Escher Style Tessellations

Transcription

1 Escher Style Tessellations The delightful designs by M. C. Escher have captured peoples imagination the world over. These are examples of what we will call Escher Style Tessellations, patterns which can be extended to the left, right, up and down to cover an entire wall. 33

2 We can imagine covering a bathroom floor with this type of design where many copies of a basic tile or tessellating piece are placed side by side to form a tiling. Tessellating Piece Sometimes the basic tile must be rotated or flipped over in order to fit together with existing pieces as in the following example. Notice in the design below that the basic tile is sometimes upright and sometimes upside down and sometimes facing right and sometimes facing left. In this chapter we will describe ways we can make our own tessellating pieces. A classification, called the Heesch Type, will be presented. The Heesch Type highlights both the basic tile and how it was made and also reveals the symmetries of the design. Finally, giving human or animal form to the abstract shapes is an opportunity for creativity and play. 34

3 Five Moves for Making Our Own Tessellating Pieces We will begin by cutting out a cardboard square which is the beginning of our tessellating piece. (See below for a discussion of more general shapes than a square which can be used as a starting shape). There are five moves (translation T, two kinds of glide reflection G and G and two kinds of rotation C and C 4 ) which can be done to the square so that the resulting shape is tessellating. 1. Move T (Translation): For the first move, translation, cut out part of the square along one side as pictured here. Then slide or translate the cutout part over to the opposite side and tape it back on the square. This is Move T. Can you see that if you had a supply of tiles like this then you could put them together side by side like a puzzle? This move also works by cutting out part of the top and translating it to the bottom (or vise versa). 2. Move G (Glide Reflection): For this move again cut some shape out of one side of the cardboard square. Then flip or reflect the cutout piece and slide it over and tape it to the opposite side of the square. This is Move G. Note: The flip used here must be over a line perpendicular to the side of the square (not parallel to the side) the wrong flip usually results in a cutout piece that can not be attached easily to the opposite side. 35

4 A supply of these shapes could also be put together in an interlocking fashion but note that the pieces must be flipped over in order to fit together. 3. Move C (Center point Rotation): This move is a little different since it involves only one side. Begin by marking the middle of one side of the cardboard square. Now cut out some shape from just one of the halves of the side. For this move, this cutout is rotated about the center point of the side and taped onto the other half of the side. Perhaps it is not so easy to see how a supply of these pieces can be fit together. It is possible. However, as you may suspect, the pieces must be rotated around in order to interlock with each other. Examples are given below. 4. Move C 4 (Corner Rotation): Again we again begin by cutting out something from one side of the square. For move C 4 this cutout is rotated around a corner of the square and taped onto a touching or adjacent side. 5. Move G (Glide Reflection, Adjacent sides): This final move, like the last, involves a side and a touching or adjacent side. After something is cut out of a side, a glide reflection carries the cutout to an adjacent 36

5 side (along a diagonal glide reflection line going through the midpoints of the two sides). Perhaps an more natural way to visualize this move is by taking the cutout piece and flipping it over (as in Move G) and then rotating the flipped piece about the corner (as in move C 4 ). This combined move makes Move G. This completes our presentation of the five moves. To make things interesting we note that two or more of the moves can be done on the same square piece of cardboard to get a variety of tessellating pieces. 37

6 Putting the Moves Together for a Tessellating Piece Heesch Types Different combinations of the five moves are possible. For example, Move T could be done on two opposite sides and Move G done on the other two sides. Another possibility is Move T on two sides and Move C on each of the remaining two sides. In fact there are 9 possible ways to make a tessellating piece using these five moves. These possibilities are diagrammed here. The Nine Heesch Types A simple classification code for Escher style tessellating tiles has been developed by the German mathematician Heinrich Heesch. According to Heesch s scheme, a letter is assigned to each side of the shape by noting how the side is related to other sides (or to 38

