Why do only three types of regular polygons tesellate the plane and what are they?

Regular polygons are polygons that have equal-length sides and equal interior angles. In order for a regular polygon to tessellate the plane, it means that it can fill a flat surface with no gaps or overlaps. There are only three regular polygons that have this property.

1. Triangle (Equilateral): A triangle is a regular polygon with three sides. It tessellates the plane because the interior angles of a triangle add up to 180 degrees. Thus, when several triangles are placed together, the sum of the interior angles around each vertex will be 360 degrees, perfectly fitting into a full rotation and allowing the triangles to fill the plane.

2. Square: A square is a regular polygon with four sides. It tessellates the plane because the interior angles of a square are all right angles, measuring 90 degrees. Since the sum of the interior angles around each vertex is 360 degrees (four times 90 degrees), the squares can be arranged edge-to-edge without any gaps or overlaps.

3. Hexagon: A hexagon is a regular polygon with six sides. It tessellates the plane because the interior angles of a hexagon measure 120 degrees. Like the triangle and square, the sum of the interior angles around each vertex is 360 degrees (six times 120 degrees), allowing hexagons to tessellate the plane without gaps or overlaps.

To determine why only these three types of regular polygons tessellate the plane, it is helpful to consider their interior angles. The angles of any polygon must evenly divide the total degrees in a full circle (360 degrees) in order for it to tessellate the plane. For example, with triangles, 180 degrees can be divided equally into three angles of 60 degrees each. Similarly, squares divide 360 degrees into four 90-degree angles, and hexagons divide 360 degrees into six 120-degree angles. These equal divisions of the interior angles allow these regular polygons to tessellate the plane perfectly.