Tessellations

 

The American Heritage Dictionary defines the word tessellate as “to form into a mosaic pattern by using small squares of stone or glass.”  The word tessellation comes from the Latin word “tessella.”  A tessella is the small pieces of glass or stone that fit together to make the design, which is also known as a mosaic.  In a geometry class, a plane tessellation is a two-dimensional design or pattern made up of one or more shapes that completely cover a surface without any gaps or overlaps (Seymour 3). 

 

Tessellations are all around us in the form of geometric designs in floor tiles, ceiling tiles, quilts, carpets and many everyday objects.  Some tessellations are very simple, like that of a child’s checkerboard game, while others are much more complex, like those by Dutch graphic design artist, M.C. Escher.  One thing that all tessellations have in common, other than the idea that they have no gaps and no overlaps, is the fact that they can go on infinitely in every direction. 

 

 

Man has been surrounded by these designs for thousands of years.  Ancient Sumerians had tessellating designs in wall mosaics that date back as early as 4000 B.C.  Moorish artists are also known for these beautiful repetitive designs because the Islamic religion forbade the images of man, animal or any real-life object in their art.  They could only use calligraphy or geometric patterns for decoration.  These limitations pushed the artists to stretch their imaginations.  They were creating intricate designs that are still as interesting and timeless today as they were thousands of years ago.  The beauty of these patterns also intrigued other cultures.  Tessellations have been found in art from Egypt, Persia, Rome, Japan, China, India, and Spain.  The media used for these tessellating patterns varied greatly, from glass, to clay tiles, to stones, to metalwork, woodwork and carpets (Seymour 9, Drexel).

 

The shapes that make up a tessellation pattern (with straight lines) are polygons.  Not all polygons can tessellate, but first we will look at the different polygons that can.  A polygon, or a shape that is made up of straight lines, is named according to how many angles and sides it contains.  The place that two of these straight lines connect on the side of a polygon is a vertex.  So a hexagon, contains six sides, six angles and six vertexes, similarly a triangle has three of each. 

 

In order for a triangle to tessellate, “the sum of the interior angles of any triangle equals 180 degrees” (Seymour 22).  There is a small demonstration that Seymour uses in his book “Introductions to Tessellations” to show that the sum of all the interior angles of any triangle equals 180 degrees.  The triangle is cut out of a piece of paper, and you are to tear off all three of the corners of it. Place all three of these corner pieces together, with the vertexes pointing in towards each other.  The shape that will be made will have a straight line on the bottom, meaning that the sum of all the interior angles of the triangle equal 180 degrees.  This will happen with any classification of triangle.  The sum of all the interior angles will be 180 degrees, whether or not it is a right, obtuse, scalene, isosceles, equilateral, equiangular, or acute triangle.

 

If you were to place three of any type of triangle in a row, with the one in the middle, upside down, you would have a nice neat row of triangles.  Another row of them could go on top and another row on the bottom.  Since the sum of all the inside angles of any triangle equal 180 degrees, they will always fit together in a tidy row.  Therefore, all triangles have the ability to tessellate (Seymour 30).  The patterns created by simply changing the color arrangement of the triangles is unending.  Well, it might eventually come to an end, but the point is that with the same line work in a tessellation, but only the colors moving around many different patterns can be created with the same triangles.

 

Quadrilaterals are another type of polygon that can tessellate. Since the sum of the interior angles of a quadrilateral is 360 degrees and there are four sides, angles and vertexes to a quadrilateral, each of the four angles equals ¼ of the 360 degrees.  If you put four of the quadrilaterals together, with one of each of the four vertexes all touching at one point, the design will tessellate.  This method only produces a tessellation if the quadrilateral is “rotated around the midpoint of its sides”--there are many ways that the four vertexes can be arranged and the pattern will not tessellate.  If the four shapes are rotated around in a circle, the pattern will create a tessellation.  Any quadrilateral can be used to make a tessellation (Think Quest). An example of this would be a checkerboard game.

