Constructing Tessellations [Redux]

Over a decade ago, I created a video to demonstrate two methods of constructing tessellations. The video was created using Adobe Flash, which proved problematic, not only because Flash is a proprietary technology (which Adobe will discontinue in 2020) rather than open source but also because my Flash methodology relied on the visual frames-based timeline technique. Such an approach ignored the underlying geometry of tessellations so I decided to redo these demonstrations using a mathematically-driven scripting technique, ideally using open source technologies. Accordingly, I invited Jonathan Dowse, a talented architect and programmer, to take on the challenge of using JavaScript and SVG (scalable vector graphics) to recreate these demonstrations.

Here are the results! Jonathan and I invite you to explore the underlying source code at GitHub and to extend these tessellations in new directions.

SVG Format by Jonathan Dowse.

Further Reading

More Information About Tessellations

The following list presents an eclectic group of resources on various aspects of tessellations. These range from general background works to examples and installations of tilings.

Web sources lacking a publication date were last accessed in March, 2017. All links open in a separate window.

Ball, Philip. “Islamic Tiles Reveal Sophisticated Maths.” News@nature, February 19, 2007. doi: 10.1038/news070219-9

Bart, Anneke and Bryar Clair. “Math and the Art of M.C. Escher.” http://mathstat.slu.edu/escher/index.php/Math_and_the_Art_of_M._C._Escher

Beyer, Jinny. Designing Tessellations: The Secrets of Interlocking Patterns. Lincolnwood, Ill.; St. Albans: Quilt Digest ; Verulam, 1999. WorldCat permalink: http://www.worldcat.org/oclc/473422657

Bohannon, John. “Quasi-Crystal Conundrum Opens a Tiling Can of Worms.” Science 315, no. 5815 (February 23, 2007): 1066–1066. doi: 10.1126/science.315.5815.1066

Frettlöh, Dirk and Edmund Harriss. “Tilings Encyclopedia.” Accessed March 18, 2017. http://tilings.math.uni-bielefeld.de/

Huffmann, Daniel. “Penrose Binning.” Somethingaboutmaps, April 30, 2015. https://somethingaboutmaps.wordpress.com/2015/04/30/penrose-binning/

Lu, Peter J., and Paul J. Steinhardt. “Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture.” Science 315, no. 5815 (February 23, 2007): 1106–10. doi: 10.1126/science.1135491

“M.C. Escher – Image Categories – Symmetry.” Official M.C. Esher website. http://www.mcescher.com/gallery/symmetry/

NRICH (University of Cambridge). “Transformations and their Properties: Tessellations.” http://nrich.maths.org/1556

Penrose, Roger. RI Channel: Forbidden Crystal Symmetry in Mathematics and Architecture. Royal Institution of Great Britain video, 2014. http://www.richannel.org/forbidden-crystal-symmetry-in-mathematics-and-architecture

Rehmeyer, Julie. “Ancient Islamic Penrose Tiles.” Science News, September 23, 2013. https://www.sciencenews.org/article/ancient-islamic-penrose-tiles-0

Weisstein, Eric W. “Tessellation.” http://mathworld.wolfram.com/Tessellation.html

Wikipedia contributors, “Euclidean tilings by convex regular polygons,” Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Euclidean_tilings_by_convex_regular_polygons&oldid=756054436

Wikipedia contributors, “Penrose tiling,” Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Penrose_tiling&oldid=767819627

Constructing Tessellations

As mentioned in above, only three types of regular tessellations exist; these use rectangles, triangles, or hexagons. Semi-regular tessellations combine two polygons that share a common edige; only a limited number of these shapes exist. To create more exotic tessellations, you can construct an irregular shape — or series of shapes — using a careful but simple methodology of polygon modification. This technique was used by M.C. Escher to create his famous “Metamorphosis” and is common in “diaper patterns.”

The video below dynamically illustrates two methods of tessellation construction. The first method shows how a single irregular tile can produce many different patterns. The second method shows how to create a metamorphosis with two different shapes that do not share a common edge through the use of transitional shapes. The demonstration requires the plug-in for Adobe Flash version 6 or later.

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A Tilings Travel Side Show

The images below feature examples of applied tessellations. These images were taken on a trip to the Andalusia region of Spain. Here, in cities such as Córdoba, Granada, and Seville, you can view stunning examples of architecture featuring tessellations. Many of these reflect the influence of the Moors, who occupied southern Spain for several centuries. Their strong geometric designs were combined with arabesque lettering and floral motifs and co-existed with ancient Roman, Christian, Jewish, and Gothic traditions in architecture and design. These tilings can be found on floors, walls, ceilings, columns, and any other available surface. Centuries after their creation, they appear absolutely modern and timeless.

  • Wall surface from the Alhambra. A clever way to add a sense of texture to a monochromatic surface.
  • Both this (leather-tooled) door and its (tiled) surround feature tilings.
  • Tessellations can be sat upon. This tiled bench is an example of a single polygon pattern.
  • On this column capital, tilings are combined with multiple layers of three-dimensional designs.
  • This floor pattern provides an example of a shift in alignment to accommodate two tiles of different sizes.
  • Incisions provide a multilayered, 3D effect to this wall surface pattern.
  • Somehow these multiple patterns on all surfaces (including the window screens) co-exist in harmony.

All images by K. Vagts.

Practical Applications

Tessellations have practical applications in many realms, from art and architecture to science, technology, and production.

