Why is it impossible for a regular polygon with more than six sides to tessellate the plane?
To tessellate a regular polygon, the internal angle must be an integral factor of 360. A regular hexagon can tessellate with three corners at a point, the minimum. The internal angles of regular polygons with more than six sides will exceed 120.
In fact, the only regular polygons to tessellate an infinite plane are triangles, squares, and hexagons.
To understand why a regular polygon with more than six sides cannot tessellate the plane, let's first define some terms:
1. Polygon: A polygon is a closed plane figure with straight sides. It is made up of line segments that connect at their endpoints to enclose a space. Regular polygons have equal side lengths and equal angles.
2. Tessellation: Tessellation, also known as tiling, is the process of covering a plane with repeating shapes without any gaps or overlaps.
Now, let's examine why a regular polygon with more than six sides cannot tessellate the plane:
1. Angle Sum Property: The sum of the interior angles of a polygon is given by the formula (n-2) x 180 degrees, where "n" represents the number of sides of the polygon. For example, a triangle (n=3) has an angle sum of (3-2) x 180 = 180 degrees, a square (n=4) has an angle sum of (4-2) x 180 = 360 degrees, and so on.
2. Vertex Angle: In a regular polygon, the interior angles are all equal. Therefore, each angle in a regular polygon with "n" sides is (n-2)/n x 180 degrees.
3. Integer Division: For a regular polygon to tessellate the plane, the vertex angles must fit perfectly around a point without any gaps. In other words, the vertex angles should evenly divide 360 degrees (a full circle) so that they can meet seamlessly. Each angle must be an integer value.
Now, let's consider some possibilities:
- Triangle (n=3): The vertex angle of a triangle is (3-2)/3 x 180 = 60 degrees. Since 360 divided by 60 equals 6, the vertex angles will fit exactly around a point, allowing triangles to tessellate the plane.
- Square (n=4): The vertex angle of a square is (4-2)/4 x 180 = 90 degrees. Since 360 divided by 90 equals 4, squares can tessellate the plane as well.
However, when we consider regular polygons with more than six sides, the vertex angles do not evenly divide 360 degrees. Here are some examples:
- Hexagon (n=6): The vertex angle of a hexagon is (6-2)/6 x 180 = 120 degrees. In this case, 360 divided by 120 equals 3, meaning that the angles will fit perfectly around a point.
- Heptagon (n=7): The vertex angle of a heptagon is (7-2)/7 x 180 = ~128.6 degrees. Here we encounter a problem since 360 divided by 128.6 equals a non-integer value (~2.8). This means that the angles of heptagons cannot meet seamlessly to tessellate the plane without gaps.
This pattern continues for larger regular polygons. As the number of sides increases, the resulting vertex angles no longer evenly divide 360 degrees, making it impossible for regular polygons with more than six sides to tessellate the plane without gaps or overlaps.
In summary, a regular polygon with more than six sides cannot tessellate the plane because its vertex angles cannot evenly divide 360 degrees, preventing the shapes from fitting seamlessly around a point.