Can some one please help me I'm stuck in this question.
Why does a regular pentagon not tessellate the plane?
http://members.cox.net/tessellations/Polygons1.html
Thank you
Of course! I'd be happy to help you with your question. To understand why a regular pentagon does not tessellate the plane, let's first define what tessellation means.
Tessellation is the process of filling a plane with repeated shapes without any gaps or overlaps. In other words, it's a way to cover a flat surface or plane using identical or similar shapes that fit together perfectly.
To determine if a regular pentagon can tessellate the plane, we need to consider its properties. A regular pentagon has five equal sides and five equal angles. We can attempt to tessellate the plane using regular pentagons, but we will encounter certain challenges.
One of the main reasons a regular pentagon cannot tessellate the plane is due to the sum of its interior angles. The interior angles of a regular pentagon measure 108 degrees each. When we try to fit several pentagons together in a tessellation, the total sum of interior angles must be a multiple of 360 degrees. However, the sum of five interior angles of a regular pentagon (5 * 108 degrees) adds up to 540 degrees, which is greater than 360 degrees. This discrepancy prevents the regular pentagon from fitting together seamlessly without leaving gaps or overlaps.
Another way to understand this is by analyzing the vertex angles. At each vertex or corner of a regular pentagon, there is an angle of 108 degrees. In a tessellation, the vertex angles of neighboring shapes should add up to a complete circle, which is 360 degrees. However, two neighboring pentagons only share one vertex angle, resulting in a total of 108 degrees. This angle cannot be evenly divided to fit together multiple times, leading to irregular gaps or overlaps between the pentagons.
In conclusion, the combination of the sum of interior angles and vertex angles in a regular pentagon prevents it from tessellating the plane without gaps or overlaps.