Glossary
Practice Quizzes

10.5 Tessellations

3 min read•june 18, 2024

are fascinating patterns that cover surfaces without gaps or overlaps. They're created using shapes that fit together perfectly, like puzzle pieces. These patterns rely on transformations like sliding, flipping, and rotating to arrange shapes in repeating, predictable ways.

Regular polygons, such as triangles, squares, and hexagons, can create tessellations on their own. Their interior angles must divide evenly into 360° to work. Other shapes can be combined to form more complex patterns, opening up endless creative possibilities.

Properties and Creation of Tessellations

Properties of tessellations

Top images from around the web for Properties of tessellations

1 of 3

Top images from around the web for Properties of tessellations

1 of 3
  • Cover a plane completely without any gaps or overlaps using one or more shapes that fit together perfectly
  • Exhibit repeating patterns where the same shape or combination of shapes is used throughout and arranged in a consistent, predictable manner across the entire plane
  • Created using transformations such as translations, rotations, reflections, and glide reflections applied to the shapes to allow them to fit together seamlessly

Transformations in tessellating patterns

  • Translations slide a shape in a specific direction by a fixed distance in a straight line without changing its orientation or size
  • Rotations turn a shape around a fixed point by a specific angle clockwise or counterclockwise while maintaining its size and shape
  • Reflections flip a shape across a line, creating a mirrored reverse image while maintaining its size and shape
  • Glide reflections combine a (sliding) and a across a line while maintaining the size and shape of the original figure
  • Complex tessellating patterns can be created by combining these transformations

Tessellations with Regular Polygons

Interior angles for regular tessellations

  • Regular polygons have equal side lengths and equal interior angles
  • Sum of interior angles of a polygon calculated using the formula: (n−2)×180°(n - 2) \times 180°, where nn is the number of sides
  • For a to tessellate a plane by itself, the must divide 360° evenly (be a factor of 360°) to avoid gaps or overlaps
  • Equilateral triangles have 60° interior angles (180°×13)(180° \times \frac{1}{3}) which divides 360° evenly (360°÷60°=6)(360° \div 60° = 6), allowing them to tessellate a plane by themselves
  • Squares have 90° interior angles (180°×12)(180° \times \frac{1}{2}) which divides 360° evenly (360°÷90°=4)(360° \div 90° = 4), allowing them to tessellate a plane by themselves
  • Regular hexagons have 120° interior angles (180°×23)(180° \times \frac{2}{3}) which divides 360° evenly (360°÷120°=3)(360° \div 120° = 3), allowing them to tessellate a plane by themselves
  • Regular pentagons, heptagons, and other regular polygons with more than six sides cannot tessellate a plane by themselves because their interior angles do not divide 360° evenly, resulting in gaps or overlaps when attempting to tessellate
  • Convex polygons, where all interior angles are less than 180°, are commonly used in tessellations due to their ability to fit together without creating inward-facing corners

Types of Tessellations

Periodic and Aperiodic Tilings

  • Periodic tilings (tessellations) have a repeating pattern that can be translated in two independent directions to cover the entire plane
  • Aperiodic tilings lack a repeating pattern that can be translated in two independent directions, yet still cover the plane without gaps or overlaps
  • Monohedral tilings use only one shape (tile) to cover the plane, such as the regular tessellations with equilateral triangles, squares, or regular hexagons
  • Dihedral tilings use two different shapes to create the pattern

Key Terms to Review (28)

