- For shapes to cover a space without gaps and overlapping, you need to first understand how shapes meet at a point. A point has 360 degrees. For the same shape to tessellate, each vertex must be a divisor of 360. For example, each vertex of a square is 90 degrees and 360 divides 90 evenly. You can show that four squares meet at one point without any gap. Squares tessellate. If a shape cannot tessellate by itself, you need to add other shapes such that the sum of the interior angles is 360 degrees.
- If you use only one shape and repeat it in a tessellation project, you are working on "regular tessellation." In Euclidean space, squares, triangles, and hexagons are the only regular polygons that tessellate because the degree measurements of each of their angle are divisible by 360. Bee hive is a real-life hexagonal regular tessellation.
- You can also use a combination of regular polygons to do tessellation. To pick the shapes, you need to add the total of the interior angles that meet at one point and make sure the total is 360 degrees. For example, each vertex of an octagon is 135 degrees. The sum of a right angle and two-135 degrees is 360 degrees. That means two regular octagons and one square tessellate. Another possible combination are two dodecagon (12-sided) and one triangle. There are many semi-regular tessellations. Experiment it with different regular polygons.
- You can also create certain non-regular shapes that can tessellate as long as their vertex is divisible by 360 evenly. Often, students like to use shapes with different colors to do their tessellation project. Even a square tessellation can make great art-work if you use colors carefully and creatively. Observe tiling in churches, roads, and architectures for ideas. Tessellation is every where in nature also.
Mathematics of Tessellation
Regular Tessellation
Semi-Regular Tessellation
Non-Regular Shapes and Color Tessllations
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