Symmetries and Tessellations

Symmetries and TessellationsAuthor(s): June OliverSource: Mathematics in School, Vol. 8, No. 1 (Jan., 1979), pp. 2-5Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211707 .

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by June Oliver, The American School of Aberdeen

Symmetries Before considering the more complex subject of tessellation, let us first look at the four basic symmetries on which this artwork is based. A symmetry is a rigid, length-preserving transformation which maps a geometric figure onto itself. Figures 1-4 each exhibit one of the four types of symmetry; these are reflection through a line (the line through which the figure is reflected is called the mirror line), rotation about a point (Fig. 2 shows a 900 or four-fold rotation, while Fig. 6 shows a 450 or eight-fold rotation), translation along a line, and glide reflection which is the composition of a translation along a line and a reflection through that line (note that if a glide reflection is present a simple franslation is present, but the converse is not true).

in each of Figures 5-7 you should be able to find more than one type of symmetry; a sine or cosine curve is an excellent illustration of all types of symmetry. It is not difficult to find other examples; symmetry is present in nature, art, architecture, mathematics and science. Some commonly found

objects which are symmetrical are flowers, snowflakes, the blades of a fan or windmill, butterflies, ink-blots, faces, and the facades of many famous buildings such as Buckingham Palace or Notre-Dame. Paper cutting of snowflakes, Christmas trees and strings of dolls are good exercises for learning about symmetry.

Tessellations M. C. Escher by no means originated the idea of the tessellation which is presently being studied by artists, crystallographers and mathematicians (the geometric connections are obvious but the underlying structure is described by the group theorists).

The Moors, conquerers of Spain in the eighth century, were masters of the art of tessellation. The walls of the thirteenth and fourteenth century Moorish palace, the Alhambra, in Granada are covered with beautiful multicoloured tiles which were used to create intricately tessellating mosaics. Figures 8 and 9 were copied from my own sketches done in the Alhambra. I chose two of the patterns which Escher had himself sketched over 40 years ago.

Fig. 1 Reflection Fig. 2 Rotation Fig. 3 Translation

Fig. 4 Glide reflection

Fig. 5 Fig. 6

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tessell

Symmetry has for a long time been a fascinating topic for mathematical discussion, and in recent years the popularity of Dutch artist, M. C. Escher (1898-1972), has brought new ideas to experimenting in the area of symmetry. Through the four basic symmetries, reflection, rotation, translation and glide reflection, Escher cleverly filled the plane with interlocking congruent figures to create beautiful repeating patterns. These plan-filling, repetitive patterns are called tessellations.

The pattern in Figure 8 illustrates translations, and 1200 (three-fold) rotations; you should be able to find three different centres of three-fold rotation. The pattern in Figure 9 illustrates translations, reflections, and 1800 (two-fold) rotations; you should be able to find four different centres of two-fold rotation and two "families" of mirror lines which will form a grid of squares. If all of the Ts were of the same colour, two four-fold rotation centres would replace two of the two-fold rotation centres.

Despite the elegance of the Moorish and Arabic mosaic designs, Escher felt that it was unfortunate that the Islamic religion forbade the making of images and therefore limited their art to abstract geometric patterns. Escher preferred to concentrate on what he felt to be more significant figures or recognisable figures borrowed from nature such as fishes, birds, reptiles and human beings.

Creating Escher-like tessellations To create a tessellation it is often useful to begin with a basic geometric figure which itself has rotational and/or

reflectional symmetry. Commonly used figures are parallelograms, special parallelograms, equilateral triangles, and regular hexagons. By repeatedly copying the chosen figure so that sides of adjacent figures coincide, the plane is filled by a lattice which exhibits symmetry. A lattice of parallelograms (Fig. 1 O) produces translations and 1 800 (two-fold) rotations centred at each of the vertices, at the midpoints of each side, and at the centre of each parallelogram. A lattice of regular hexagons (Fig. 1 1) produces translations in six different directions, reflections (six families of mirror lines can be found), six-fold rotations at the centre of each hexagon, two-fold rotations at the midpoint of each side, and three-fold rotations centred at each vertex.

Draw your own lattices using rectangles, squares, or equilateral triangles and find all the existing symmetries. Now try to fill the plane with regular pentagons or regular octagons leaving no gaps and allowing no overlaps. Is it possible?

The patterns of Figures 10 and 1 1 are not very interesting, but with a few modifications of the

Fig. 8 Fig. 9

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boundaries and a little artistic licence an Escher-like tessellation may be created.

Consider a lattice of the simplest geometric figure: the square. This lattice exhibits translational, reflectional, and rotational (two-fold and four-fold) symmetries. To simplify the task we will concentrate only upon the translational symmetry (the other symmetries will be destroyed as the boundaries are modified).

