Tessellations & how they fit together.

Tessellations are beautiful creations in geometry.  But what makes a two dimensional shape tessellate?  Why don't all shapes tessellate? Can non straight edged shapes tessellate? This page will answer these common questions.

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Where to spot tessellations

The most common shapes used for tessellations, or tessellating patterns are triangles, squares and hexagons.  In fact, these are the only REGULAR polygons that will tessellate. 

You will find these used commonly in floor tile patterns. You can also see these on ceilings when ceiling tiles are used.

Is there a 'rule' for tessellations?

"The shape chosen, must be repeated over and over to cover a flat surface without resulting in overlaps or gaps."

It's also nice to know that;

  • EVERY triangle will tessellate and,
  • EVERY quadrilateral will tessellate.

Can you think why this would be true?

Classification of Tessellations

Regular

These are patterns completed using ONLY Regular Polygons. The only regular polygons that work are the triangle, square and hexagon.

Semi-Regular

This is a pattern created using two or more REGULAR polygons, but the exciting thing is that even regular polygons that cannot tessellate on their own, are combined with other regular polygons, and the 'new shape' is what actually tessellates!

Non-Regular

This is a tessellations where any shapes are used together to create a tessellating pattern.

Do tessellations only occur with straight edged shapes?

That's the million dollar question.  Some mathematicians don't agree that curves should be included with tessellations, but others say, if it can be made up of any shape that can repeat, fill a flat surface with no resulting spaces or overlaps - it is a tessellation.  For now, I'm going to say yes, because I think they are very beautiful!

World renowned artists have used tessellations in spectacular ways. The artist M.C. Escher created the image below.

You can learn how to non regular tessellations here or you can print off pre-made classic geometric coloring pages to star designing your own.

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