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This paper was written as an assignment for Ian Walton's Math G - Mathfor liberal Arts Students - at Mission College. If you use material from thispaper, please acknowledge it.
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Tessellations
TheAmerican Heritage Dictionary defines the word tessellate as “to form intoa mosaic pattern by using small squares of stone or glass.”
Tessellationsare all around us in the form of geometric designs in floor tiles, ceilingtiles, quilts, carpets and many everyday objects. Some tessellations are very simple, like that of achild’s checkerboard game, while others are much more complex, like thoseby Dutch graphic design artist, M.C. Escher. One thing that all tessellations have in common, other thanthe idea that they have no gaps and no overlaps, is the fact that they can goon infinitely in every direction.
Manhas been surrounded by these designs for thousands of years.
Theshapes that make up a tessellation pattern (with straight lines) arepolygons. Not all polygons cantessellate, but first we will look at the different polygons that can.
Inorder for a triangle to tessellate, “the sum of the interior angles ofany 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 theinterior angles of any triangle equals 180 degrees. The triangle is cut out of a piece of paper, and you are totear off all three of the corners of it. Place all three of these corner piecestogether, with the vertexes pointing in towards each other.
Ifyou were to place three of any type of triangle in a row, with the one in themiddle, upside down, you would have a nice neat row of triangles.
Quadrilateralsare another type of polygon that can tessellate. Since the sum of the interiorangles of a quadrilateral is 360 degrees and there are four sides, angles andvertexes to a quadrilateral, each of the four angles equals ¼ of the 360degrees. If you put four of thequadrilaterals together, with one of each of the four vertexes all touching atone 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 waysthat the four vertexes can be arranged and the pattern will nottessellate. If the four shapes arerotated around in a circle, the pattern will create a tessellation.
Anotherway to think about how quadrilaterals can tessellate is by looking at the factthat any quadrilateral can be divided into two triangles.
Wouldwe be able to make tessellating patterns from any pentagon, since we know howfar we can go with a triangle and quadrilateral? As it turns out, if you place three regular pentagonstogether, with vertexes touching, there is a slight gap.
Thereare certain irregular pentagons that are able to tessellate without the help ofa parallelogram. Up until 1968,mathematicians thought that all the irregular pentagons that could tessellatehad been discovered. Several morewere found that year and then more in 1975, when Scientific American wrote anarticle about tessellating pentagons. More were discovered again in 1977 and in 1985 (PerplexingPentagons). This seems to indicatejust how fascinated man is with these repeating patterns.
Onevery familiar tessellation is that of the hexagon. The shape seems to naturallylend itself to tessellations. Mother Nature demonstrated this in beehives, and we liked the look somuch that we made chicken wire in the same pattern. A hexagon can be divided in triangles by drawing threelines from any one vertex across the shape to the three vertexes on the otherside. We already know that theinterior angles of a triangle equal 180 degrees, so now we can add the fourtriangle measurements together. The sum of the four triangles is 720 degrees. There are 6 angles on the inside of the hexagon, so if youdivide the 6 angles by 720 degrees, you find out that each of the interiorangles is 120 degrees. 120 degreesdivides into 360 degrees exactly 3 times, so we know that 3 hexagons fittogether perfectly and will tessellate (Seymour 49).
Asthe number of sides on a polygon goes up, so does the sum of the interiorangles. There are 3 regular tessellationsthat have only one shape in each of them. They are the triangle tessellation,the quadrilateral tessellation and the hexagonal tessellation. There are manypossible combinations of shapes that can fit together, whose interior anglesadd up to 360 degrees, but for a tessellation to have only one shape inside it,there are only those three kinds (Seymour 50).
Inorder for us to have a way to talk about tessellations and the combinations ofpolygons that they are made from, a numbering system is used.
Mathematicianshave determined that there are a total of 21 different arrangements of 17possible combinations of regular polygons that will make a tessellatingpattern. Since the smallestpossible angle in any regular polygon is 60 degrees, 1/3 of a triangle, wecan’t have more than six polygons meeting at any vertex.
Thisexpression can be simplified to—
Similarly,the arrangements for four polygons give us the equation—
Andfor five polygons,
Andfor six polygons,
Thesefour equations have 17 possible solutions, which make up the 21 possiblearrangements (Seymour 245).
Inaddition to all the possible polygon combinations that can be used to make atessellation, the shapes can be moved or transformed in different symmetricalways to create interest in the pattern. One way to move the shapes around is to slide them up and down or leftand 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 thatthe shape had been “rotated” or turned. It is turned at a certain point called the “center ofrotation.” The shapes canalso be reflected or flipped so that they mirror each other.
Everythingthat has been talked about up to this point is the basic information needed tocreate tessellations. Since threedifferent polygons (triangle, quadrilateral, and a hexagon) will tessellate, agrid of any one of those three shapes can be used to make a tessellation.
