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A hexagon is a six-sided polygon, and in terms of regular polygons, all sides and angles are equal. One of the fascinating properties of a regular hexagon is its ability to tessellate the plane perfectly without any gaps or overlaps.
Regular hexagons fit together edge-to-edge, much like tiles on a floor, illustrate a regular tessellation. Each internal angle in a regular hexagon is precisely 120 degrees, as can be calculated using the formula for interior angles.

• To tessellate, there can be different vertex configurations using hexagons alone, or combined with other polygons. But when using hexagons solo, every vertex consists of three meeting hexagons, creating a seamless tiling of the plane.

These properties make them not only geometrically interesting but also practically useful in design and art.

The dodecagon is a twelve-sided polygon which is another interesting figure in geometry. When we're talking about regular dodecagons, each side and each angle are equal.
The interior angle of a regular dodecagon is calculated to be 150 degrees. While a single regular dodecagon doesn't tessellate the plane on its own like a hexagon does, it plays a role in more complex tessellations.

• A dodecagon can work with other polygons to possibly form tessellations, but needs to be paired carefully due to the angle size.
• When considering for a semi-regular tessellation, it becomes apparent that combinations with hexagons and dodecagons must be balanced to ensure the sum of angles meeting at a vertex equals 360 degrees.

This requirement is challenging given their angle size, and thus, directly affects the ability to use them in such tessellations efficiently.

Interior angles are the angles found on the inside of a polygon at each vertex. Understanding these angles is crucial when analyzing tessellations.
The general formula to calculate the interior angle of a regular polygon with n sides is: \[(n-2) \times \frac{180}{n}\]For example, in a hexagon, plugging 6 into the formula, you get: \[(6-2) \times \frac{180}{6} = 120°\]Similarly, for a dodecagon (12 sides), it is calculated as: \[(12-2) \times \frac{180}{12} = 150°\]

• These calculations are vital in ensuring that a combination of polygons at a vertex in tessellations sums exactly to 360 degrees.

Without this calculated precision, making any tessellation, regular or semi-regular, becomes impossible.

Vertex configuration refers to the arrangement of polygons around a shared vertex in a tessellation. This configuration is central to forming a semi-regular tessellation.
Each vertex in a semi-regular tessellation must have the same pattern of polygons around it. Therefore, understanding the sum of the angles that meet at a vertex is crucial. It should always total 360 degrees encompassing:

• Each vertex configuration can be represented by a sequence of numbers: for example, a vertex with three hexagons would be labeled as (3.6.3.6).
• This representation helps visualize and ensure that different polygon types correctly fit together without overlaps.

When either a hexagon or a dodecagon is involved, the point is to assess whether their respective angles combine efficiently at vertices.
In our case, since combining hexagons and dodecagons does not yield a practical configuration equating to 360 degrees, a semi-regular tessellation isn't feasible using these two shapes together.