Why does a regular pentagon not tessellate the plane?
A regular pentagon does not tessellate the plane if the vertices or edges do not meet. In tesselations all the edges and vertices meet.
A regular pentagon does not tessellate the plane because it fails to meet one of the conditions required for a shape to tessellate. In order for a shape to tessellate, it must have the following properties:
1. The shape must be a polygon.
2. The angles of the shape must add up to 360 degrees.
3. The shape's sides must fit together with no gaps or overlaps.
A regular pentagon satisfies the first two conditions as it is a polygon and its angles add up to 540 degrees (each angle measures 108 degrees). However, it fails to satisfy the third condition. If we try to put regular pentagons next to each other to fill the plane, there will inevitably be gaps or overlaps between the shapes.
To visualize this, you can try drawing regular pentagons on a piece of paper and attempting to fit them together without any gaps or overlaps. You will notice that it is impossible to completely cover the plane without breaking one of the conditions for tessellation.
In contrast, there are shapes that can tessellate the plane, such as triangles, squares, and hexagons. These shapes fulfill all three conditions and can be arranged to seamlessly cover a flat surface without leaving any gaps or overlaps.