 

Another way to think about how quadrilaterals can tessellate is by looking at the fact that any quadrilateral can be divided into two triangles.  We have already looked at how all triangles can tessellate, so the same must be true about a shape that can be divided into two triangles.  The quadrilateral does not need to be divided into two even, or congruent triangles, because the sum of the interior angles of any triangles is 180 degrees (Seymour 32).  This indicates that tessellating designs can be made from all quadrilaterals, including a rhombus, parallelogram, trapezoid, rectangle, kite, square, and scalene shapes.

 

Would we be able to make tessellating patterns from any pentagon, since we know how far we can go with a triangle and quadrilateral?  As it turns out, if you place three regular pentagons together, with vertexes touching, there is a slight gap.  If you tried to place a fourth regular pentagon in that space, there would be an overlap.  Therefore, regular pentagons all by themselves are not able to tessellate.  But if you add an additional shape to the pentagons, it works.  Regular pentagons, attempting to tessellate, create a parallelogram shaped gap, so pentagons need parallelograms to go with them before a successful tessellation can be made (Seymour 48).

 

There are certain irregular pentagons that are able to tessellate without the help of a parallelogram.  Up until 1968, mathematicians thought that all the irregular pentagons that could tessellate had been discovered.  Several more were found that year and then more in 1975, when Scientific American wrote an article about tessellating pentagons.  More were discovered again in 1977 and in 1985 (Perplexing Pentagons).  This seems to indicate just how fascinated man is with these repeating patterns.

 

One very familiar tessellation is that of the hexagon. The shape seems to naturally lend itself to tessellations.  Mother Nature demonstrated this in beehives, and we liked the look so much that we made chicken wire in the same pattern.   A hexagon can be divided in triangles by drawing three lines from any one vertex across the shape to the three vertexes on the other side.  We already know that the interior angles of a triangle equal 180 degrees, so now we can add the four triangle measurements together.  The sum of the four triangles is 720 degrees.  There are 6 angles on the inside of the hexagon, so if you divide the 6 angles by 720 degrees, you find out that each of the interior angles is 120 degrees.  120 degrees divides into 360 degrees exactly 3 times, so we know that 3 hexagons fit together perfectly and will tessellate (Seymour 49).

 

As the number of sides on a polygon goes up, so does the sum of the interior angles.  There are 3 regular tessellations that have only one shape in each of them. They are the triangle tessellation, the quadrilateral tessellation and the hexagonal tessellation. There are many possible combinations of shapes that can fit together, whose interior angles add up to 360 degrees, but for a tessellation to have only one shape inside it, there are only those three kinds (Seymour 50). 

 

In order for us to have a way to talk about tessellations and the combinations of polygons that they are made from, a numbering system is used.  The numbers represent the number of side on the polygons and the order that the numbers are in represent their arrangement around a given vertex, in a clockwise direction. For example, 3.3.3.3.3.3 represents 6 triangles and 3.3.3.4.4 is 3 triangles with 2 squares (Think Quest).  This technique for number the arrangements was devised by Owen Jones when he traveled to the Middle East in the 1840’s to document hexagonal tilings and symmetry on the walls of Islamic Mosques (Field 93).

 

Mathematicians have determined that there are a total of 21 different arrangements of 17 possible combinations of regular polygons that will make a tessellating pattern.  Since the smallest possible angle in any regular polygon is 60 degrees, 1/3 of a triangle, we can’t have more than six polygons meeting at any vertex.  Similarly, there can’t be less than three polygons meeting at any vertex.  The measure of any angle of a regular polygon with n sides and n angles is shown by the expression                                          The sum of the angles around any vertex is always 360 degrees, so if we look at three polygons with the sides labeled as                                      we have the following equation—

 

 

This expression can be simplified to—

 

 

Similarly, the arrangements for four polygons give us the equation—

 

 

And for five polygons,

 

 

And for six polygons,

 

 

These four equations have 17 possible solutions, which make up the 21 possible arrangements (Seymour 245).

