Tilings: covering the plane

A tiling (or tessellation) is a pattern of shapes that covers a flat surface , no gaps, no overlaps, forever in every direction. Honeycombs. Bathroom floors. Brick walls. The jali screens of the Taj Mahal. Tilings are everywhere, and the mathematics behind them is gloriously simple.

Idea

The vertex rule. At every point where tile corners meet, the angles must add up to exactly $360°$. If they fall short, you get a gap. If they overshoot, the tiles overlap.

Regular tilings. A regular tiling uses copies of just one regular polygon (all sides equal, all angles equal). Three regular polygons , and only three , can tile the plane all by themselves:

A regular pentagon has an interior angle of $108°$: $3 \times 108° = 324°$ (gap), $4 \times 108° = 432°$ (overlap). So regular pentagons do not tile the plane alone.

Why these three? The interior angle of a regular $n$-gon must divide $360°$ for whole copies to fit around a vertex. The divisors of $360$ that match interior angles of regular polygons happen to be exactly $60°, 90°$ and $120°$.

All triangles tile. All quadrilaterals tile. A wonderful surprise: take any triangle (even a scalene one). Pair it with a copy rotated by $180°$ , you get a parallelogram. Parallelograms always tile. So any triangle tiles. Similarly, any quadrilateral (even non-convex!) tiles, by rotating $180°$ about the midpoint of each side.

Semi-regular tilings. Mix two or more regular polygons so each vertex looks the same. Example: at every vertex meet two squares and three triangles ($2 \times 90° + 3 \times 60° = 180° + 180° = 360°$ , works!). These are called the Archimedean tilings; there are exactly $8$.

Real-world tilings.

• Honeycomb , hexagons, because hexagons use the least wall length for a given area (a fact bees seem to know).
• Brick walls , staggered rectangles, for strength.
• Indian jali screens , six-pointed stars and hexagons.
• Penrose tilings , kite and dart shapes that tile without ever repeating exactly. Discovered in $1974$ by Roger Penrose.

Worked examples

Example 1. Can regular octagons tile the plane alone?

The interior angle of a regular octagon is $\tfrac{(8-2)\times 180°}{8} = 135°$. We need $135° \times k = 360°$, giving $k = \tfrac{360}{135} = \tfrac{8}{3}$ , not a whole number. So no, octagons alone do not tile. (But octagons + squares do , that's a famous Archimedean tiling.)

Example 2. Show that any triangle tiles the plane.

Take any triangle $T$. Make a copy $T'$ and rotate it $180°$ about the midpoint of one side. Glued together, $T$ and $T'$ form a parallelogram. Parallelograms tile the plane in obvious rows and columns. So triangles tile.

Example 3. A vertex of a tiling has angles $90°, 90°, 60°, 60°, 60°$. Check that it works.

Sum $= 90 + 90 + 60 + 60 + 60 = 360°$. Yes , two squares and three equilateral triangles meet at every vertex of this Archimedean tiling.

Example 4. Why are honeycombs hexagonal rather than triangular or square?

Among the three regular tilings, hexagons minimise the total wall length for a given total area. Bees use the least wax , an evolutionary optimum noticed already by Pappus of Alexandria around $300$ CE.

Try it yourself

1. List the three regular polygons that tile the plane and verify the $360°$ condition at each vertex.
2. The interior angle of a regular dodecagon ($12$-gon) is $150°$. Does it tile alone? Why or why not?
3. Cut a paper scalene triangle and try to tile a sheet of paper with copies of it. (Use both the triangle and its $180°$-rotation.)
4. A vertex meets $4$ squares and $1$ hexagon. Does it tile? Check angles.
5. Take a non-convex (concave) quadrilateral and try to tile with copies. Surprisingly, it works!
6. Combine regular hexagons and equilateral triangles so each vertex has two hexagons and two triangles ($2\times 120° + 2\times 60° = 360°$). Sketch the tiling.
7. Why does a regular pentagon fail to tile alone? Compute the gap angle.
8. Look at the floor of your kitchen, bathroom or classroom. Identify the basic tile and the vertex pattern.

Activity

Cut $10$ congruent equilateral triangles from coloured paper. Tile a small area of a sheet , can you do it in more than one way? Now cut $5$ regular hexagons. Tile again. Finally, mix triangles and hexagons. Photograph or sketch your three patterns and label the angle sum at one vertex of each.