Aperiodic tiling: Aperiodic tiling refers to a way of covering a surface with tiles in such a manner that the pattern does not repeat itself. This means that, unlike periodic tilings where the arrangement can be shifted to create identical sections, aperiodic tilings create a unique and non-repeating structure. These tilings can illustrate fascinating mathematical properties and have applications in various fields, including materials science and art.
Cairo pentagonal tiling: Cairo pentagonal tiling is a unique type of tessellation made up of convex pentagons that can completely cover a plane without any gaps or overlaps. It was discovered by Robert Ammann in 1977 and is notable for having five different types of pentagons that fit together in a specific arrangement, showcasing the complexity and beauty of geometric patterns in tessellation.
Convex polygon: A convex polygon is a polygon in which all interior angles are less than 180 degrees and any line segment drawn between two points within the polygon lies entirely inside it. This characteristic ensures that no vertices point inwards, making convex polygons a fundamental shape in geometry, particularly in the study of polygons, perimeter, and tessellations.
Dihedral tiling: Dihedral tiling refers to a specific type of tessellation that uses shapes exhibiting dihedral symmetry, which means they can be rotated and reflected while maintaining their appearance. This type of tiling is particularly interesting because it involves regular polygons arranged in a way that creates a repeating pattern, showcasing both rotational and reflective symmetry. Dihedral tiling is a fundamental concept in the study of tessellations, connecting geometry and art through patterns that can be visually appealing and mathematically significant.
Edge-to-edge: Edge-to-edge refers to a method of tessellation where shapes fit together perfectly along their edges without any gaps or overlaps. This concept is crucial in creating a seamless pattern where the individual tiles or shapes connect directly, ensuring a smooth transition between them and maintaining a continuous surface. Edge-to-edge arrangements are fundamental in various artistic and architectural designs, as they allow for intricate and visually appealing patterns.
Glide reflection: A glide reflection is a transformation that combines a translation with a reflection over a line that is parallel to the direction of the translation. This type of transformation is significant in understanding symmetrical patterns and creating tessellations, as it preserves the overall structure while altering individual components. Glide reflections play a crucial role in generating complex designs by repeating shapes and ensuring that they fit together seamlessly without gaps or overlaps.
Hexagon: A hexagon is a polygon with six sides and six angles. This shape can be regular, where all sides and angles are equal, or irregular, where they vary in length and degree. Hexagons are significant in various mathematical concepts, including perimeter calculations, tessellation patterns, and area measurements, making them a versatile and interesting shape in geometry.
Interior angle: An interior angle is the angle formed between two sides of a polygon that meet at a vertex, lying within the boundaries of that polygon. These angles play a crucial role in understanding the properties and classifications of polygons, including their perimeter and area calculations. Additionally, interior angles are essential when analyzing how shapes fit together in tessellations, where the angles determine the arrangement and overall design of the pattern.
M.C. Escher: M.C. Escher was a Dutch graphic artist known for his mathematically inspired artworks that often feature impossible constructions, explorations of infinity, and intricate tessellations. His work combines art and mathematics, making complex geometric patterns visually captivating while challenging our perception of space and reality.
Monohedral tiling: Monohedral tiling is a type of tessellation where a single shape is used repeatedly to cover a surface without any gaps or overlaps. This concept emphasizes the use of congruent shapes, meaning all the tiles are identical, leading to interesting patterns and arrangements. Monohedral tiling is often explored in the context of geometry and art, as it reveals the beauty of symmetry and uniformity in spatial arrangements.
Octagon: An octagon is a polygon with eight sides and eight angles. This shape has unique properties that relate to the concepts of perimeter and area, making it significant in geometry. Octagons can also play a role in tessellations, where they can fit together with other shapes to cover a plane without gaps or overlaps, showcasing their versatility in various mathematical contexts.
Penrose tiling: Penrose tiling refers to a non-periodic tessellation created by Roger Penrose that uses a set of shapes to cover a plane without repeating patterns. This unique arrangement of shapes produces a structure that exhibits a form of order while avoiding periodicity, making it an interesting subject in the study of symmetry and mathematics.
Periodic Tiling: Periodic tiling refers to a method of covering a surface with tiles in such a way that the arrangement of tiles repeats itself in a regular pattern. This concept is closely linked to the idea of tessellations, where shapes are arranged without gaps or overlaps, creating an infinite visual design. In periodic tiling, the repetition can occur in one or more directions, resulting in a structured and predictable design that can be mathematically analyzed.
