Lisbon’s Patterns and the 17 Ways to Symmetry

Opposites may attract in infinite ways, symmetry though can only be achieved in 17 ways. And this exact number was proved by Evgraf Fedorov in 1891.

Symmetry is the main attribute of any pattern. Of any kind – geometrical, social, behavioral, you name it. Symmetry is, among others, deeply connected to our memorizing patterns.

It is obvious that we cannot detect a pattern and thus perceive its symmetry, without storing it somewhere in memory. For example, the perception of bilateral symmetry requires that we store the two mirror shapes, compare them and then perceive their symmetry. By the same argument the perception of a single musical note, requires the storing of many samples of sound – the words of Zito Giuseppe, a physicist fascinated by randomness.

But there is another subtle connection between memory and pattern. Once we have found a pattern, we can free the memory from the burden of storing all the details, we remember now the pattern as a single object. This helps us in storing more complex things.

A Graphic Pattern of Lisbon

This design looks in one way, if seen frontal, and in another way, if seen in diagonal.

The geometric pattern is achieved through rotation, translation and mirroring effects, and covers the surface of around 1,800 square meters of a central square of Lisbon. This central square of Lisbon, Praça da Restauradores, is a black and white pavement of 3-limestone-width lines, crossing each other in straight or 45º angles. The central module of the design does not repeat in the same position. Also, the angle from where you look at the pattern might radically change the appearance of the whole.


The pattern belongs to Portuguese architect João Abel Manta (born in 1928), best known for his cartoons, widely published in the local press, and which stand for his political views and interventions, and also for various modern housing projects in Lisbon, as well as for his paintings, ceramics, tapestry, mosaics, illustration, and graphic arts. Manta has got as many aesthetic views as techniques: modernist in architecture, close to Impressionism in paintings and abstract in his urban works. The facet of the cartoon covers a long period of his work, from approximately 1954 to 1991, being particularly intense between 1969 and 1976. For about seven years his works were published regularly in newspapers like Diário de Lisboa, Diário de Notícias, and O Jornal, critically and deeply ironically dealing with Portuguese reality. His political intervention was intensive especially  in 1974 and 1975, straight after the fall of the dictatorship, times when he questioned the identity of a country in turmoil in drawings such as A Difficult Problem, where a group of outstanding figures from the past, from Karl Marx to Trotsky and Sartre, stare inquisitively at a small map of Portugal on a blackboard. João Abel Manta will somehow always be associated, in a very particular way, to the best and worst that Portugal went through during those years.

The 17 Ways

Evgraf Fedorov was a Russian mathematician, crystallographer and mineralogist and is considered as the one who demonstrated the assumption. The same maximum number for symmetries, 17, was, also independently derived by George Polya in 1924. The proof that the list of wallpaper groups was complete only came after the much harder case of space groups had been done.

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