Dutchgraphic artist M.C. Escher tells how he starts his tessellations, or“dividing a plane” as he puts it in his book, The RegularDivision of the Plane, that he wrote in 1957. His interest in repeating patterns started when he traveledto northern Italy and Alhambra, Spain as a young man in 1922 (Bool 52).
Eschertells that he begins “dividing a plane regularly” by having two sets of parallellines. The distance at which thelines intersect and the angles at which they intersect will reveal somethingabout the creatures that will emerge from the page later.
Heenvisions one creature being black and the other is white, allowing the mostpossible contrast between the two. The black and white silhouettes begin to assume a “particular, andnot arbitrary shape.” Escher then attempts to create a form that the observer can recognize asfamiliar (Bool 158). By adding a few details at a time, the metamorphosisbegins. What started out as a fishat the top of the page becomes a “shape” and from that shape a birdemerges. The transformation has taken place. Escher says, “obviously this can be done theother way around as well” (Bool 158). It seems so easy for this famous artist to explain how hemade these incredible works of art. His explanation is so simple and inadequate; that he gives theimpression that doesn’t completely understand how he makes the creaturescome to life. As if he thought his simple explanation would help a MissionCollege 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 anyformal mathematics training, says, “By keenly confronting the enigmasthat surround us, and by considering and analyzing the observations that Imade, I ended up in the domain of mathematics. Although I am absolutely withouttraining or knowledge in the exact sciences, I often seem to have more incommon with mathematicians than with my fellow-artists” (Bool 55). Itwould have been interesting to see how Escher art would have been influenced ifhe had received a formal (mathematics) education.
In1956 Escher met and became friends with a mathematics teacher, BrunoErnst. Ernst was fascinated withEscher’s images and the mathematical quality of his work. By means of anextensive analysis of Escher’s prints and sketches, Ernst was able tocategorize all of Escher’s work according to mathematical themes.
Ernst’sfirst attempt at identifying themes was unsuccessful. At first he thought thatthe different areas would be mirrors, regular polyhedrons, spirals, and Mobiusstrips. This plan would notwork because it would not lead to any great insights. The math teacher thoughthe could only make progress if he included the “intentional” contentof the art, or the meaning of the art. The themes that Ernst decided toclassify Escher art into are: penetration of worlds, illusion of space, theregular division of the plane, perspective, regular solids and spirals, theimpossible and the infinite. Hehad to have many consultations with Escher during this time to find out whatmessage the art was trying to give to viewer (Bool 135).
Thefirst 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 surprisedwhen he notices that the world behind the mirror, which looks so real, isactually ‘intangible’; however, he soon ceases to consider thisfalse reality as being strange. Over time, the surprise disappears and themirror becomes a tool used to help them see themselves as other see them. ToEscher, the mirror image is no ordinary matter. Escher is fascinated by themixture of reality (the mirror itself and everything surrounding it) with theother reality (the reflection in the mirror)” (Bool 136).
Professorand writer, Douglas R. Hofstadter discusses Escher’s ability todemonstrate “self reference” and “self presentation” inhis Pulitzer Prize winning book, Gödel, Escher, Bach: an Eternal GoldenBraid. Hofstadter, now aprofessor at the University of Indiana, debates the question of consciousnessand the possibility of artificial intelligence. He attempts to discover what"self" really means, by using art and music to illustrate fine pointsin mathematics. The artwork of Escher is used as an example mathematics and artblending at multi levels.
Inhis book, the following humorous phone conversation takes place. Achilles istalking to his friend the Tortoise. They are discussing the mathematical notionof figure and ground: how, by defining one subset of a given set, youimplicitly define another subset of that same set -- the part that is not includedin the first subset. In the visual arts, Escher’s Mosaic lithographs bestexemplify 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 UnaccompaniedViolin, where the listeners' imagination fill in "between the notes"as the violin plays, and one often imagines hearing the accompanying piano. Butthe form of the dialogue pulls the very same trick, as the reader can easilyimagine the Tortoise answering Achilles at the other end of the line! (Cohen)
Escher’suntrained mathematical way to express himself in his art is fascinating andinspiring. The following pages are the underlying grid plans that I used and mytry at making a tessellating pattern using Escher’s simple advice andexplanations of his work. I couldeasily use a computer to make my tessellation, but Escher didn’t haveone, so I’m going to do it the way he did it. Tessellating symmetricalpatterns are a perfect example of the marriage of art and math.
Works Cited
American Heritage Dictionary of the English Language
Bool, F.H., M.C. Escher, HisLife and Complete Graphic Work Harry Abrahams Inc Publishers, New York 1981
Cohen’s Bookshelf
Drexel University MathForum,
Field, Michael andGolubitsky, Martin Symmetry inChaos Oxford University Press,Oxford 1992
Seymour, Dale Introductionto Tessellations Dale SeymourPublications, Palo Alto, California 1989
Think Quest, TotallyTessellated