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. 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. If we imagined ourselves inside this fundamental triangle surrounded by those three mirrors, we could see the entire sidewalk reproduced to infinity. Let’s focus on a triangular sector bounded by three mirrors (an eighth of a square), which is called fundamental domain. For example, the sidewalk in a net mesh shown in Figure 2 such as a paper grid is preserved by several reflections. When several mirrors intersect, such as in a kaleidoscope, we also indicate the number of mirrors that intersect at each point.

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.

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.įigure 1: Sidewalk in Rossio (Lisbon), Copacabana (Rio de Janeiro) and Belém (Lisbon). The classification of plane symmetries à la Conway and Thurston is discussed in Section 2 with the use of elementary arithmetic. 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. See the correction by Marston Conder at the end of the vignette.īesides the language in terms of these aspects of symmetry, Conway established a notation with symbols with mnemonic correspondence (, figures, and ). 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.

This explanation was discovered by Bill Thurston and widely publicised by John H. 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. 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. Over thousands of years symmetric patterns have been used to create fabrics, baskets, floors, wallpapers and wrapping papers, and so on.Īt 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.

Symmetry has always fascinated and served humankind in architecture, arts, engineering and science. Originating author is Ana Cannas da Silva.

Star reflection symmetry