Determine Whether The Shape Below Will Tessellate The Plane

Tessellation, a fascinating area of geometry, delves into the art of covering a plane with repeating geometric shapes without any gaps or overlaps. Understanding which shapes can tessellate, and how they do so, is a core concept in mathematics, design, and even nature. This article provides a comprehensive guide on how to determine whether a shape will tessellate the plane, exploring the underlying principles, methods, and examples to solidify your understanding.

Understanding Tessellation

Tessellation, also known as tiling, is the process of covering a surface with one or more geometric shapes, called tiles, with no gaps or overlaps. Tessellations can be found in various contexts, from the intricate patterns of honeycombs in nature to the decorative designs in art and architecture. The ability of a shape to tessellate depends on its geometric properties and the angles formed at its vertices.

Types of Tessellations

Regular Tessellations: These are tessellations made up of only one type of regular polygon (a polygon with all sides and angles equal). There are only three regular polygons that tessellate the plane: the equilateral triangle, the square, and the regular hexagon.
Semi-Regular Tessellations: These tessellations use two or more different regular polygons. The arrangement of these polygons at each vertex must be identical. There are only eight possible semi-regular tessellations.
Irregular Tessellations: These tessellations are formed by irregular polygons (polygons with sides and angles that are not all equal) or other shapes. These tessellations can be periodic (repeating in a predictable pattern) or aperiodic (non-repeating).

Criteria for Determining Tessellation

Determining whether a shape will tessellate involves understanding the geometric conditions necessary for a successful tiling of the plane. Here are the key criteria:

Angle Sum at a Vertex: The sum of the angles around any vertex in a tessellation must equal 360 degrees. This is a fundamental requirement because the tiles must completely surround the vertex without any gaps or overlaps.
Geometric Properties of the Shape: The shape's angles and sides play a crucial role. For polygons, the angles must be such that when they meet at a vertex, their sum is 360 degrees. For irregular shapes, transformations like translations, rotations, and reflections may be necessary to achieve tessellation.
Symmetry and Transformations: Symmetry is a significant factor in tessellations. Shapes with symmetry are often easier to tessellate. Transformations such as translations (sliding), rotations (turning), and reflections (flipping) can be used to arrange the tiles to cover the plane.

Methods to Determine Tessellation

To determine whether a shape will tessellate the plane, several methods can be employed:

1. Angle Sum Method

The angle sum method is particularly useful for polygons. It involves calculating the interior angles of the polygon and checking if a combination of these angles can add up to 360 degrees at a vertex.

For Regular Polygons: Calculate the interior angle using the formula:

Interior Angle = (n - 2) * 180 / n, where n is the number of sides.

If 360 is divisible by the interior angle, then the regular polygon tessellates.
- Equilateral Triangle (n=3): Interior Angle = 60 degrees. Since 360/60 = 6, the equilateral triangle tessellates.
- Square (n=4): Interior Angle = 90 degrees. Since 360/90 = 4, the square tessellates.
- Regular Pentagon (n=5): Interior Angle = 108 degrees. Since 360/108 is not an integer, the regular pentagon does not tessellate.
- Regular Hexagon (n=6): Interior Angle = 120 degrees. Since 360/120 = 3, the regular hexagon tessellates.
For Irregular Polygons: Measure the interior angles of the polygon. If a combination of these angles can sum up to 360 degrees at a vertex, the irregular polygon may tessellate.

2. Transformation Method

The transformation method involves using translations, rotations, and reflections to fit copies of the shape together. This method is particularly useful for irregular shapes where angle sums alone may not be sufficient to determine tessellation.

Translations: Check if sliding the shape along a plane can create a repeating pattern without gaps or overlaps.
Rotations: Determine if rotating the shape around a point can create a pattern that fills the plane.
Reflections: See if reflecting the shape across a line can help create a tessellating pattern.

3. Vertex Configuration Method

The vertex configuration method is used for semi-regular tessellations. It involves listing the polygons that meet at a vertex in a specific order. The same configuration must occur at every vertex for the tessellation to be semi-regular.

For example, the vertex configuration 3.6.3.6 indicates that at each vertex, an equilateral triangle (3 sides) and a regular hexagon (6 sides) alternate.

4. Experimental Method

The experimental method involves physically or digitally arranging copies of the shape to see if they can cover a plane without gaps or overlaps. This method is particularly useful for complex or irregular shapes.

Physical Tiling: Cut out multiple copies of the shape from paper or cardboard and try to arrange them to cover a surface.
Digital Tiling: Use computer software to create and manipulate copies of the shape to test for tessellation.

Examples of Tessellating Shapes

To further illustrate these methods, let's look at examples of shapes that tessellate and those that do not.

