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Imagine taking a walk and spotting various shapes in the architecture around you—hexagonal windows, triangular gables, or square tiles. Many of these shapes are actually examples of regular polygons, crucial elements in the world of geometry. A regular polygon is a flat shape with straight sides that are all equal in length, and with angles that are also all equal. If you've ever seen an equilateral triangle or a square, then you're already familiar with some of the simplest forms of regular polygons.

The beauty and harmony of these shapes come from their symmetry, which allows for uniformity and ease when using them in designs and constructions. In mathematics, understanding these shapes is fundamental as they serve as the building blocks for more complex geometric concepts.

The angles inside any polygon are what we call interior angles. It's like standing at a corner inside a room and looking at the angle the walls make; that's your interior angle right there. Now, for regular polygons, all these inside angles are equal, giving the shape a uniform look. Knowing how to calculate these angles is a handy skill.

For a regular polygon with 'n' sides, the sum of the interior angles is always \( 180(n-2) \) degrees. By dividing that sum equally among the 'n' angles, since they are all the same in a regular polygon, we get \( 180(n-2)/n \) degrees for each interior angle. This formula is pivotal for solving many geometry problems and understanding how shapes can fit together, which brings us to tessellation.

When discussing tessellation, we're delving into the art and science of tiling. Tessellation occurs when a shape, or a combination of shapes, is repeated over and over again covering a plane without any gaps or overlaps—much like the tiles in your bathroom or the pattern on your favorite quilt.

For a single type of polygon to tessellate, each interior angle must be a factor of 360 degrees, because the angles at a point (where the vertices meet) must add up to 360 degrees for a flat, seamless surface. As we previously discovered, a regular nine-sided polygon has interior angles that do not allow for this perfect fit. Therefore, while tessellation is a common and fascinating occurrence in mathematics and design, not every shape is up to the task. This principle greatly influences patterns in architecture, art, and even nature.