Constructions and Tilings · Lesson 11 of 12
Tiling the Entire Plane
“Explore how repeating shapes cover the plane and connect the patterns to art, buildings, and nature.”
• Distinguish tiling a bounded region from tiling the plane. • Explain square, equilateral-triangle, and regular-hexagon tilings. • Use angles around a point to check local fit. • Recognise mixed-shape and non-regular tilings. • Find repeating units in everyday and natural patterns.
From a Region to the Entire Plane
Until now we have tried to cover a particular rectangle, silhouette, or altered grid. A plane has no edge to reach. Tiling the plane means arranging copies of shapes so the pattern can continue in every direction without gaps or overlapping interiors.
A covering of the entire plane by shapes with no gaps and no overlapping interiors. A finite drawing can show a patch of a pattern that continues beyond its boundary.
A page can show only a finite patch, so inspect how its edges would extend. A pattern that fills one small outline may fail when more copies are added. For a plane tiling, the repeating arrangement must remain consistent as we continue it.
Squares give a familiar example. Place them in aligned rows and columns. Every interior vertex is shared by four squares, and each contributes a 90° corner. Their total is 360°, so the squares fit around that point. The matching sides also line up all along the boundaries.
Squares, Equilateral Triangles, and Regular Hexagons
Other regular polygons can also create repeating plane patterns. We have already studied the angles of equilateral triangles and regular hexagons. Their corner angles let us check how several copies meet at a common point.
Equilateral triangles have 60° angles. Six corners fit around an interior meeting point because 6 × 60° = 360°. The sides match when the triangles are congruent, and repeating the two triangle orientations fills the spaces between rows.
Regular hexagons have 120° interior angles. Three corners fit around each meeting point because 3 × 120° = 360°. Their matching edges extend into the familiar honeycomb pattern. The angle calculation explains local fit, while the repeated diagram shows how that fit continues across the plane.
| Regular tile | Corner angle | Number meeting at a vertex | Angle total |
|---|---|---|---|
| Square | 90° | 4 | 360° |
| Equilateral triangle | 60° | 6 | 360° |
| Regular hexagon | 120° | 3 | 360° |
Problem
Why do four squares fit at a vertex, whereas three leave a gap?
- 1.One square contributes 90° at that corner.
- 2.Three give 3 × 90° = 270°, leaving 360° − 270° = 90°.
- 3.Four give 4 × 90° = 360°, filling the full turn.
- 4.The four matching square edges form the usual grid pattern; adding a fifth at the same vertex would overlap.
Problem
How many regular hexagon corners fill a full turn?
- 1.One interior angle is 120°, from the two equilateral-triangle angles at the vertex.
- 2.Two hexagons contribute 240°, leaving 120°.
- 3.Three contribute 360°, exactly filling the meeting point.
- 4.The repeated hexagon diagram confirms that the matching side edges also continue without gaps.
More Than One Shape
A plane tiling does not have to use just one shape. Different polygons can fill the gaps left by each other. At a shared vertex their angles still need to complete a full turn, and their boundary edges must fit.
The source includes a mixed polygon pattern. In the example above, squares fill gaps between octagons. This supports the same central idea: a gap around one shape may be covered by another suitable shape. We are observing the fit, not adding a separate polygon-angle formula lesson.
An angle total of 360° at one point is an important local check, but it is not automatically a proof that an arbitrary collection of shapes tiles the entire plane. You must also check compatible edge lengths and that the arrangement can continue without a later mismatch.
Non-Regular and Artistic Tilings
A tile need not be a regular polygon. A rectangle that is not a square can tile the plane in rows, for example. More elaborate patterns can use curved outlines that interlock with their neighbours. The geometry concerns how the boundaries fit, not whether the tile looks like a familiar polygon.
The artist M. C. Escher used tiling ideas in patterns resembling animals. The outline of a tile can suggest a fish or bird while fitting a matching neighbouring outline. A useful design experiment begins with a tile that already fits: cut a shape from one side and move the identical cutout to the opposite side without changing its orientation. The removed space on one translated copy matches the added part on its neighbour.
The source also presents an irregular tiling example reported in 2023. This reminds us that tiling remains an active area of mathematical exploration; it does not mean that all tiling questions have been answered. We can appreciate a complex pattern while first practising the simple reasoning behind its local fits and repeats.
Problem
Can congruent 2 cm by 3 cm rectangles tile the plane?
- 1.Place rectangles in one row with their 3 cm edges horizontal and 2 cm edges vertical.
- 2.Repeat the row above and below, aligning the boundaries.
- 3.All matching sides meet exactly; each corner contributes a right angle and four meet around an interior vertex.
- 4.The pattern can continue indefinitely. A tile can therefore tile the plane without being a regular polygon.
Tilings in Buildings and Nature
Floor tiles, wall patterns, and designs on surfaces provide everyday examples of repeated coverage. You may see one tile shape, several shapes together, or an arrangement whose repeating unit contains more than one tile. Look for the smallest part whose repetition explains the pattern.
The front faces of bee hives and some wasp nests have hexagonal cell patterns. The cells hold eggs, larvae, pupae, or stored food. Their fitted arrangement uses the region without leaving gaps between the cell openings, connecting a natural pattern to the hexagon tiling we studied.
This geometric observation does not by itself explain the full biological process by which insects build the cells, nor prove that one shape is best for every possible task. Observe the fit carefully and separate what the pattern shows from questions it raises. Sketch a nearby floor or wall tiling, identify a unit, and explain how copies meet.
Seeing a finite patch with no gaps is not always enough to establish a plane tiling. Ask whether it extends consistently in every direction. Similarly, a 360° angle total at one vertex must be accompanied by matching edges and a repeatable arrangement.
Check Your Understanding
Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.
Quiz
What distinguishes a plane tiling from filling one bounded region?
How many 60° equilateral-triangle corners fill a full turn?
How many 120° regular-hexagon corners meet in the standard tiling?
Can a plane tiling use more than one shape?
Which statement is justified by a honeycomb front-face pattern?
Practice Problems
- Sketch a square tiling and label the four angles at one interior vertex.
- Sketch an equilateral-triangle patch and explain the 360° angle total.
- Sketch a hexagon tiling and identify three cells meeting at a vertex.
- Create a mixed-shape pattern or analyse the octagon-and-square example. Explain how the shapes fit.
- Find a tiling in a building or nearby design and identify a repeating unit.
- Explain why a non-square rectangle can tile the plane.
- State one geometrical observation and one unanswered process question about natural hexagonal cells.
Four angles of 90° total 360°. Equal square edges align in rows and columns.
Key Takeaways
• Plane tiling means gap-free, overlap-free coverage that continues throughout the plane. • Four squares, six equilateral triangles, or three regular hexagons can meet around a full turn. • Angle totals help check fit but must be combined with matching boundaries. • Mixed shapes and non-regular tiles can also produce plane tilings. • Artistic outlines can interlock while still obeying geometric coverage rules. • Buildings and natural hexagonal cells connect tiling to observable patterns.