Tessellations

Mathematics, Art, and Recreation

Robert Fathauer

• Publisher: A K Peters/CRC Press
• Year: 2020
• eBook ISBN: 9780429589232
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Tessellations: Mathematics, Art and Recreation aims to present a comprehensive introduction to tessellations (tiling) at a level accessible to non-specialists. Additionally, it covers techniques, tips, and templates to facilitate the creation of mathematical art based on tessellations. Inclusion of special topics like spiral tilings and tessellation metamorphoses allows the reader to explore beautiful and entertaining math and art.

The book has a particular focus on 'Escheresque' designs, in which the individual tiles are recognizable real-world motifs. These are extremely popular with students and math hobbyists but are typically very challenging to execute. Techniques demonstrated in the book are aimed at making these designs more achievable. Going beyond planar designs, the book contains numerous nets of polyhedra and templates for applying Escheresque designs to them.

Activities and worksheets are spread throughout the book, and examples of real-world tessellations are also provided.

Key features

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• Introduces the mathematics of tessellations, including symmetry

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• Covers polygonal, aperiodic, and non-Euclidean tilings

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• Contains tutorial content on designing and drawing Escheresque tessellations

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• Highlights numerous examples of tessellations in the real world

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• Activities for individuals or classes

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• Filled with templates to aid in creating Escheresque tessellations

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• Treats special topics like tiling rosettes, fractal tessellations, and decoration of tiles

Chapter 1

Introduction to Tessellations

A tessellation is a collection of shapes that fit together without gaps or overlap to cover the infinite mathematical plane. Another word for tessellation is tiling, and the individual shapes in a tessellation are referred to as tiles. In a more general sense, a tessellation covers any surface, not necessarily flat or infinite in extent, without gaps or overlaps.

A tessellation is a type of pattern, which can be defined as a two-dimensional design that (generally) possesses some sort of symmetry. Symmetries in one, two, and three dimensions have been classified mathematically, and these classifications apply to all types of patterns, including tessellations. Symmetry and transformation in tessellations are the topics of Chapter 3.

Tilings in which the individual tiles are recognizable, real-world motifs will often be referred to as Escheresque tessellations in this book (after Dutch graphic artist M.C. Escher), and the tiles in them as Escheresque tiles. These sorts of designs are a relatively recent phenomenon, dating back a little over a century. Tilings in which the individual tiles are geometric shapes rather than recognizable motifs will often be referred to as geometric tessellations in this book. Types and properties of geometric tessellations will be explored in the next chapter, and some particular types of geometric tessellations will be treated in greater depth in Chapters 6–10. Chapters 11–17 are devoted to planar Escheresque tessellations. Chapters 18 and 19 deal with specialized topics, and Chapters 20–25 deal with applying Escheresque tessellations to three-dimensional surfaces. The terms “tiling” and “tessellation” will be used interchangeably in this book.

Historical examples of tessellations

The use of patterns by people is ubiquitous and must surely predate recorded history in virtually every society. There is a basic human impulse to decorate, personalize, and beautify our surroundings by applying patterns to textiles, baskets, pavers, tiles, and even our own bodies. As a result, tessellations are found in virtually all cultures from very early on. Tessellations have been and continue to be used in tribal art from around the world for decoration of baskets (Figure 1.1), blankets, masks, and other objects.

Figure 1.1. Contemporary African basket in which tessellated regions are used for decoration.

The word “tessellation” comes from the Greek word tessares, meaning four, and the Latin tessellare, meaning to pave with tesserae. Tessera (plural form tesserae) are small squares or cubes of stone or glass used for creating mosaics. The Ancient Romans often used tesserae to form larger geometric figures, as well as figures such as animals or people (Figure 1.2). In addition, they used geometric tiles like rhombi (sometimes called diamonds) to form tessellations as decorative floor coverings.

Figure 1.2. Roman mosaics and tilings in Pompeii, Italy.

Beautiful examples of tessellations and related patterns in architecture dating back several centuries can be found in monumental buildings such as cathedrals and palaces around the world. These often have patterns and styles characteristic of particular cultures. Islamic art has an especially rich tradition in the use of distinctive and sophisticated geometric tessellations, starting around the 9th century and continuing to the present. Examples from the Middle East, Europe, and China are shown in Figures 1.3, 1.4, and 1.5, respectively.