Reflection: Reflection is a geometric transformation that flips a shape over a line, creating a mirror image of the original figure. This transformation preserves the shape and size but alters the orientation, resulting in corresponding points that are equidistant from the line of reflection. In the context of tessellations, reflection plays a crucial role in creating symmetrical patterns and understanding how shapes fit together without gaps or overlaps.
Regular polygon: A regular polygon is a flat shape with straight sides that are all equal in length and angles that are all equal in measure. This geometric property means that regular polygons can be classified based on the number of sides they have, which also influences their perimeter, area, and how they can fit together to create tessellations.
Regular tessellation: A regular tessellation is a pattern formed by repeating a single type of regular polygon without any gaps or overlaps. These shapes fit together perfectly in a plane, creating a visually appealing design that exhibits both symmetry and regularity. Regular tessellations can be categorized into three types based on the polygons used: triangles, squares, and hexagons, which are the only regular polygons capable of tessellating the plane on their own.
Rotation: Rotation refers to the circular movement of a shape or object around a central point or axis. This key concept is fundamental in understanding how shapes can be manipulated within a plane, allowing for the creation of visually interesting patterns and designs. In the context of tessellations, rotation plays a crucial role in determining how shapes fit together seamlessly without gaps or overlaps, often creating mesmerizing geometric patterns.
Semi-regular tessellation: A semi-regular tessellation is a type of tiling pattern that is created using two or more different regular polygons, arranged in such a way that each vertex of the pattern has the same arrangement of polygons. This kind of tessellation exhibits both symmetry and regularity, making it visually appealing and mathematically interesting.
Symmetry: Symmetry refers to a balanced and proportionate similarity in the arrangement of parts on opposite sides of a dividing line or around a central point. It plays a crucial role in understanding patterns and designs, as it helps create harmony and visual appeal in various forms, including art and geometric shapes.
Tessellation: Tessellation refers to the covering of a plane with one or more geometric shapes, called tiles, without any overlaps or gaps. This concept is crucial in various fields such as mathematics and art, where it showcases patterns and symmetries, and emphasizes how shapes can fit together harmoniously. The study of tessellations reveals important properties of shapes and their relationships in two-dimensional spaces.
Tessellations: A tessellation is a pattern of shapes that fit together perfectly without any gaps or overlaps. These patterns can cover a plane infinitely using one or more geometric shapes.
Tiling: Tiling refers to the covering of a surface using one or more geometric shapes, called tiles, without any overlaps or gaps. This process is fundamental in creating tessellations, where tiles are arranged in a repeating pattern that fills a plane completely. Tiling can involve various shapes, such as squares, triangles, and hexagons, and plays a crucial role in art, architecture, and mathematical concepts related to symmetry and geometry.
Transformation: Transformation refers to the process of changing the position, size, or orientation of a shape on a plane. This concept is crucial in creating tessellations, as it involves translating, rotating, or reflecting shapes to fill a plane without any gaps or overlaps. Understanding transformations helps in recognizing how patterns can be systematically arranged and manipulated to create visually appealing designs.
Translation: Translation in geometry is the process of moving a shape or tessellation from one position to another without rotating, resizing, or otherwise deforming it. It involves sliding the shape along a straight line.
Translation: Translation refers to the process of moving a shape or object from one position to another in a plane without altering its size, shape, or orientation. This concept is crucial for understanding how figures can be systematically arranged in patterns, allowing for the creation of tessellations that fill a plane without gaps or overlaps.
Triangle: A triangle is a polygon with three edges and three vertices, and it is one of the simplest shapes in geometry. The sum of the internal angles of a triangle always equals 180 degrees, which is a fundamental property that connects it to various mathematical concepts like perimeter, area, and graphing relationships.
Vertex: A vertex is a point where two or more curves, lines, or edges meet. In different contexts, it can represent a significant feature such as the peak of a parabola, a corner of a polygon, or a key point in graph theory. Understanding the concept of a vertex helps in analyzing the properties and relationships of various mathematical structures.
Vertex-to-vertex: Vertex-to-vertex refers to the connections made between vertices in geometric shapes, particularly in the context of tessellations. This concept is essential when determining how shapes fit together without gaps or overlaps, as each vertex of one shape must align with a vertex of another shape to create a seamless pattern. Understanding vertex-to-vertex relationships helps in exploring symmetry, tiling patterns, and the overall structure of tessellations.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Glossary
Glossary
10.6 Area