored. The Cairo tiling is also the dual tiling of the snub square tiling (one of the eight semiregular Archimedean tilings), which is why it inherits that tiling's four-fold rotational symmetry rather than the six-fold symmetry a hexagonal tiling would have, despite the visual resemblance.

What changes if the pentagon doesn't have to be convex

Rao's 2017 completeness proof is specifically about convex pentagons. Drop that restriction and allow concave (non-convex) pentagons, and there are additional, less-studied tilings — the classification problem for the fully general non-convex case remains open. This is a useful reminder about how tiling theory is actually built: it is not a single sweeping theorem but a sequence of narrower, precisely-scoped questions (regular polygons, then convex polygons of each side count, then increasingly general shapes), each answered by its own separate proof.

Frequently asked questions

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