Tessellations and Tilings

Introduction

Few geometric ideas are as immediately satisfying as a tessellation. Take a single shape, copy it, and fit the copies together so that they cover an entire flat surface with no gaps and no overlaps. The result is a tiling — a pattern that could, in principle, extend to infinity. We encounter tessellations every day: in bathroom floors, brick walls, honeycombs, and the hexagonal columns of basalt at the Giant's Causeway. Yet behind this familiar concept lies a surprisingly deep mathematical story. Which shapes can tile the plane? Why do some regular polygons work and others fail? And what happens when we abandon periodicity altogether?

This chapter explores tessellations from the simplest regular cases through to the extraordinary aperiodic tilings discovered in the twentieth century. Along the way, we will see connections to regular polygons, Platonic Solids, and the golden ratio.

What Is a Tessellation?

A tessellation (from the Latin tessella, a small square tile used in Roman mosaics) is an arrangement of shapes that covers a flat surface completely, with no gaps between shapes and no overlaps. More precisely, a tessellation of the Euclidean plane is a countable collection of closed regions (called tiles) whose union is the entire plane and whose interiors are pairwise disjoint.

There are only a few requirements:

1. The tiles must cover every point in the plane.
2. No two tiles may overlap (though they share edges and vertices).
3. The arrangement must be edge-to-edge — where two tiles meet, they share a complete edge, not a partial one (this is the standard convention, though more general tilings relax it).

A tessellation is periodic if it has a translational symmetry: you can slide the entire pattern by some fixed distance in some direction and it maps exactly onto itself. Most of the tessellations we encounter in everyday life are periodic. But as we shall see, not all tessellations need be.

Regular Tessellations

A regular tessellation is one composed entirely of congruent copies of a single regular polygon, arranged edge-to-edge, with the same number of polygons meeting at every vertex. How many regular tessellations exist?

The answer comes from a beautifully simple argument about angles.

The Interior Angle Argument

Recall from the chapter on regular polygons that the interior angle of a regular n-gon is:

Interior angle = (n − 2) × 180° ÷ n

At any vertex of a tessellation, the angles of all tiles meeting at that point must sum to exactly 360° — no more (that would cause overlap) and no less (that would leave a gap). If k copies of a regular n-gon meet at each vertex, we need:

k × (n − 2) × 180° ÷ n = 360°

Rearranging:

k × (n − 2) = 2n

k = 2n ÷ (n − 2)

Since k must be a positive integer (you cannot have a fractional number of tiles at a vertex), we need (n − 2) to divide 2n evenly. Let us test the possibilities:

For n ≥ 7, the value of k falls below 3, and you need at least 3 tiles to surround a point (two tiles meeting at a vertex would only cover a wedge). Therefore exactly three regular tessellations exist: the triangular, the square, and the hexagonal.

The Three Regular Tessellations

Triangular tessellation (3, 3, 3, 3, 3, 3). Six equilateral triangles meet at every vertex. The vertex configuration is written 3.3.3.3.3.3, or simply 3⁶. This is the densest arrangement of the simplest polygon, and it possesses six lines of symmetry through each vertex.

Square tessellation (4, 4, 4, 4). Four squares meet at every vertex, giving the familiar grid pattern. Its vertex configuration is 4⁴. The square tiling is the basis of Cartesian coordinates and graph paper — a fact explored further in our chapter on coordinate geometry.

Hexagonal tessellation (6, 6, 6). Three regular hexagons meet at every vertex, configuration 6³. This is the tessellation found in honeycombs and is notable for being the most efficient: of all ways to partition the plane into regions of equal area, the hexagonal tiling has the shortest total perimeter. This was conjectured by the Roman scholar Marcus Terentius Varro in 36 BC and proved by Thomas Hales in 1999.

Why Pentagons Do Not Tessellate

The failure of the regular pentagon to tessellate deserves special attention, because it connects to a much larger theme in geometry.

The interior angle of a regular pentagon is 108°. Three pentagons meeting at a vertex would give 3 × 108° = 324°, leaving a 36° gap. Four pentagons would give 4 × 108° = 432°, which exceeds 360° by 72°. There is no integer number of regular pentagons that sums to exactly 360°.

This is not merely a curiosity. The pentagon's inability to tile the plane is mirrored in three dimensions by the regular dodecahedron's inability to fill space. Regular tetrahedra, cubes, and octahedra can be combined to fill three-dimensional space (as discussed in the chapter on Platonic Solids), but dodecahedra cannot. The dodecahedron's twelve pentagonal faces carry the same angular incompatibility into the third dimension.

Semi-Regular (Archimedean) Tessellations

If we allow two or more different types of regular polygon in the same tiling, but still require that every vertex has the same arrangement of polygons (in the same cyclic order), we obtain the semi-regular or Archimedean tessellations. These were studied by Johannes Kepler in his 1619 work Harmonices Mundi.

