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.

Excel and FileMaker Exchange

When I was the “knowledge manager” for an architecture firm, my responsibilities included populating a company-wide FileMaker solution with decades’ worth of data from a variety of sources, including many Excel, .txt, and .csv files.  This experience exposed me to the various methods for exchanging data between Excel and FileMaker.  Later, for a course in technical writing, I wrote a chapter of a hypothetical manual on FileMaker-Excel data exchange.  I posted the chapter online and then received many queries about “where is the rest of the manual? ”

That proposed manual is unlikely to become a reality, meanwhile both Excel and FileMaker have launched many newer versions.  But below is a summary of the current methods for exchanging data among these two programs as well as guidelines for deciding which option is best for a particular task.  The information is current for FileMaker Pro 15 and recent versions of Excel.

Excel versus FileMaker: When Should You Use Them?

These two tools perform many of the same functions and produce reports and other output that can appear virtually identical.  So the decision to use one or the other can seem difficult.  One deal breaker is timeframe – setting up a simple data table that you will use once for a quick analysis or presentation and then never touch again is faster in Excel.  Another is output – you can force Excel to mimic (or be exported to or embedded in) a great many presentations types but FileMaker’s layout capabilities can produce alternatives to the spreadsheet mode more rapidly and more reliably.

In comparing the two, remember that the equivalent of a cell in an Excel spreadsheet is a single incidence of a data field for a particular record.  If you present FileMaker data in the table layout mode (the equivalent of a spreadsheet view), a record is the equivalent of an Excel row with each column representing a distinct field.

Use Excel when….

  • The cells in a column (the equivalent of a FileMaker data field) can vary by data type or appearance, particularly on an ad hoc basis.
  • You have complicated, nested calculations, especially if they aren’t applied consistently across a row or a column.
  • You are doing scenario testing but don’t want to touch the original data that is stored in a data warehouse.
  • You need a great many options for charting (FileMaker offers a nice charting option but it is not as full-featured as Excel).
  • You want to embed tables or charts in other Microsoft Office programs through technologies such as OLE (object linking and embedding); FileMaker dispensed with OLE several releases prior to 15.
  • Your primary task is to analyze and manipulate data.
  • You have simple lists to sort and filter.
  • You lack skills in Microsoft scripting languages and have little need to automate processes.
  • You’re in the initial stages of planning a database – a spreadsheet can serve as an effective tool for determining the fields you would want in a database (you can import that information into FileMaker).

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Use FileMaker when….

  • Your time horizon potentially verges on medium to long and you will use and reuse the data multiple times.
  • You want to protect the integrity of your data.
  • You have data sets that would require the use of multiple, linked Excel spreadsheets to manage, e.g. you need to view relations among data sets or a 3D view.
  • You need to use subsets of your data in a wide range of outputs, from reports to personalized letters to labels.
  • You seek more control over document layout, graphics, and typography (FileMaker isn’t InDesign or Quark but it provides more graphics control than does Excel).
  • Your primary task is to store data and present it in multiple formats (including exports to Excel).
  • You’d like to automate tasks and routines through an accessible,visual scripting language.  The FileMaker scripting language is easy to learn and the Script Workspace enables management of script libraries.

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Exchanging Data between Excel and FileMaker

The basic options for exchanging data between these two software are listed below.  The FileMaker website provides a summary of exchange options as well as information on how FileMaker manages Excel data.
Where available, links to the relevant FileMaker 15 documentation are provided.  Some tips are included.

Option 1:  Create a FileMaker solution directly from an Excel Spreadsheet

Related FileMaker documentation

This option opens an Excel spreadsheet directly in FileMaker. You will be given the option to have the first row in the spreadsheet import as either a database record or as the names of database fields.

  • If you have multiple worksheets or a “named range” (a subset of cells that you’ve assigned a name to) in the source spreadsheet, FileMaker gives you a choice of what to import.
  • FileMaker will import determine the field “type” (text, numbers, dates) based on the spreadsheet contents.  It will import calculations as direct data rather than as FileMaker calculation or summary fields.
  • FileMaker assumes every row containing data should import as a standard database record with uniform fields.  It can’t determine, for example, whether a footnote or an extraneously populated cell is not part of the desired data import.  So it is best to eliminate all extraneous contents prior to importing the data.  Headers, footers, and charts (check) do not import.

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Option 2: Create a FileMaker solution and import the Excel data

Related FileMaker documentation

This option imports data from Excel – or any other compatible source – into an existing FileMaker solution.  This is a good solution for when you have a FileMaker database already established and you want to reuse that database structure or add new records to an existing data set.

  • This option gives you several options for importing – for example, you can ignore both certain Excel and FileMaker content and you usually should ignore the first potentially importable record (e.g. first Excel row) if it contains field names.
  • You can perform multiple imports from the same Excel sheet into the same FileMaker table and you can do so on a recurring basis.  With these, you can treat the imports as additions to the existing data set or as updates or overrides.  Be very careful with these options as you can overwrite existing data that you may not want lost or else end up with lots of duplicates.  Reimporting will not reflect any subsequent changes to the data that you may have made to the initially imported data.
  • You can automate the process by a FMP script, which is handy if you’re augmenting an existing database on a routine basis.