Shapes That Tessellate

Equilateral Triangle:
- Each interior angle is 60 degrees.
- Six triangles can meet at a vertex (6 x 60 = 360 degrees).
- Tessellates through translations.
Square:
- Each interior angle is 90 degrees.
- Four squares can meet at a vertex (4 x 90 = 360 degrees).
- Tessellates through translations.
Regular Hexagon:
- Each interior angle is 120 degrees.
- Three hexagons can meet at a vertex (3 x 120 = 360 degrees).
- Tessellates through translations.
Parallelogram:
- The sum of adjacent angles is 180 degrees.
- Two pairs of adjacent angles meet at a vertex (2 x 180 = 360 degrees).
- Tessellates through translations.
Any Triangle:
- The sum of the angles in a triangle is 180 degrees.
- Two triangles can be combined to form a quadrilateral, which can then tessellate.
Any Quadrilateral:
- The sum of the angles in a quadrilateral is 360 degrees.
- Quadrilaterals can always tessellate by rotating them 180 degrees around the midpoint of their sides.

Shapes That Do Not Tessellate (Alone)

Regular Pentagon:
- Each interior angle is 108 degrees.
- No integer number of pentagons can meet at a vertex to total 360 degrees.
- Does not tessellate alone.
Regular Heptagon:
- Each interior angle is approximately 128.57 degrees.
- No integer number of heptagons can meet at a vertex to total 360 degrees.
- Does not tessellate alone.
Regular Octagon:
- Each interior angle is 135 degrees.
- Although two octagons can meet at a vertex (2 x 135 = 270 degrees), there is a 90-degree gap that cannot be filled by another octagon.
- Does not tessellate alone, but can be part of a semi-regular tessellation.

Practical Examples and Applications

Understanding tessellations is not just an academic exercise; it has practical applications in various fields:

Art and Design: Tessellations are used to create visually appealing patterns in art, textiles, wallpaper, and decorative tiling. Artists like M.C. Escher have famously used tessellations in their work.
Architecture: Architects use tessellations in flooring, wall designs, and structural designs. Tessellated patterns can provide both aesthetic appeal and structural stability.
Engineering: Engineers use tessellations in designing lightweight and strong structures. For example, hexagonal tessellations are used in aircraft wings and other aerospace applications.
Computer Graphics: Tessellation is used in computer graphics to create detailed and realistic 3D models. It allows for the creation of smooth surfaces from polygonal meshes.
Science: Tessellations are observed in natural phenomena such as honeycombs, crystal structures, and cell arrangements.

Step-by-Step Guide to Determining Tessellation

To provide a clear and practical guide, here is a step-by-step approach to determining whether a shape will tessellate:

Identify the Shape: Determine whether the shape is a regular polygon, an irregular polygon, or another type of shape.
Calculate Interior Angles (for Polygons):
- For regular polygons, use the formula: Interior Angle = (n - 2) * 180 / n.
- For irregular polygons, measure each interior angle.
Check Angle Sum at a Vertex:
- Determine if a combination of the shape's interior angles can add up to 360 degrees. This is a necessary condition for tessellation.
Apply Transformations (if necessary):
- If the angle sum condition is met, explore whether translations, rotations, or reflections can be used to arrange copies of the shape to cover the plane without gaps or overlaps.
Experimental Verification:
- Create physical or digital models of the shape and try to arrange them to cover a surface. This can help visualize and confirm whether the shape tessellates.
Consider Vertex Configurations (for Semi-Regular Tessellations):
- If using multiple shapes, list the polygons that meet at a vertex in a specific order. The same configuration must occur at every vertex for the tessellation to be semi-regular.

Advanced Topics in Tessellation

For those interested in delving deeper into the subject, here are some advanced topics in tessellation:

Aperiodic Tessellations: These are tessellations that do not have a repeating pattern. The most famous example is the Penrose tiling, which uses two different rhombus shapes to create a non-periodic tessellation.
Escher-Type Tessellations: These tessellations involve transforming shapes into recognizable figures such as animals or objects. M.C. Escher was a master of this type of tessellation.
3D Tessellations: These are tessellations in three-dimensional space, such as the packing of spheres or polyhedra. They have applications in materials science and crystallography.
Tessellations on Non-Euclidean Surfaces: Tessellations can also be explored on curved surfaces such as spheres or hyperbolic planes. These tessellations have different properties than those on a flat plane.

Common Mistakes to Avoid

When determining whether a shape will tessellate, it's important to avoid these common mistakes:

Assuming Angle Sum is Sufficient: While the angle sum condition is necessary, it is not sufficient. The shapes must also be able to fit together without gaps or overlaps.
Ignoring Transformations: For irregular shapes, it's important to consider translations, rotations, and reflections. Simply looking at angle sums may not provide a complete picture.
Overlooking Vertex Configurations: When using multiple shapes, ensure that the vertex configurations are consistent throughout the tessellation.
Not Experimenting: Sometimes, the best way to determine whether a shape tessellates is to try it out physically or digitally. Don't rely solely on theoretical calculations.

Conclusion

Determining whether a shape will tessellate the plane involves understanding geometric principles, applying various methods, and avoiding common mistakes. By using the angle sum method, transformation method, vertex configuration method, and experimental method, you can effectively analyze shapes and determine their tessellating capabilities. Whether you're an artist, architect, engineer, or simply a math enthusiast, understanding tessellations can provide valuable insights and creative possibilities. As you explore this fascinating area of geometry, you'll discover the beauty and complexity of patterns that cover the world around us.