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We are now concerned with tessellations of regular polygons, since regular polygons are the simplest types of polygons. Let us start with the simple case of a tessellation using only one type of polygon. Tessellations of only one type of polygon are called regular tessellations.
Trial and Error
However, when you try to put together some regular pentagons, you find that they cannot perfectly fill the space around a point. Thus, regular pentagons cannot tessellate by themselves. The following image explains the situation:
What about regular hexagons? Regular hexagons can tessellate easily, as it turns out.
Now consider regular heptagons. When trying to fit regular heptagons around a point or vertex, we see that the situation we had with regular pentagons happens again, overlap. Thus, like pentagons, heptagons do not perfectly fill the space around the point and, therefore, do not tessellate.
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