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Grades K–2 · Mathematics

How can repeating tiles cover a floor without holes?

A huge floor can be covered by repeating one small tile shape. Square tiles have straight sides that can touch all the way along. Where four tiles meet, their four corners fit around one point. No little space is left between those corners.

N27 visual explanationThe squares form a repeating floor pattern. The displayed window is fully covered without gaps or overlapping interiors. The pattern can continue beyond its border. This is about covering a surface, not counting rows or mirror symmetry.A floor can repeat without leaving holesA window of a continuing patternThe tiles meet with no gaps or overlaps.The border shows only part of the repeating floor.

In the diagramA floor can repeat without leaving holesA window of a continuing patternThe tiles meet with no gaps or overlaps.The border shows only part of the repeating floor.

The square tiles cover the displayed window without gaps or overlaps; the pattern can continue beyond its border.

More squares can join the same way, making new rows beside the old ones. No patch of floor is left bare, and no tile needs to lie on top of another. A covering like this is called a tessellation. The picture shows a small part of a pattern that could keep going.

Triangles can fit together to cover a floor too. Touching circles leave little spaces between them. A pattern can look neat and repeat many times without covering the whole surface. For a tiled floor, both the sides and the spaces where corners meet matter.

About this explanation

Starting knowledge: Familiarity with squares, triangles and circles is sufficient; an adult may read the text aloud.

Connect this idea: Structures & Materials

Sources

  • University of Cambridge NRICH — Tessellating Triangles
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