Tessellations A FUN WAY TO COMBINE MATH AND ART WHAT IS A TESSELLATION? .......................................................................................................2 WHAT IS A TILING? ........................................................................................................................2 MAKING A CLASS QUILT OUT OF TESSELLATED PIECES OF PAPER........................3 MAKING A CLASS QUILT CONTINUED FROM PAGE 3 ............................................................................6 MAKING A TESSELLATION OF THE ESHER TYPE .............................................................2 WHO IS M.C. ESHER? ..........................................................................................................................2 AN EXAMPLE OF HOW ESHER-TYPE TESSELLATION IS MADE........................................................3 MAKING YOUR OWN ESHER-TYPE TESSELLATION ..........................................................................5 Works Cited............................................................................................................................................9 1 What is a tessellation? Making a tessellation of the Esher type • A tessellation is made when shapes are repeated Who is M.C. Esher? continuously on a plane, with The beauty of tessellations can be seen in the no gaps or overlaps. The pieces fit together like a jigsaw artwork of M.C. Escher (1898-1972), who is also puzzle. • Examples: known as Maurits Cornelius Escher. M.C. Escher was born in Leeuwarden, Netherlands to a hydraulic engineer, G.A. Escher (Ernst 7). His father wanted him to become an architect, however What is a tiling? after traveling to many places, he created many • Another word for a tessellation is a tiling. Find out more here: fascinating landscapes, portraits, and geometric What is a Tiling? designs. The work M.C. Escher is most famous for What is the difference between regular tessellations and semi- are his tessellations: “The late Dutch artist M.C. regular tessellations? Escher was famous for, among other things, artistic • If you use one regular polygon (sides are equal in length and and perplexing tessellations whose nonpolygonal interior angles are congruent), such as a square, to cover a fundamental regions seemed to be beyond the grasp plane without any gaps or overlapping, then it is called a of artists, geometers, and laymen alike” (Teeters pure or regular tessellation, and when more than one polygon is 307). With his detailed and perfectly made used to tessellate a plane, than it is called a semi-regular tessellations, it is really hard to say whether M.C. tessellation. In more detail, “A semi-regular tessellation is a Escher should be considered an artist or a tessellation of regular polygons of more than one kind meeting mathematician. In fact, I would consider him to be side to side and vertex to vertex in such a way that the both an artist and a mathematician. From the words same polygons, in the same cyclic (circular) order of Escher himself, “Although I am absolutely surrounded every vertex” (Britton 110). Not every without training or knowledge in the exact sciences, polygon can form a regular or semi-regular tessellation. I often seem to have more in common with 2 • It has been proven that there mathematicians than with my fellow artists” (Potter are only three regular polygons that tessellate a plane with only and Ribando 28). M.C. Escher’s motivation for one shape, and they are squares, equilateral triangles, creating his unique tessellations came after visiting and hexagons. The reason for this is that, in order for a the Alhambra in Spain in the early 1920’s: regular polygon to be used in tessellating a plane, the “Although inspired by the Moorish mosaics, Escher measure of its interior angle in degrees must divide 360° (a preferred recognizable, animate figures to purely complete revolution) exactly (Britton 77). As discussed geometrical shapes. With extraordinary earlier, regular polygons have congruent interior angles, so inventiveness, he created tessellating shapes that for a square every angle measures 90°, and if you form resemble birds, fish, lizards, dogs, humans, 4 squares around a point, the sum of the four angles will butterflies, and the occasional creature of his own measure 360°, therefore proving it will tessellate a invention” (Britton 122). After learning a little plane without leaving any gaps. The same is true for about M.C. Escher, we can now take a closer look equilateral triangles, if six equilateral triangles meet at a at how exactly a tessellation in the style of Escher point, the end result will form 360° (60° x 6 = 360°). For the is made. same reasons we can form a regular tessellation with hexagons, because their An Example of how Esher-type interior angles measure 120°, tessellation is made and if three of them meet at a A significant example to show how some of point they will again form 360° (120° x 3 = 360°). M.C. Escher’s drawings are made from regular Making a class quilt out of polygons, we can look at is his symmetry drawing tessellated pieces of paper No. 104 of tadpoles. The figures in Escher’s • After the students have explored and played with the drawings are derived from polygons that tessellate geometric shapes, they should select which geometric shapes a plane (Haak 648). The tadpoles in Escher’s they will use to create a design that tessellates and covers the drawing` drawing are formed from squares. entire sheet of paper; …continued on page 6 ...continued on page 4 3 I produced a copy of this photo and (footnote1) have highlighted and placed letters to show how the squares are formed. Point “A” connects a point where the heads of two white lizards meet. Point “B” is where four front legs meet (two black and two white). Point “C” is where the heads of two black lizards meet, and point “D” is where four hind legs meet (two black and two white). The pattern of this square “ABCD” repeats with rotation and translation symmetries. If you notice in the same picture I have duplicated, I have rotated that same square “ABCD” 180°along point “C,” to form the square “CDAB.” As for the translation, you can see that original square “ABCD” slides in vertical and horizontal directions. “After an investigation of the symmetries of Escher’s artwork… the next logical step is to explore the methods of producing original pictures of that type” (Haak 649). The method of producing an original picture of the Escher type is actually easier than it sounds, especially when we understand the transformations in Euclidean Geometry. 1 Photocopy of Escher’s drawing No. 104 of tadpoles was taken from the article Transformation Geometry and the Artwork of M.C. Escher by Sheila Haak 4 Making your own Esher-type tessellation Anyone can gain an appreciation for geometry by producing an artwork of their own. I will show an easy way to produce a unique and interesting tessellation of the Escher-type. Being that there are four different types symmetries in tessellations, I will explain how to make the most simple of them all, a translational tessellation. We will start with the square, because like Escher’s drawing we must have an underlying grid of a polygon. I will assume that from this point we know how to make a tiling with squares, and that would be the first step. We then need to make a template of the shape we will make out of a square (the square should be the same size as our tiling so there will be no gaps or overlapping) that will be used to tessellate the tiling we made. The next step is to draw and then cut out pieces of the square that will be taped to the opposite side of the square to form our figure (what is drawn and cut on one side must be taped on the exact opposite side and only pieces from the square must be used in order for the template to be used in a tessellation). (footnote2) Once we have our template completed, we can use it to trace onto our tiling. When we are tracing our template, it must line up with the squares on your tiling so that no gaps or spaces are left. We can notice that while tracing the template the figures fit together like pieces of a jigsaw puzzle. After our design covers the entire paper, we can make the final touches, such as coloring every other bird and adding the details to the bird. If we remember what a translation tessellation is, “…you will see that the finished design is repeated in both a horizontal 2 picture of bird template was taken from Symmetry and Tessellations by Jill Britton 5 and vertical pattern” (Stephens 20). (footnote2) Our finished product may look like this tessellation of birds. Making a class quilt continued from page 3 …it is normal for some shapes to extend beyond the paper. Each student will be given an 8 1/2 –by-11-inch sheet of white paper to create his or her own symmetrical tessellating design using a choice of materials, either pattern blocks, attribute blocks, or tangrams; they all work well for making tessellations (Moyer 142). These manipulatives (i.e. pattern blocks, attribute blocks, or tangrams), consist of squares, triangles, hexagons, rhombuses, etc., and they work well in creating tessellations because they can create a pattern by either sliding, reflection, or rotation. These shapes of squares, triangles, and hexagons as discussed earlier fit together and can create a 360° degree angle; and if using the right combination will make a nice tessellation leaving no gaps or overlapping. It is important to let the students use their creativity to make their own choices