A tessellation is a pattern made by fitting copies of a shape together so that they cover a surface completely with no gaps and no overlaps, and could carry on for ever. A tiled floor, a brick wall and a honeycomb are all tessellations.
Look at any point where tiles meet. The angles round that point must add up to exactly 360 degrees, because a full turn is 360 and the tiles must close up without leaving a gap or overlapping. That single fact decides everything in this topic.
Which regular polygons tessellate?
For a regular polygon the question becomes simple arithmetic: does the interior angle divide exactly into 360? If you get a whole number, that many tiles fit round a point. If you get a fraction, it never works.
| Regular polygon | Interior angle | 360 ÷ angle | Tessellates? |
|---|---|---|---|
| Equilateral triangle | 60° | 6 | Yes |
| Square | 90° | 4 | Yes |
| Regular pentagon | 108° | 3.33... | No |
| Regular hexagon | 120° | 3 | Yes |
| Regular heptagon | 128.57° | 2.8 | No |
| Regular octagon | 135° | 2.67 | No |
Exactly three regular polygons tessellate on their own: the equilateral triangle, the square and the regular hexagon. From the heptagon onwards the interior angle is between 120 and 180, so two tiles leave a gap and three would overlap. Two is not enough and three is too many.
The surprising part
Irregular shapes do far better than you would expect. Every single triangle tessellates, whatever its shape, and so does every single quadrilateral, including ones that look completely useless for the job.
Any triangle has angles adding to 180. Two copies, one turned 180 degrees, make a parallelogram, and parallelograms fill a plane. At each meeting point you get each of the three angles twice, and twice 180 is exactly 360.
The angles of any quadrilateral add to 360. So arrange four copies round a point with a different angle at that point each time, and the four angles present are exactly the four angles of the shape, adding to 360 by definition.
Rotating and flipping are allowed, and for most irregular shapes they are essential. An awkward quadrilateral will not tile if every copy faces the same way, but tiles perfectly if alternate copies are turned through 180 degrees.
Three regular shapes tile a plane, so why is a honeycomb hexagonal? Because for a given amount of wax, a hexagon encloses more space than a square or triangle with the same perimeter, so hexagonal cells hold more honey for less wall.
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