Symmetry Step by Step

Originating author is Ana Cannas da Silva. Symmetry has always fascinated and served humankind in architecture, arts, engineering and science. Over thousands of years symmetric patterns have been used to create fabrics, baskets, floors, wallpapers and wrapping papers, and so on.
At the end of the 19th century, the Russian mathematician and mineralogist Yevgraf Fyodorov established that there are types of symmetry for patterns in the plan [WPG]. That is, we can have exactly different wallpapers in terms of replications of symmetry, and no more! Notably, all these types of symmetries can be found in decorative arts in antiquity.

The existence of these types was proved in a geometrical way, using knowledge of addition of fractions and a little of topology, in the 1980s. This explanation was discovered by Bill Thurston and widely publicised by John H. Conway. Conway created a language for the ideas of Thurston in terms of the four aspects of symmetries: kaleidoscope, gyration, miracle and wonder, that we briefly describe in section 1.

See the correction by Marston Conder at the end of the vignette.

Besides the language in terms of these aspects of symmetry, Conway established a notation with symbols with mnemonic correspondence (, figures, and ). Each type of symmetry is thus associated, by Conway and Thurston, with a symbolic signature (symbols that identify the type of symmetry), which is more informative and inviting than the old crystallographic notation.
The classification of plane symmetries à la Conway and Thurston is discussed in Section 2 with the use of elementary arithmetic. We close our paper in Section 3 with the study of the symmetries of some Portuguese cobblestone sidewalks, and the boardwalk in Rio de Janeiro (center of Figure 1) which is perhaps the most famous example internationally.

1. Aspects of Symmetry

• The aspect of symmetry called kaleidoscope indicates the presence of reflection symmetry and has symbol a star, For example, an ordinary chair has only a symmetry by reflection due to a bisector plane hence its symmetry signature is . When several mirrors intersect, such as in a kaleidoscope, we also indicate the number of mirrors that intersect at each point. For example, the sidewalk in a net mesh shown in Figure 2 such as a paper grid is preserved by several reflections. Let’s focus on a triangular sector bounded by three mirrors (an eighth of a square), which is called fundamental domain. If we imagined ourselves inside this fundamental triangle surrounded by those three mirrors, we could see the entire sidewalk reproduced to infinity. Aiming at characterizing these symmetries we have to determine the angles of this triangle, or equivalently determine the number of mirrors that intersect at each vertex. There are mirrors intersecting at the center of a square , mirrors intersecting at each vertex, and at the middle of each line segment, therefore the symmetry signature is .

• A gyration symmetry which is not a reflection symmetry is characterized by the (minimum positive) rotation angle which is a fraction of degrees. For example, the sidewalk in Figure 3 is preserved by rotation. Any of its rotation centre is equivalent to one of the three distinct ones shown in the figure, whose angles are , and . The associated signature is composed of the denominators of the fractions: .

• The aspect of the symmetry called miracle, represented by a cross, , occurs when the path form a pattern to its image does not cross any mirror.

• The aspect wonder, denoted by , occurs when a pattern does not present any of the previous aspects. For example the sidewalk shown in Figure 5 is conceived by repeating horizontally and vertically a pattern extracted from the Restauradores square sidewalk, presents zero () of the previous aspects (Oh!), and hence has signature .

2. Patterns in a Plane

A metaphor can help us to find out the 17 patterns in terms of the signatures. Let’s imagine that the restaurant Symmetry which offers the following menu

, , … have a % discount when offered to the right of a . For example, costs euros nevertheless the combo costs only euros.

Let’s imagine that in difficult times, the budget for lunch is only euros. What can we have for lunch using the entire budget? For example, the option costs euros, and the option also costs euros.

Magic theorem for planes: the types of patterns for planes are equivalent to combinations of symbols whose total cost is exactly equal to .

What are the types of patterns and where does the magical theorem come from?

Using the magical theorem, the answer to the first question is no more than an exercise. It should be remarked that in the absence of stars, each digit costs at least one half and it is always less than . Consequently, the first list of signatures without star is

In fact, assuming that , , represent numbers, the combo costs when costs . Hence the previous list is at the origin of the associated list of signatures which start by :

If you can add fractions, you can obtain the list with mixed signatures:

We thus find out that there are at most types of these patterns.
Concerning the second question, the reader should refer to the text [NOS] in order to find out the origins of the magical theorem. The book [CBG] contains a more complete analysis and fascinating illustrations. This book also explains in which spaces lie the patterns whose costs are less than , and the ones whose costs are greater than .

3. Mathematics of the cobblestone

Portuguese cobblestone pavement was conceived in the 19th century as an initiative to keep prisoners in the castle of Sao Jorge busy. Since then, it became iconic for the city of Lisbon and it is the pavement style which is most used for the sidewalks in the historical areas in Portugal.

Its most famous pattern ondas do mar largo (roughly “waves of the open sea”) was used in 1849 in King Pedro IV of Portugal square, commonly known as Rossio, and was exported to Copacabana in 1906 with great success. See Figure 1. Curiously, King Pedro IV of Portugal was the first emperor, Pedro I of Brazil.

In ondas do mar largo we can find equivalent mirrors and two types (black and white) of gyration, where the space between the waves is the thinnest. Therefore the signature is . See Figure 6.

Can we find all types of symmetries in the Portuguese sidewalks?

A study was launched in Lisbon investigating the types of symmetries present in Portuguese cobblestone pavements aimed at completing the list. The type was found in the city of Guimaraes. And the following patterns were not detected yet:

Whereas the type is illustrated in Figure 5, four types are still to be identified in the details of the patterns of Figure 7. More examples can be found in [PEF].

We encourage the reader to find such patterns on sidewalks in his neighbourhood. The author thanks the emails of contribution from around the world!

4. References

[CBG] Conway, J. H., H. Burgiel, C. Goodman-Strauss, The Symmetries of Things, A K Peters, 2008.

[NOS] Cannas da Silva, A., Um Novo Olhar Sobre Simetria, http://www.math.ethz.ch/~acannas/Outreach.

[PEF] Padroes em Falta, http://www.math.ethz.ch/~acannas/Outreach.

[SPP] Simetria Passo a Passo – Matematica nas Calcadas de Lisboa, http://www.math.ist.utl.pt/simetria.

[WPG] Wallpaper Group, http://en.wikipedia.org/wiki/Wallpaper group.

Department of Mathematics, ETH Zurich, 8092 Zurich Switzerland e Departamento de Matematica, Instituto Superior Tecnico, 1049-001 Lisboa, Portugal

Work partially supported by the Foundation for Science and Technology (FCT Portugal) and by the Gulbenkian Foundation (Portugal); Lisbon photos by Joao Ferrand; Figure 7 photos by Dror Bar-Natan.

Correction (by Marston Conder, University of Auckland): The first person to realise/discover this explanation was not Thurston or Conway, but Murray Macbeath (A.M. Macbeath), some 20 years earlier, in his work on crystallographic groups (in a wider context). His observations have been “re-discovered” by a number of people, including Thurston, but Macbeath deserves the credit for the explanation, which is partly algebraic. Conway publicised it, and developed a new notation for it.