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Option 3: Link to ODBC data via a driver

Related FileMaker Documentation

You can create links between an Excel spreadsheet and a FileMaker database through a ODBC (Open Database Connectivity) driver (you can do something similar with FileMaker and Microsoft Word or with other data sources such as SQL or Oracle databases).

Unlike Options 1 and 2 described above, ODBC linking enables dynamic linking between multiple data sources of different types.  That can reduce the risk of dealing with outdated information while enabling either a “client” document to tap into a “data warehouse” without modifying the warehouse.  It also enables you to take advantage of each software’s most powerful tools – for example, using Excel’s charting capabilities to present FileMaker data in various ways.

The challenge is that this mode of linking takes a lot of configuration and can be finicky.  If you operate in an environment with a lot of firewalls, you may face challenges getting access to ODBC sources.  You will need to read FileMaker’s guide on the topic and do a lot of testing.  But this can be worthwhile for any type of long-term relationship between multiple resources (for quick data “dating,” the other options may be better uses of your time!).

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Option 4: XML

Related FileMaker Documentation

Both Excel and FileMaker can exchange data through the Extensible Markup Language (XML) and Extensible Stylesheet Language (XSL) formats.  These are powerful languages for twisting and turning data in insightful ways but they involve a learning curve.  They are overkill for basic tabular data exchange such as can be achieved by the options described.  However, they can be useful for extracting metadata from date sources and for dealing with summary data, prepping data for use in Excel pivot tables, and other high-level data uses.  XSL, Document Type Definition (DTD), and other related standards can identify quality-control issues and verify character encodings.  Because most applications nowadays support XML and XSL, you can use these standards to build data exchange frameworks among many different software, such as desktop publishing software, for a systematic deployment of data from a specific source.

If you use FileMaker Advanced (a high-powered developer version of FileMaker), you can generate an XML-formatted report of a FileMaker database and open that in Excel and other applications.

For guidance on XML-based exchange between FileMaker and Excel (and other applications), look not only at the FileMaker Community resources but also more general online “communities” such as Stack Overflow).

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Option 5: Exporting Data from a FMP Database

Related FileMaker Documentation

With the same ease that Options 1 and 2 provide from importing data from an Excel spreadsheet into FileMaker, you can easily export data from FileMaker into Excel.  FMP provides the following options:

File Menu…Save/Send Records As > Excel…

This provides the option to export either the current record or the current found set (which may or may not be all the database records) and to add metadata such as worksheet name (the name of the spreadsheet’s tab), title, subject, and author.  You also can email the results as an attachment.  The resulting spreadsheet is a table with the FileMaker field names appearing as column heading names and the metadata options converted to Excel document properties.

File Menu…Export > Excel….

This option also gives you the option to export metadata but it also opens up a dialog window that lets you select the fields to export and the order in which the fields will appear as columns (the FileMaker default is to include only basic text, numeric, and date fields but you can add summary, calculation, or global fields, although these will present the same values in many records).  You also can apply the current layout’s data formatting to the data (handy if you want to skip the formatting in Excel) and you can export data from related tables (although you’ll likely only import the value of the first related record if multiple records are attached).

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Option 6: Plain Text Formats

Both FileMaker and Excel can read and write common text interchange formats such as tab-separated (with a .tab or .txt file extension) or comma-separated (with a .csv or .txt file extension).  These file formats treat every Excel line or FileMaker record as a separate line of text, with field or column information separated by tabs or commas.  They do not contain content other than text nor do they retain formatting information and they import field names (column header content) as generic terms.

Because FileMaker enables reliable direct exchange with Excel, an interim import or export via plain text formats isn’t ordinarily required.  However, if you experience conflicts or failed direct exchange, you could try using these file formats as a diagnostic; sometimes, this is useful with files created in older versions of a software.  And some people who are concerned with long-term archival access recommend storage in these formats because they may outlive proprietary file formats.  For example, it is not possible to open up a Filemaker solution created in an older version (pre-FileMaker 7) in version 15 without having going through a series of interim upgrades that require access to older versions of the software.  If you only need to preserve the raw data, backups in these formats can provide a means of future access.

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Scripting and Automating the Process

Related FileMaker Documentation (Opens as an Acrobat PDF file)

For many of the options above, you can use FileMaker’s scripting features to automate the process of sharing data with Excel spreadsheets.  The Scripts Step window displays options for importing and export records as well as steps for converting Excel files to FileMaker files and saving records in Excel format.  With scripts, you not only can automate the data exchange options but you can build more complex routines of working with multiple exchange processes and modifying data and controlling record sets before or after the exchange process.  For example, you could initiate a complicated Find strategy to nail a particular subset of records that you routinely want to export.  You also can build in steps to provide guidance and helpful hints for users, which is nice for non-programmers executing routines that they haven’t designed.

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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.