Tessellations - Everything You Ever Wanted to Know

The Incredible Tessellations Page

tes⋅sel⋅la⋅tion [tes-uh-ley-shuhn] [1]

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Definitions

"A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps."[2]

 

"To form into a mosaic pattern, as by using small squares of stone or glass." [From Latin tessellātus, of small square stones, from tessella, small cube, diminutive of tessera, a square; see tessera. Dictionary.com]

 

"A collection of plane figures that fills the plane with no overlaps and no gaps." [3]

"Designs featuring animals, birds, etc, which can fill the page, without over-lapping, to form a pattern." [4]

escher pic

M. C. Escher, Reptiles, Lithograph, 1943. Click for the full size image.

The King of Tessellations, M. C. Escher

Maurits Cornelis Escher (1898-1972) is a graphic artist known for his art tessellations. His art is enjoyed by millions of people all over the world. He created visual riddles, playing with the pictorially logical and the visually impossible.

 

He is most famous for his so-called "impossible structures", such as Ascending and Descending, Relativity, his Transformation Prints, such as Metamorphosis I, Metamorphosis II and Metamorphosis III, Sky & Water I or Reptiles. What made Escher's pictures so appealing was that he used tessellations to create optical illusions. He also gave them depth by adding shade.

 

M.C. Escher, during his lifetime, made 448 lithographs, woodcuts and wood engravings and over 2000 drawings and sketches [You can buy a book at the bottom of this page that includes them all]. Like some of his famous predecessors, - Michelangelo, Leonardo da Vinci, Dürer and Holbein-, M.C. Escher was left-handed.

 

sky and water
M. C. Escher, Sky & Water I, woodcut, 1938

M.C. Escher illustrated books, designed tapestries, postage stamps and murals. He was born in Leeuwarden, the Netherlands, as the fourth and youngest son of a civil engineer. After finishing school, he traveled extensively through Italy, where he met his wife Jetta Umiker. They settled in Rome, where they stayed until 1935. During these 11 years, Escher would travel each year throughout Italy, drawing and sketching for the various prints he would make when he returned home.

 

Many of these sketches he would later use for various other lithographs and/or woodcuts and wood engravings. He played with architecture, perspective and impossible spaces. His art continues to amaze and wonder millions of people all over the world. In his work we recognize his keen observation of the world around us and the expressions of his own fantasies. M.C. Escher shows us that reality is wondrous, comprehensible and fascinating.

 

Examining one of his woodcuts, Sky & Water I (left above), we see fish in the sea and as you go up, the space between the fish transform into black ducks. The tessellations are the fish shapes in white next to the duck shapes in white. Technically, the shapes at the top and bottom of his woodcut are no longer tessellations because they spread apart and the space around them no longer resemble fish or ducks.

Another Tessellation Artist, Robert Fathauer

robert fathauer

Robert Fathauer stands next to his art, "Twice iterated Knot." He entered this in the American Mathematical Society Exhibition.

Robert Fathauer, born in 1960, creates his tessellations using a computer. Robert has an interest in mathematics and art and has been a great fan of Escher.

 

Says Dr. Fathauer, "if there's anything one can be certain of in this world it's mathematics. It's the one discipline where results can be proven to be true. At the same time, there is great beauty and elegance in mathematics. Conversely, art is the discipline where beauty is the traditional goal, but art also strives to get at deep truths. Both disciplines appeal to me for these reasons, and it seems natural to combine them." [5]

 

Robert Fathauer received his doctorate from Cornell University in Electrical Engineering and joined the research staff of the Jet Propulsion Laboratory in Pasadena, California. Later in 1993 he founded his own company called Tessellations to produce tessellation puzzles and offer them for sale. Dr. Fathauer now promotes mathematical art at exhibitions and conferences. His products look excellent for any classroom teacher.

Tessellation Artist/Mathematician Roger Penrose

Some people say that he is the expert in recreational math. Roger Penrose, a professor of mathematics at the University of Oxford in England, pursues an active interest in recreational math which he shared with his father. While most of his work pertains to relativity theory and quantum physics, he is fascinated with a field of geometry known as tessellation, the covering of a surface with tiles of prescribed shapes.

 

penrosePenrose received his Ph.D. at Cambridge in algebraic geometry. While there, he began playing with geometric puzzles and tessellations. Penrose began to work on the problem of whether a set of shapes could be found which would tile a surface but without generating a repeating pattern (known as quasi-symmetry). "Eventually Penrose found a solution to the problem but it required many thousands of different shapes. After years of research and careful study, he successfully reduced the number to six and later down to an incredible two." [6] He called these shapes Penrose tiles.

 

Believe it or not, but the shapes he came up with are like the chemical substances that form crystals in a quasi-periodic manner. Not only that, but these quasi-crystals make excellent non-scratch coating for frying pans.

 

Penrose and Escher have been influences on each other. Penrose first met Escher at the International Congress of Mathematicians in Amsterdam. Penrose saw some of Escher's work there and began playing with tessellations and came up with what he calls a tri-bar. A tri-bar is a triangle that looks like a three-dimensional object, but could not possibly be three-dimensional in real life. He published his work in the British Journal of Psychology. Escher read his article.

 

penrose tileSays Penrose, "One was the tri-bar, used in his lithograph called Waterfall. Another was the impossible staircase, which my father had worked on and designed. Escher used it in Ascending and Descending, with monks going round and round the stairs. I met Escher once, and I gave him some tiles that will make a repeating pattern, but not until you’ve got 12 of them fitted together. He did this, and then he wrote to me and asked me how it was done—what was it based on? So I showed him a kind of bird shape that did this, and he incorporated it into what I believe is the last picture he ever produced, called Ghosts." [7]

Integrating Math and Art Using Tessellations

Triangles, squares and hexagons are the only regular shapes which tessellate by themselves. If you put many of these shapes together side-by-side, they form a tessellation. You can have other tessellations of regular shapes if you use more than one type of shape. You can even tessellate pentagons, but they won't be regular ones.

