Tessellations Glide Reflections

8/18/2019 Tessellations Glide Reflections

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Escher Style Tessellations

The delightful designs by M. C. Escher have captured peoples’ imagination the worldover.

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.


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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.

Sometimes the basic tile must be rotated or flipped over in order to fit together withexisting pieces as in the following example. Notice in the design below that the basic tileis sometimes upright and sometimes upside down and sometimes facing right andsometimes facing left.

In this chapter we will describe ways we can make our own tessellating pieces. Aclassification, called the Heesch Type, will be presented. The Heesch Type highlights boththe 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.

TessellatingPiece


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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 beused 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 squareso 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 picturedhere. 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 themtogether 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. Thenflip 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 cannot be attached easily to the opposite side.


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A supply of these shapes could also be put together in an interlocking fashion butnote 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 themiddle of one side of the cardboard square. Now cut out some shape from justone 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. Itis possible. However, as you may suspect, the pieces must be rotated around inorder 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 atouching 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


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side (along a diagonal glide reflection line going through the midpoints of the twosides).

Perhaps an more natural way to visualize this move is by taking the cutout pieceand 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 notethat two or more of the moves can be done on the same square piece of cardboard to get avariety of tessellating pieces.


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Putting the Moves Together for a Tessellating Piece – Heesch Types

Different combinations of the five moves are possible. For example, Move T could bedone on two opposite sides and Move G done on the other two sides. Another possibilityis 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 arediagrammed here.

A simple classification code for Escher style tessellating tiles has been developed by theGerman mathematician Heinrich Heesch. According to Heesch’s scheme, a letter isassigned to each side of the shape by noting how the side is related to other sides (or to

The Nine Heesch Types


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itself). The code letter T is assigned to sides related to their opposites by translation. Theletter 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 letterstaken 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 becalled Type CTCT. In abbreviated form, here again are the nine Heesch Types.

TTTT TGTG (or GTGT) TCTC (or CTCT)GGGG GCGC (or CGCG) CCCCC4 C4 C 4 C4 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 4CC and C 4C4G’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 itsHeesch 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 thetessellating figure come together. For example, note in the figure where the bird’sforehead and feet come together. These four corners are circled in the figure below.


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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 eachof the figures is related to adjoining figures. Looking at side by side copies of the basicfigure, are they translated or are the flipped or rotated? Looking at above and belowcopies 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 youcan 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.


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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 (seefinal page of this chapter for examples).


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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 havetranslational symmetry. (For all of these symmetries we consider that the pattern isextended 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 aremarked with circles). A piece of tracing paper can help to see these symmetries.

Type C 4C 4C 4C 4 Type TCTC


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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 theHeesch type and the types of symmetry present?


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Examples: Creative Interpretations of a Tessellating Piece

Incomplete: Examples needed.

Documents

Tessellations Glide Reflections