Start with a square (Fig. 12) with sides labelled A, B, C and D. Translate the square up, down, left and right, keeping track of each side. Similarly translate each new square always keeping track of the sides. As this process is continued (ad infinitum) the lattice is completed. Notice that sides A and C and sides B and D always fall adjacent. This means that any modification to side A will effect side C and any modification to side B will effect side D.

In Figure 13a, sides A and C are modified in exactly the same way, and in Figure 13b sides B and D are modified in the same way. Now using tracing paper (or by cutting a piece of card to the desired shape) this new modified version of the "square" may be copied onto each square in the completed lattice of Figure 12.

The final touches are added to produce the tessellating starfish (Fig. 14).

Starting with the same simple square, let us now concentrate on the four-fold rotational symmetry present in a lattice of squares. As the square is rotated about one of its vertices keep track of where the vertices fall as well as where the sides fall.

Rotate by 900 the square of Figure 1 5 about the vertex labelled 1 keeping track of sides and angles. The new block of four squares may then be translated as a unit to the right, to the left, up, and down to produce the completed lattice in Figure 1 5. Notice that now sides A and B and sides C and D always fall adjacent, so modifications to sides A and C will effect sides B and D respectively. Also notice that because of the way the pattern is generated, four-fold rotations are present only at vertices 1 and 3, while two-fold rotations are present at vertices 2 and 4.

To demonstrate how the sides may be modified one of M. C. Escher's designs is used. In Figure 16a, side A is modified; using tracing paper the new boundary is pivoted 900 about vertex 1 and copied onto side B. The same is done in Figure 16b in order to modified sides C and D.

Fig. 10 Fig. 11

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Again using tracing paper (or cut card) copy the new modified square onto the squares in the completed lattice of Figure 1 5. The motion of the tracing paper in order to fit the figures together is always a rotational motion; the paper is never lifted off of the lattice to be turned over. Adding the final touches results in one of Escher's lizard designs. See if you can find all centres of rotation (there are four).

Additional tessellations made in similar fashion are shown in the following figures (Figs 18-20). The sides of the basic parallelogram for the fish are modified in the same way as for the starfish, except that the modification to side A (and therefore C) is rotationally symmetric about the midpoint; this allows two-fold rotation. The spiderweb pattern is the simplest of all as no boundary modifications are made. The placement of the spider destroys most of the symmetry which was present in the hexagonal lattice of Figure 11. What symmetries are left?

Creating Escher-like tessellations involves patience, experimentation, practice and inspiration. The abstract patterns of Arabic design (Figs 9 and 10) require less artistic talent and inspiration yet are just as enjoyable. Whether you would like to create your own drawings or just look at the creations of others, it is worthwhile to investigate the works of Escher (his works also

include designs showing optical illusions and geometric impossibilities) and the designs of Arabic mosaics. A list of reference books follows:

References

Escher, M. C. (1972) The Graphic Works of M. C. Escher, Pan Books, London.

Haak, Sheila, "Transformation Geometry and the Artwork of M. C. Escher", Mathematics Teacher, Vol. 69 (Dec. 1976), 647-52.

Locher, J. L. (Ed) ( 1971 ) The World of M. C. Escher, Harry N. Abrams, Inc., New York.

MacGillavry, Caroline H. (1976) Fantasy and Symmetry, Harry N. Abrams, Inc., New York.

Further reading

Bourgoin, J. (1973) Arabic Geometrical Pattern and Design, Dover Publication, New York.

Bourgoin, J. (1977) Islamic Patterns, Dover Publication, New York. Ernst, Bruno, (1976) The Magic Mirror of M. C. Escher, Ballantine

Books, New York. Holiday, Ensor, (1977) Altair Design Book, Longman Group Limited,

London.

Maletsky, Evan M., "Designs with Tessellations", Mathematics Teacher, Vol. 69 (April 1974), 335-8.

Ranucci, Ernest R. "Master of Tessellations: M. C. Escher, 1898- 1972", Mathematics Teacher, Vol. 69 (April 1974), 299-306.

Teeters, Joseph L., "How to Draw Tessellations of the Escher Type", Mathematics Teacher, Vol. 69 (April 1974), 307-10.

Ranucci, E. R. and Teeters, J. L. (1977) Creating Escher-type Drawings, Creative Publications, Palo Alto, Ca.

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Figures 8, 9, 14, 19 and 20 are taken from the author's original sketches.

Figure 17 is reproduced with permission from Escher Foundation - Haags Gemeentemuseum - The Hague.

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Symmetries and Tessellations