 

In addition to all the possible polygon combinations that can be used to make a tessellation, the shapes can be moved or transformed in different symmetrical ways to create interest in the pattern.  One way to move the shapes around is to slide them up and down or left and right. This kind of movement does not change the orientation of the shape, just to location. This kind of transformation is referred to as “translation.” If the orientation were changed, you would say that the shape had been “rotated” or turned.  It is turned at a certain point called the “center of rotation.”  The shapes can also be reflected or flipped so that they mirror each other.  And the last way to transform the shape would be to flip it then to slide it along a straight line, which would be called a “glide reflection.”  These four ways to manipulate the shapes on a plane are all ways to keep the image symmetrical.  Symmetry is a very important aspect of tessellations (Seymour 70).

 

Everything that has been talked about up to this point is the basic information needed to create tessellations.  Since three different polygons (triangle, quadrilateral, and a hexagon) will tessellate, a grid of any one of those three shapes can be used to make a tessellation.   Another way to start a tessellation would be to use “dot paper” as the underlying grid.  The dots are used to represent certain vertexes of polygons.  The dots can be arranged in a square pattern or in a triangular pattern to make the grid.  The grid pattern can be combined with the dot pattern to make a more elaborate underlying grid (Seymour 104).  This idea of “tweaking” or transforming existing polygons and lines by way of symmetry and coming up with my own tessellation is fascinating to me and I have included the steps that I took to create my own original tessellation. 

 

Dutch graphic artist M.C. Escher tells how he starts his tessellations, or “dividing a plane” as he puts it in his book, The Regular Division of the Plane, that he wrote in 1957.  His interest in repeating patterns started when he traveled to northern Italy and Alhambra, Spain as a young man in 1922 (Bool 52).  The beauty of the graphics that decorated churches and mosques inspired him.  He was already a well-known graphic artist, but not famous. This inspiration lead to an obsession that consumed M.C. Escher for the rest of his life.  He was no longer interested in expressing images that he observed, but in  “constructing images that dealt with the regular division of the plane, limitless space, rings and spirals in space, mirror images, inversions, polyhedrons, relativities, the conflict between flat and special and impossible constructions” (Bool 52).

 

Escher tells that he begins  “dividing a plane regularly” by having two sets of parallel lines.  The distance at which the lines intersect and the angles at which they intersect will reveal something about the creatures that will emerge from the page later.  The size of the parallelograms on the paper determines the size of each creature and that size will remain a constant through the entire metamorphosis.  Escher refers to the 4 types of transformations or movements that the shapes can possibly make: translation, rotation, reflection or glide reflection. Escher never refers to his repeating pattern as “tessellations,” but always “division of plane.”  This may be due to the fact that his essays, which are included in the book M.C. Escher, His Life and Complete Works, were translated from Dutch to English in 1982 and there may be no direct translation for the word “tessellation.”

 

He envisions one creature being black and the other is white, allowing the most possible contrast between the two.  The black and white silhouettes begin to assume a “particular, and not arbitrary shape.”   Escher then attempts to create a form that the observer can recognize as familiar (Bool 158). By adding a few details at a time, the metamorphosis begins.  What started out as a fish at the top of the page becomes a “shape” and from that shape a bird emerges. The transformation has taken place.   Escher says, “obviously this can be done the other way around as well” (Bool 158).  It seems so easy for this famous artist to explain how he made these incredible works of art.  His explanation is so simple and inadequate; that he gives the impression that doesn’t completely understand how he makes the creatures come to life. As if he thought his simple explanation would help a Mission College student duplicate these amazing images!  I am, however, willing to make a humble attempt.

 

Escher, a graphic artist who got failing grades in high school and never received any formal mathematics training, says, “By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I made, I ended up in the domain of mathematics. Although I am absolutely without training or knowledge in the exact sciences, I often seem to have more in common with mathematicians than with my fellow-artists” (Bool 55). It would have been interesting to see how Escher art would have been influenced if he had received a formal (mathematics) education.

 

In 1956 Escher met and became friends with a mathematics teacher, Bruno Ernst.  Ernst was fascinated with Escher’s images and the mathematical quality of his work. By means of an extensive analysis of Escher’s prints and sketches, Ernst was able to categorize all of Escher’s work according to mathematical themes.   Ernst’s aim was to discover some structure and coherence in Escher’s marked mathematical tendencies. When asked by his new friend to explain why his art portrayed such calculated mathematical images, Escher had no explanation about it and did not know how the obsession came about.

