Tessellation, also known as tiling, is a mathematical concept that involves covering a plane with a repeated pattern of shapes without any gaps or overlaps. These shapes can be polygons, such as triangles, squares, or hexagons, or even more complex shapes. The resulting pattern is called a tessellation.
The concept of tessellation has been around for centuries and has been observed in various cultures throughout history. Ancient civilizations, such as the Egyptians and the Greeks, used tessellations in their art and architecture. The famous Alhambra palace in Spain is adorned with intricate tile patterns that form tessellations.
Tessellation is typically introduced in elementary or middle school mathematics, around grades 3 to 6. It is a topic that can be explored further in high school geometry.
Tessellation involves several key concepts and steps:
Regular Polygons: Understanding the properties of regular polygons, such as triangles, squares, and hexagons, is essential for creating tessellations. These polygons have equal sides and angles.
Translation: The shapes in a tessellation are repeated by translating them, which means moving them without rotating or reflecting. This ensures that the pattern fits together seamlessly.
No Gaps or Overlaps: The shapes in a tessellation must fit together perfectly, without any gaps or overlaps. This requires careful arrangement and alignment of the shapes.
Infinite Extension: A tessellation can be extended infinitely in all directions, creating a pattern that repeats indefinitely.
There are three main types of tessellation:
Regular Tessellation: In a regular tessellation, the same regular polygon is used to cover the plane without any gaps or overlaps. Examples include a honeycomb pattern made of hexagons or a checkerboard pattern made of squares.
Semi-Regular Tessellation: In a semi-regular tessellation, different regular polygons are used to cover the plane without any gaps or overlaps. The pattern repeats, but not all the polygons are the same. Examples include a pattern made of triangles, squares, and hexagons.
Irregular Tessellation: In an irregular tessellation, different shapes are used to cover the plane without any gaps or overlaps. The pattern does not repeat. Examples include a pattern made of animals or objects.
Tessellations have several interesting properties:
Symmetry: Tessellations often exhibit different types of symmetry, such as rotational, reflectional, or glide reflectional symmetry. This adds to their aesthetic appeal.
Congruence: The shapes in a tessellation are congruent, meaning they have the same size and shape.
Infinite Repetition: Tessellations can be extended infinitely in all directions, creating a never-ending pattern.
Tessellations can be created by following these steps:
Start with a regular polygon or a combination of regular polygons.
Arrange the polygons in a way that they fit together without any gaps or overlaps.
Extend the pattern infinitely in all directions.
There is no specific formula or equation for tessellation. It is more of a geometric concept that involves arranging shapes in a specific way.
As mentioned earlier, there is no specific formula or equation for tessellation. Instead, it requires an understanding of geometric concepts and the ability to arrange shapes without any gaps or overlaps.
There is no specific symbol or abbreviation for tessellation.
There are several methods for creating tessellations:
Translation Method: This involves translating a shape to create a repeating pattern.
Reflection Method: This involves reflecting a shape to create a repeating pattern.
Rotation Method: This involves rotating a shape to create a repeating pattern.
Combination Method: This involves combining translation, reflection, and rotation to create a repeating pattern.
Example 1: Create a regular tessellation using equilateral triangles.
Solution: Start with an equilateral triangle. Translate it to create a row of triangles. Repeat this row infinitely in both directions.
Example 2: Create a semi-regular tessellation using squares and equilateral triangles.
Solution: Start with a square. Place an equilateral triangle on one side of the square. Repeat this pattern infinitely in all directions.
Example 3: Create an irregular tessellation using animal shapes.
Solution: Choose different animal shapes, such as cats, dogs, and birds. Arrange them without any gaps or overlaps to create a unique pattern.
Create a regular tessellation using hexagons.
Create a semi-regular tessellation using pentagons and squares.
Create an irregular tessellation using your favorite shapes.
Question: What is tessellation?
Answer: Tessellation is a mathematical concept that involves covering a plane with a repeated pattern of shapes without any gaps or overlaps.
Question: What grade level is tessellation for?
Answer: Tessellation is typically introduced in elementary or middle school mathematics, around grades 3 to 6.
Question: How do you create a tessellation?
Answer: Tessellations can be created by arranging shapes, such as polygons, without any gaps or overlaps, and extending the pattern infinitely in all directions.
Question: What are the types of tessellation?
Answer: The types of tessellation include regular tessellation, semi-regular tessellation, and irregular tessellation.
Question: What are the properties of tessellation?
Answer: Tessellations have properties such as symmetry, congruence, and infinite repetition.
Question: Are there formulas or equations for tessellation?
Answer: There are no specific formulas or equations for tessellation. It is more of a geometric concept that involves arranging shapes in a specific way.