 

The benefits of making connections to other subjects using art are well documented. Carol Goodrow, a first grade teacher saw improvement in math skills by making connections through other areas. For example, during year-end benchmark testing, her class completed sections on numeration more quickly, yet scored as well or better, than past classes. In addition, Goodrow reported that many students demonstrated a better understanding of fractions. "My fraction committee, a group of the most capable math students, computed the class's [running] mileage on their own, working with ½'s, ¼'s and 3/4's. They represented the fractions by models, but they could also compute them in their heads." [8]

 

animated pegasusCanadian Math teacher Jill Britton uses Escher tessellations to help students learn the mathematics term, congruent. While examining Escher's picture, Tessellation 105, she says, "When the students study a pegasus in its parent square, they discover how Escher modified the square to obtain his creature. Each "bump" on the upper/lower side is compensated for by a congruent "hole" on the lower/upper side. The same is true of the left/right sides. Corresponding modifications are related by translation. The area of the parent square is maintained." [9]

 

On the left you see an animated pegasus image that illustrates perfectly how Escher achieved the tessellations through congruent shapes on each side. Jill has used this animated GIF image to show us how Escher accomplished this task.

Fractals - Asymmetrical shapes

Definitions: "A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole ..."

 

"A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales."

"Fractals are endlessly repeating patterns that vary according to a set formula, a mixture of art and geometry. Fractals are any pattern that reveals greater complexity as it is enlarged."

 

This is the best definition on fractals I've found on the web:

fractal

This fractal was created by Melissa D. Binde. Her website is no longer online. As you can see, there is an increasing level of complexity. The black space on the right become fractals themselves.

"Imagine flying in a space shuttle looking at the coast of Britain. From such a great distance the coast looks perfectly straight, going from north to south. But, as you approach the earth something tells you the coast is not perfectly straight... Of course! As you go into the upper atmosphere you realize that it has thousands of bays, harbors, capes, and peninsulas that you could not see from a distance. Thinking that now you have a detailed picture of the coast, you turn towards one of the harbor beaches, which seems to be straight... However, as you get closer to it, you see that it too has thousands of smaller bays, harbors, capes, and peninsulas! Wondering if this will ever end you decide to get even closer... Eventually you wind up on the beach looking at the coast through a microscope. You can now see every grain of sand clearly, but, it too has thousands of indentations and extrusions! Benoit Mandelbrot called shapes like this fractals. Fractals are figures with an infinite amount of detail. When magnified, they don’t become more simple, but remain as complex as they were without magnification. In nature, you can find them everywhere. Any tree branch, when magnified, looks like the entire tree. Any rock from a mountain looks like the entire mountain." [10]

How Are Tessellations and Fractals Alike and Different?

The Same:

Both tessellations and fractals involve the combination of mathematics and art. Both involve shapes on a plane. Sometimes fractals have the same shapes no matter how enlarged they become. We call this self-similarity. Tessellations and fractals that are self-similar have repeating geometric shapes.

 

How they are different:

Tessellations repeat geometric shapes that touch each other on a plane. Many fractals repeat shapes that have hundreds and thousands of different shapes of complexity. The space around the shapes sometimes, but not always become shapes in the design. The space around shapes in tessellations become repeating shapes themselves and play a major part in the design.

Tessellation and Fractal Lessons

Links

escher tie
Buy the M. C. Escher Metamorphosis Tie! See also this tie.

Books

Introduction to Tessellations. - This clear introduction to tessellations and other intriguing geometric designs help students explore polygons, regular polygons and combinations of regular polygons, Escher-type tessellations, Islamic art designs, and tessellating letters.

Designing Tessellations : The Secrets of Interlocking Patterns. - Inspired by artist M. C. Escher, Jinny Beyer introduces quilters to the fascinating world of symmetry and then clearly shows how to experiment with shapes and images to create sensational, tessellating designs.

The Magic Mirror of M.C. Escher. - Escher was a master of the third dimension. Mathematician Bruno Ernst is stressing the magic spell Escher's work invariably casts on those who see it. Ernst visited Escher every week for a year, systematically talking through his entire aeuvre with him.

M.C. Escher: His Life and Complete Graphic Work. - Illustrated are 448 (of the 449) original woodcuts, wood engravings, lithographs, linocuts and mezzotints by Maurits Cornelis Escher.

M. C. Escher. - Renowned artist M.C. Escher is not a surrealist drawing us into his dream world, but an architect of perfectly impossible worlds who presents the structurally unthinkable as though it were a law of nature.

Origami Tessellations: Awe-Inspiring Geometric Designs. - In this book, Eric Gjerde presents easy-to-follow instructions that introduce the reader to the incredible beauty and diversity of origami tessellations.

Mosaic and Tessellated Patterns: How to Create Them.- This book introduces the basic types of tessellations and presents many examples.

 

Videos

Tessellations: How to Create Them [VHS] - The 27-minute DVD breaks down the mystery of creating tessellations into simple steps.

Art In The Classroom Series: Tessellations - Students discover how to create amazing patterns by sliding, rotating and flipping shapes, and by contrasting colors.

 

Puzzles and Games

Magnetic Tessellation Puzzle. - Each of the four shapes in this set can be transformed into a different geometric shape.

In the Garden Classroom Pack - Included lesson guide covers symmetry in tessellations and arcs of circles. Have fun while learning about geometry and tessellations with this unique puzzle that combines art and math!

GEOMETRIC TESSELLATIONS. - Put together puzzle pieces to form your own tessellation. Created by Worldclass Learning.

 

 


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