There are exactly eight semi-regular tessellations (excluding the three regular ones and their mirror images). Each is uniquely identified by its vertex configuration — the list of polygon types, in cyclic order, around each vertex:

The proof that exactly eight exist relies on the same angular constraint: the polygon angles at each vertex must sum to exactly 360°, and the vertex arrangement must be able to propagate consistently across the entire plane. Several vertex configurations that satisfy the 360° condition locally turn out to be impossible to extend globally — they create contradictions a few tiles away from the starting vertex.

Notable Semi-Regular Tilings

The trihexagonal tiling (3.6.3.6) is particularly significant. Alternating triangles and hexagons create a pattern known in Japanese tradition as kagome ("basket-eye"). In physics, the kagome lattice appears in certain magnetic materials that exhibit geometric frustration — the triangular arrangement prevents all neighbouring magnetic moments from aligning, leading to exotic quantum states.

The truncated square tiling (4.8.8) is familiar as the pattern of many tiled floors and pavements. It can be understood as a square grid in which each corner has been "cut off" (truncated), turning each square into a regular octagon and revealing a small square at each former vertex. This connects to the idea of truncation in polyhedra, explored in the chapter on regular and semi-regular polyhedra.

Dual Tessellations

Every tessellation has a dual, constructed by placing a point at the centre of each tile and connecting points whose tiles share an edge. The dual of a tessellation is itself a tessellation.

The duals of the three regular tessellations are instructive:

• The dual of the triangular tiling is the hexagonal tiling, and vice versa. This mirrors the duality between the icosahedron (triangular faces) and the dodecahedron (pentagonal faces) among the Platonic Solids, and more directly the duality between the octahedron and the cube — a theme explored in Dual Polyhedra.
• The dual of the square tiling is another square tiling, offset by half a cell. The square tiling is self-dual, just as the tetrahedron is the self-dual Platonic Solid.

The duals of the eight semi-regular tessellations are called the Laves tilings (or Catalan tilings, by analogy with the Catalan Solids in three dimensions). While the Archimedean tilings have regular polygons but different polygon types at each vertex, the Laves tilings have congruent (but generally non-regular) polygon tiles, with different vertex types. The duality swaps the roles of faces and vertices, exactly as it does for polyhedra.

This pattern of duality — swapping faces and vertices, swapping regularity of tiles for regularity of vertices — is one of the deep structural themes running through geometry. It appears in two dimensions (tessellations), three dimensions (polyhedra), and beyond.

Non-Periodic Tilings

All the tessellations discussed so far are periodic: they have translational symmetry. But in the 1960s, mathematicians began asking a provocative question: can a set of tiles cover the plane only non-periodically?

Penrose Tilings

In 1974, the mathematical physicist Roger Penrose discovered that just two tile shapes — now known as kites and darts — can tile the entire plane, but only in a non-periodic way. No matter how the tiles are arranged (following certain edge-matching rules), the resulting pattern never repeats.

The kite and dart are both quadrilaterals whose angles and side lengths are governed by the golden ratio φ = (1 + √5) ÷ 2 ≈ 1.618. The ratio of the area of the kite to the area of the dart is φ. In any sufficiently large Penrose tiling, the ratio of kites to darts also approaches φ. The golden ratio, which we encounter in the pentagon, the dodecahedron, and the Fibonacci sequence, is thus woven into the very fabric of Penrose tilings.

Penrose tilings display fivefold rotational symmetry — a property forbidden in periodic tilings (the crystallographic restriction theorem states that periodic tilings of the plane can only have 2-fold, 3-fold, 4-fold, or 6-fold rotational symmetry). This fivefold symmetry connects Penrose tilings back to the pentagon and to the concept of forms that cannot repeat periodically — the same angular incompatibility that prevents pentagons from tessellating periodically reappears here in a more subtle guise.

Properties of Penrose Tilings

Several remarkable properties distinguish Penrose tilings:

Self-similarity. A Penrose tiling can be "deflated" — each tile subdivided into smaller kites and darts — to produce another valid Penrose tiling at a smaller scale. This process can be repeated indefinitely, revealing a fractal-like hierarchy of structure. Conversely, tiles can be grouped ("inflated") into larger kites and darts. The ratio between scales is always φ.

Local isomorphism. Any finite patch of tiles that appears in one Penrose tiling appears in every Penrose tiling. Despite the fact that uncountably many distinct Penrose tilings exist, they are all locally indistinguishable. Any finite region you examine could belong to any of them.

Forbidden symmetries. While no single point of perfect fivefold symmetry need exist in a given Penrose tiling, the diffraction pattern (Fourier transform) displays sharp, bright spots arranged with perfect fivefold symmetry. The crystallographic restriction theorem forbids fivefold symmetry in periodic tilings — Penrose tilings achieve it precisely because they are aperiodic.

Conclusion

Tessellations begin with a deceptively simple question — which shapes fit together? — and lead to some of the most profound ideas in mathematics. The angular arithmetic that limits regular tessellations to exactly three is the same arithmetic that limits Platonic Solids to exactly five. The pentagon's refusal to tile the plane foreshadows the dodecahedron's inability to fill space. And when we relax the requirement of periodicity, Penrose tilings reveal the deep role of the golden ratio in organising non-periodic order.

The next chapter examines the geometric paths defined by constraints — Loci.