7 itself). The code letter T is assigned to sides related to their opposites by translation. The letter G (or G ) means related to the opposite (or adjacent) side by glide reflection. Finally, C means midpoint rotation and C 4 means corner point rotation. The four letters taken in order from the four sides form the code name for the particular Heesch Type. Heesch types are given under each of the examples in the above diagram. Also note that the starting point for the four letter code is unimportant so that Type TCTC could also be called Type CTCT. In abbreviated form, here again are the nine Heesch Types. TTTT TGTG (or GTGT) TCTC (or CTCT) GGGG GCGC (or CGCG) CCCC C 4 C 4 C 4 C 4 G G G G G G CC (or CCG G ) Note: Since each move involves two opposite sides, two adjacent sides or just one side, the possible combinations can be worked out. However, there are two logical possibilities which do not, in fact, form tessellating shapes (namely, C 4 C 4 CC and C 4 C 4 G G ) since a supply of either of these type of pieces cannot actually be put together. Analyzing Tessellations We illustrate on the following example how to analyze a tessellation to figure out its Heesch type. 1. Begin by identifying the corners of the beginning or parent square (or, more generally, the beginning quadrilateral). The corners will be the points where four copies of the tessellating figure come together. For example, note in the figure where the bird s forehead and feet come together. These four corners are circled in the figure below. 39

8 2. Using tracing paper, now trace around the shape. Using the traced shape, it is often possible to recognize how a side is related to other side (by T, G, C or C 4 ). Further insight into how the tessellating piece was made is provided by noticing how each of the figures is related to adjoining figures. Looking at side by side copies of the basic figure, are they translated or are the flipped or rotated? Looking at above and below copies of the basic figure can you see how they are related (translation, reflection or rotation)? Recognizability of the Shapes You can see in them battles and human figures, strange facial features and items of clothing, and an infinite number of other things whose forms you can straighten out and improve. Leonardo da Vinci Leonardo da Vinci was describing what an artist sees when looking at cloud formations, but the same opportunity for creativity is presented with a tessellating shape. Escher used the term recognizability for his fascination with the creative possibilities for giving human or animal form to the abstract shapes of tessellating pieces. 40

9 Abstract Recognizable Cari Hollrah, a seventh grade student, saw many different faces in the same tessellating piece for this Oklahoma State Grad Prize winning tessellation. Stretching the imagination to find the many creatures who inhabit an abstract tessellating piece is great fun! What Creatures Do You See Here? Raunchy, cute, gross, cuddly! All describe the creatures a good imagination can find (see final page of this chapter for examples). 41

10 Parent Quadrilaterals (in addition to Squares) We have described five moves which can be done to a square and which give tessellating shapes. We say that the parent quadrilateral for these tessellating pieces is a square since we begin with a square piece of cardboard. Actually, other parents besides squares are possible. For example, we could start with a cardboard parallelogram and make a Type TTTT tessellating piece in just the same way as with a square (cut out a part of a side and translate over). In this section a summary of the possibilities for parent quadrilaterals for the different Heesch types is given. For each of the five moves, there is a natural condition which the parent quadrilateral must satisfy. Move Move T Move G Move G Move C Move C 4 Condition on Quadrilateral Opposite sides parallel, same length Opposite sides parallel, same length Adjacent sides same length No condition (works on any side) 90 o angle between adjacent sides of same length Putting these conditions together give the most general parent quadrilateral for each Heesch type. For example, for type TGTG, move T requires that two opposite sides are parallel and move G requires that the other two opposite sides are parallel. Thus we can begin with a cardboard parallelogram (parent) and make a Type TGTG tessellation piece. Notice that if a parent parallelogram works, then using a rectangle or square or rhombus would also work since these shape are special cases of parallelograms. The next table shows the most general parent quadrilateral for each Heesch type. Type TTTT TCTC TGTG GCGC GGGG CCCC C 4 C 4 C 4 C 4 G G G G G G CC Most General Parent Quadrilateral Parallelogram Parallelogram Parallelogram Parallelogram Rectangle Any quadrilateral Square Kite One pair of adjacent sides equal It is remarkable that any quadrilateral can be the parent of a tessellating shape. As the table indicates, if we start with any four sided shape and cut and tape a midpoint rotation on each side (Move C), we get a tile piece that can cover the bathroom floor. Here is an 42

11 example of such a floor (along with copies of the parent quadrilateral and tessellating piece). Parent Quadrilateral Tessellating Piece Symmetry of Escher Style Tessellations Escher style tessellations often have overall symmetry. All such tessellations have translational symmetry. (For all of these symmetries we consider that the pattern is extended in all directions.) Notice that the first example below has rotational symmetry of order 4 and the second one rotational symmetry of order 2 (the centers of rotation are marked with circles). A piece of tracing paper can help to see these symmetries. Type C 4 C 4 C 4 C 4 Type TCTC 43

12 This final example has glide reflectional symmetry (the glide line is shown). Type TGTG Do you notice that there is a simple and direct connection between the moves used in the Heesch type and the types of symmetry present? 44

13 Examples: Creative Interpretations of a Tessellating Piece Incomplete: Examples needed. 45