In design and architecture, tessellation refers to the paving of walls, floors, or other surfaces with a pattern of small tiles (tesserae) made of ceramics, glass, metal leaf, stone, or other materials. These tesserae normally are cut into geometric shapes that fit together perfectly in either simple or complex designs in a seemingly infinite pattern while providing continuous surface coverage. This is an ancient technique that you can see in buildings and wall murals in Greece, Italy, Turkey, India, and many other countries. Tessellations are particularly prominent in Islamic art, which forbids representational images of God; therefore, its designs favor abstract forms with mathematical underpinnings.

Although tesserae often consist of abstract shapes, primarily symmetrical rectangles, hexagons, octagons, and other polygons, they also can consist of figurative elements, as in the work of artists like Kolomon Moser (1868-1918) and M.C. Escher (1898-1972). Escher is famed for his tessellations composed of horses, butterflies, birds, and imaginary creatures (which in the 1990s formed the basis of a popular line of upholstery fabrics!). Many of his designs “morph” different shapes, such as hexagons evolving into creatures.

Many contemporary artists and craftspeople apply tessellations in their work. These designs, which can contain representational elements, are often called diaper designs. Those with a 3D aspect often incorporate principles of origami.

Although often considered an art or design application, tessellations can be found in nature, as in the patterns of snowflakes, honeycombs, and cracks in dry earth. Scientists have determined that beehives are composed of hexagons because that is the most efficient way for bees to construct their homes.

Tessellations appear in various scientific and engineering disciplines. Chemical discoveries show that certain carbon molecules take the shape of a truncated isocahedren. Geodesic domes are both theoretical 3D geometric constructs and built structures. The mathematics of various tessellation types underscore efficiency-focused processes of machining and manufacturing.

Tessellations: Origins and Types

The word tessellation originates from the ancient Greek word tessares, meaning four or four-cornered. The Latin word tessera means cube or die and tessella refers to small squares laid in a mosaic. From these ancient words derive similar terms in various practical applications.

In mathematics, tessellation refers to the study of “tiling” or how regular shapes can be placed to fill an infinite space with no gaps and no overlapping shapes. This is a mathematical discipline which has been evolving since the early 17th century and formally recognized in the 19th century.

Types of Tessellations

Tessellations can be divided into several categories; sample subsets include the following:

    • Regular Tessellations
      Regular tessellations form patterns consisting of a single shape. Only three types of regular tessellations exist: triangles, squares, and hexagons. These shapes by themselves can fill a surface because their interior angles are exact divisors of 360°. Of these shapes, only the squares line up with one another without requiring rotating or shifting.

Tessellations using one type of polygon

Figure 1: Tessellations using one type of polygon.

    • Semi-Regular Tessellations
      Semi-regular tessellations combine two types of polygons that share a common vertice. For example, a regular hexagon with a 1″ side can line up with a 1″ square. Nine types of semi-regular tessellations exist.

Tessellations using two types of polygons.

Figure 2. Tessellations using two types of polygons.

  • Replicating Shapes (Rep-Tiles)
    Rep-tiles consist of congruent shapes that are rotated to create ever-larger versions of the shaped in an infinite series. Often called polyforms, rep-tiles are implicit in such phenomena as the classic illustration of the Golden Mean and the Penrose Tile.
  • 3D Tessellations
    Tessellations can take 3 dimensional forms as in truncated octahedrons and in geodesic domes. Such forms can combine combinations of shapes; only five are regular polyhedra (i.e. platonic) shapes.
  • Non-Periodic Tessellations
    Non-periodic (aperiodic) tilings have no regular, repetitious patterns but rather evolve as they expand over a plane. The Dutch artist M.C. Escher produced well-known examples of such tilings, such as his graphic of birds that morph into triangles.

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Practical Applications of Tessellations

Tessellations have practical applications in many realms, from art and architecture to science, technology, and production.

In design and architecture, tessellation refers to the paving of walls, floors, or other surfaces with a pattern of small tiles (tesserae) made of ceramics, glass, metal leaf, stone, or other materials. These tesserae normally are cut into geometric shapes that fit together perfectly in either simple or complex designs in a seemingly infinite pattern while providing continuous surface coverage. This is an ancient technique that you can see in buildings and wall murals in Greece, Italy, Turkey, India, and many other countries. Tessellations are particularly prominent in Islamic art, which forbids representational images of God; therefore, its designs favor abstract forms with mathematical underpinnings.

Although tesserae often consist of abstract shapes, primarily symmetrical rectangles, hexagons, octagons, and other polygons, they also can consist of figurative elements, as in the work of artists like Kolomon Moser (1868-1918) and M.C. Escher (1898-1972). Escher is famed for his tessellations composed of horses, butterflies, birds, and imaginary creatures (which in the 1990s formed the basis of a popular line of upholstery fabrics!). Many of his designs “morph” different shapes, such as hexagons evolving into creatures.

Many contemporary artists apply tessellations in their work. These designs, which can contain representational elements, are often called diaper designs.

Although often considered an art or design application, tessellations can be found in nature, as in the patterns of snowflakes, honeycombs, and cracks in dry earth. Scientists have determined that beehives are composed of hexagons because that is the most efficient way for bees to construct their homes.

Tessellations appear in various scientific and engineering disciplines. Chemical discoveries show that certain carbon molecules take the shape of a truncated isocahedren. Geodesic domes are both theoretical 3D geometric constructs and built structures.

Tessellations

One definition of a tessellation is: "the careful juxtaposition of elements into a coherent pattern"1 The first iteration of my personal website was named Tessellation Design because that juxtaposition underscores much of what interests me in information design and librarianship as well as other intriguing topics, including systems, scripting, patterns, abstract designs, puzzles, architecture, and geometry and other mathematic disciplines.

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1 Webster’s New International Dictionary, 3rd Edition.