Figure 1.3. Door in the Sultan Ahmed (Blue) Mosque, Istanbul, Turkey.

Figure 1.4. Mosaic from the floor of Saint-Sernin Basilica, a Romanesque building in Toulouse, France.

Figure 1.5. Panel on the quarters of the “Dragon Lady” in the Forbidden City, Beijing, China.

During the Renaissance, artists and mathematicians alike were fascinated by polyhedra and tessellations. Albrecht Dürer (1471–1528), a painter and printmaker, designed a tessellation, incorporating pentagons and rhombi [Jardine 2018]. Johannes Kepler (1571–1630), most famous for discovering the laws of planetary motion, designed tessellations of regular and star polygons (Figure 1.6) that are included in his 1619 book Harmonices Mundi [Grünbaum 1987].

Figure 1.6. Tessellations by Johannes Kepler incorporating regular and star polygons (redrawn).

Tessellations in the world around us

A tessellation is a mathematical construction in which the individual tiles meet along a line of zero width. In any real-world tessellation, there will be some finite width to that line. In addition, no real-world plane is perfectly flat. Nor is any real shape perfectly geometric; e.g., no brick is a perfect rectangular solid. Finally, any real-world tessellation will obviously have a finite extent, while a mathematical tessellation extends to infinity. When we talk about tessellations in the real world, then, it should always be kept in mind that we are talking about something that does not strictly meet the definition of a tessellation given above. Another way of looking at it is that a tessellation that we associate with a physical object is a mathematical model that describes and approximates something in the real world. Many tessellations in nature are highly irregular, as seen in the example of Figure 1.7. A variety of other examples will be presented in Chapter 4.

Figure 1.7. A tessellation created by the cracking of a lava flow in Hawaii. Many of the individual tiles can be approximated as pentagons and hexagons.

Tessellations have been widely used in architecture, not only purely for decoration but also by necessity from the nature of building materials like bricks (Figure 1.8). These same types of uses continue today and are so widespread that one is likely to find numerous examples on a short walk around most modern cities. Tessellations are also widely used in fiber arts, as well as games and puzzles. These sorts of uses are explored in more detail in Chapter 5.

Figure 1.8. Tessellated brickwork on the steeple of a church in the Friesland region of the Netherlands.

Escheresque tessellations

Lifelike motifs were incorporated early on in decorative patterns and designs, and some of these might be considered tessellations. Koloman Moser, an Austrian designer and artist who lived from 1868 to 1918, is generally credited with creating the first tessellations, in which the individual tiles depict recognizable, real-world objects. In addition to painting, Moser designed jewelry, glass, ceramics, furniture, and more. As a leading member of the Vienna Secession movement, he was dedicated to fine craft...

Table of contents

• Cover
• Half-Title
• Title
• Copyright
• Contents
• About the Author
• Preface
• 1 Introduction to Tessellations
• 2 Geometric Tessellations
• 3 Symmetry and Transformations in Tessellations
• 4 Tessellations in Nature
• 5 Decorative and Utilitarian Tessellations
• 6 Polyforms and Reptiles
• 7 Rosettes and Spirals
• 8 Matching Rules, Aperiodic Tiles, and Substitution Tilings
• 9 Fractal Tiles and Fractal Tilings
• 10 Non-Euclidean Tessellations
• 11 Tips on Designing and Drawing Escheresque Tessellations
• 12 Special Techniques to Solve Design Problems
• 13 Escheresque Tessellations Based on Squares
• 14 Escheresque Tessellations Based on Isosceles Right Triangle and Kite-Shaped Tiles
• 15 Escheresque Tessellations Based on Equilateral Triangle Tiles
• 16 Escheresque Tessellations Based on 60°–120° Rhombus Tiles
• 17 Escheresque Tessellations Based on Hexagonal Tiles
• 18 Decorating Tiles to Create Knots and Other Designs
• 19 Tessellation Metamorphoses and Dissections
• 20 Introduction to Polyhedra
• 21 Adapting Plane Tessellations to Polyhedra
• 22 Tessellating the Platonic Solids
• 23 Tessellating the Archimedean Solids
• 24 Tessellating Other Polyhedra
• 25 Tessellating Other Surfaces
• References
• Glossary of Terms
• Index