 

Ernst’s first attempt at identifying themes was unsuccessful. At first he thought that the different areas would be mirrors, regular polyhedrons, spirals, and Mobius strips.   This plan would not work because it would not lead to any great insights. The math teacher thought he could only make progress if he included the “intentional” content of the art, or the meaning of the art. The themes that Ernst decided to classify Escher art into are: penetration of worlds, illusion of space, the regular division of the plane, perspective, regular solids and spirals, the impossible and the infinite.  He had to have many consultations with Escher during this time to find out what message the art was trying to give to viewer (Bool 135).

 

The first category, penetration of worlds, refers to Escher attraction to mirrors. Ernst says that a “child who looks into a mirror for the first time is surprised when he notices that the world behind the mirror, which looks so real, is actually ‘intangible’; however, he soon ceases to consider this false reality as being strange. Over time, the surprise disappears and the mirror becomes a tool used to help them see themselves as other see them. To Escher, the mirror image is no ordinary matter. Escher is fascinated by the mixture of reality (the mirror itself and everything surrounding it) with the other reality (the reflection in the mirror)” (Bool 136).  In 1920 he made a large pen and ink drawing of the inside of a church. In the center of the church is a large reflective object, a chandelier. Most of the interior of the church is reflected in the sphere; even Escher himself and his easel with drawing paper can be seen in the reflection. So the church and the sphere are in the same place, they are both in both places and they “interpenetrate” (Bool 136)

 

Professor and writer, Douglas R. Hofstadter discusses Escher’s ability to demonstrate “self reference” and “self presentation” in his Pulitzer Prize winning book, Gödel, Escher, Bach: an Eternal Golden Braid.  Hofstadter, now a professor at the University of Indiana, debates the question of consciousness and the possibility of artificial intelligence. He attempts to discover what "self" really means, by using art and music to illustrate fine points in mathematics. The artwork of Escher is used as an example mathematics and art blending at multi levels.

 

In his book, the following humorous phone conversation takes place. Achilles is talking to his friend the Tortoise. They are discussing the mathematical notion of figure and ground: how, by defining one subset of a given set, you implicitly define another subset of that same set -- the part that is not included in the first subset. In the visual arts, Escher’s Mosaic lithographs best exemplify this, where the shapes that form the background for a group of black "phantasmagorical beasts" define another set of figures, in white. The musical example that Hofstadter uses is Bach's Sonatas for Unaccompanied Violin, where the listeners' imagination fill in "between the notes" as the violin plays, and one often imagines hearing the accompanying piano. But the form of the dialogue pulls the very same trick, as the reader can easily imagine the Tortoise answering Achilles at the other end of the line! (Cohen)

 

Escher’s untrained mathematical way to express himself in his art is fascinating and inspiring. The following pages are the underlying grid plans that I used and my try at making a tessellating pattern using Escher’s simple advice and explanations of his work.  I could easily use a computer to make my tessellation, but Escher didn’t have one, so I’m going to do it the way he did it. Tessellating symmetrical patterns are a perfect example of the marriage of art and math.

 

“He who wonders discovers that this in itself is wonder.”  M.C. Escher

 

 

Works Cited

 

American Heritage Dictionary of the English Language

 

Bool, F.H., M.C. Escher, His Life and Complete Graphic Work  Harry Abrahams Inc Publishers, New York  1981

 

Cohen’s Bookshelf    http://www.forum2.org/tal/books/geb.html

 

Drexel University Math Forum,   http://mathforum.com/sum95/suzanne/historytess.html  April 15, 2002

 

Field, Michael and Golubitsky, Martin  Symmetry in Chaos  Oxford University Press, Oxford 1992

 

Perplexing Pentagons  http://ccins.camosun.bc.ca/~jbritton/jbperplex.htm

 

Seymour, Dale Introduction to Tessellations  Dale Seymour Publications, Palo Alto, California 1989 

 

Think Quest, Totally Tessellated  http://library.thinkquest.org/16661/simple.of.non-regular.polygons/quadrilaterals.html     April 15, 2002