Determine Whether The Given Figure Tessellates The Plane.

Determining Whether a Given Figure Tessellates the Plane

Tessellations, also known as tilings, are captivating arrangements of shapes that cover a plane without any overlaps or gaps. Understanding whether a given figure can tessellate the plane is a fundamental concept in geometry, with applications ranging from art and design to architecture and computer graphics. This comprehensive guide delves into the methods and principles used to determine tessellation capability, exploring various shapes and their properties.

What is a Tessellation?

A tessellation is a pattern of shapes that covers a surface completely without any overlaps or gaps. Think of tiling a floor – if the tiles fit perfectly together, covering the entire floor without leaving any spaces, you've created a tessellation. This concept extends beyond simple squares and rectangles; various shapes, regular and irregular, can create tessellations, subject to specific criteria.

Key Characteristics of Tessellations:

Complete Coverage: The shapes must cover the entire plane without leaving any uncovered areas.
No Overlaps: The shapes cannot overlap each other.
Uniformity (Optional): While not a requirement for all tessellations, many exhibit uniformity, meaning the pattern repeats regularly. This type of tessellation is often called a periodic tessellation. However, aperiodic tessellations also exist, which lack a repeating pattern.

Determining Tessellation Capability: Regular Polygons

Let's begin by examining regular polygons – shapes with equal sides and equal angles. Regular polygons that tessellate the plane are limited:

Equilateral Triangles: Six equilateral triangles meet at a single point, forming a 360-degree angle. This allows for complete coverage of the plane without gaps or overlaps.
Squares: Four squares meet at a single point, forming a 360-degree angle. This straightforward arrangement readily tessellates the plane.
Regular Hexagons: Three regular hexagons meet at a single point, also creating a 360-degree angle. This is another example of a readily tessellable regular polygon.

Why Other Regular Polygons Fail:

Other regular polygons, such as pentagons, heptagons, octagons, and so on, cannot tessellate the plane. This is due to their internal angles. The angles must add up to 360 degrees at each vertex for a complete tessellation. Let's consider a pentagon:

A regular pentagon has interior angles of 108 degrees. If you try to arrange pentagons around a single point, you'll find that three pentagons (324 degrees) fall short, while four pentagons (432 degrees) exceed 360 degrees. Therefore, pentagons cannot form a tessellation. This logic extends to other regular polygons whose interior angles are not factors of 360.

Determining Tessellation Capability: Irregular Polygons and Other Shapes

Moving beyond regular polygons, the determination of tessellation capability becomes more nuanced. While regular polygons adhere to strict mathematical rules, irregular shapes offer greater flexibility but also require more careful analysis.

Using Interior Angles:

While the sum of angles at a vertex must still be 360 degrees, the individual angles can vary. The key is to ensure that a combination of angles from multiple shapes sums to 360 degrees at each vertex.

Example: Consider a quadrilateral with angles of 90, 90, 135, and 45 degrees. This quadrilateral can tessellate, as two of these shapes can be arranged to make a 360 degree angle using the 90, 90, 135 and a combination of the 45 with another 45 degree angle, making 360.

Analyzing Shape Properties:

For irregular shapes, examining their properties is crucial. Look for features such as:

Symmetry: Symmetrical shapes often tessellate more easily.
Matching Sides and Angles: The ability to fit shapes together seamlessly relies on the compatibility of their sides and angles.
Rotational Symmetry: Shapes with rotational symmetry (e.g., a square) can facilitate efficient tessellations.
Translational Symmetry: Repeating a shape via translations (shifting it along a path) is a common approach in tessellations.

Techniques for Determining Tessellation:

Several techniques can help determine whether a given figure can tessellate the plane:

Visual Inspection and Manipulation: A hands-on approach, using cutouts or digital tools, allows for experimentation and visualization. Try arranging the shape in different configurations to see if a complete coverage without overlaps is achievable.
Mathematical Analysis: For more complex shapes, employing mathematical calculations of angles and side lengths is necessary. This involves checking if various combinations of angles at a vertex sum to 360 degrees.
Computer-Aided Design (CAD) Software: CAD software provides powerful tools for creating and manipulating shapes, facilitating the design and analysis of tessellations. This approach is especially helpful for complex, irregular shapes and large-scale tessellations.

Common Mistakes in Determining Tessellation:

Several common mistakes occur when assessing tessellation potential:

Focusing solely on one vertex: While the sum of angles at a single vertex must be 360 degrees, ensuring this condition is met at every vertex is crucial for a successful tessellation.
Ignoring gaps or overlaps: Thoroughly check the entire pattern for any gaps or overlaps that would invalidate the tessellation.
Assuming all shapes tessellate: Many shapes, even those with seemingly compatible angles, will not tessellate due to unforeseen compatibility issues when trying to extend the pattern across the entire plane.

Examples of Tessellating Shapes and Patterns:

Beyond regular polygons, a vast array of shapes and patterns exhibit tessellation properties:

Penrose Tilings: These aperiodic tilings use a combination of two distinct shapes, demonstrating that tessellations don't necessarily require repetitive patterns.
Escher's Tessellations: The renowned artist M.C. Escher created many intricate and beautiful tessellations, often involving transformations of shapes to create fascinating visual effects.
Islamic Geometric Patterns: Islamic art extensively uses tessellations, employing intricate geometric designs to create stunning and complex patterns.

Applications of Tessellations:

The concept of tessellations extends far beyond theoretical geometry. Its applications span various fields:

Art and Design: Tessellations inspire artistic creations, used in paintings, mosaics, textiles, and other mediums.
Architecture: Tessellations provide efficient and aesthetically pleasing ways to cover surfaces in buildings and other structures.
Computer Graphics: Tessellations are used in computer graphics to model and render surfaces efficiently. This approach reduces computational complexity for visually complex objects.
Nature: Hexagonal patterns in honeycombs and the arrangement of cells in some organisms are natural examples of tessellations.

Conclusion:

Determining whether a given figure tessellates the plane involves a combination of visual inspection, mathematical analysis, and a deep understanding of geometric principles. While regular polygons offer straightforward criteria, irregular shapes require a more nuanced assessment. Through careful examination of angles, symmetry, and shape compatibility, one can determine a figure's tessellation capabilities. The concept of tessellations is not only a fascinating area of geometric study but also a powerful tool with widespread applications across diverse fields. Understanding tessellation opens doors to a world of creative possibilities and efficient solutions in design, art, architecture, and computational fields.