Figure 3

Louis Vuitton’s Two Dimensional Crystal

Summary: Symmetrical motifs are used in a whimsical way by the world of fashion to convey subliminal messages.

The fashion world may not be familiar with the scientific notion of ‘symmetry’. Yet, the most dominant concept in most of their endeavor revolves around the ‘left-and-right’ repetition dominant in the human body. Each and every human being is aware of this and it is indeed the most dominant motif in the early art, sculpture and architecture. It might be related to the origin of the Greek work ‘sym-metron’: harmony of measurements or proportions. A common term to describe this symmetry is ‘mirror’ symmetry as it relates to the two mirror images, and also to the two-sided (bilateral) symmetry present in a myriad of biological forms.
From this mundane beginning the mathematical concept of symmetry evolved into the repetition of motifs around a circle or around a central point that remained fixed to generate the abstraction of ‘point symmetry’. Examples of this symmetry are prevalent in our daily experience as symmetrically decorated plates with a motif repeated around the circle (Fig.1).

Fig.1. Center ceramic plate: example of point symmetry around a central point. Lace (yellow square): plane symmetry, repetition in two dimensions. Lower right inset: schematic of 4mm symmetry on lace repeating ‘motif’.

The epitome of this symmetry in  Art is the magnificent repeated patterns in the rose windows of medieval cathedrals. They represent truly underappreciated examples of the interplay between mathematical symmetry and artistic expression, particularly when combined with the stained-glass content that was used to convey so much of the world vision of the medieval societies (Fig.2)

Fig.2. Point symmetry in Art. Examples of point symmetry in cathedral windows. Further details of this link between symmetry, artistic expression and the rose windows of cathedrals can be found in

There is another type of symmetry that the fashion world is intimately connected to, although they might not realize it. This is ‘plane symmetry’, this is the symmetry used by ‘print designers’ to develop motifs, designs and patterns that fill the surface of printed fabrics. The vast majority of printed fabrics have a motif that repeats in the two dimensions of the plane that constitute the fabric. This type of symmetry is characterized by the absence of a ‘fixed point’: the design (unique) motif is displaced in the two directions of the fabric to fill the space (Fig. 1-table cloth lace). In three dimensions, this is named ‘Space symmetry’ and it is the symmetry that generates the internal regularity of crystals, where the repeating motif is an atomic arrangement of atoms. Crystal regularity, i.e. crystal symmetry, can be found in two and three dimensions.
This brings me to the title of this note. I had partially noticed before that the dominant motif in the pattern used to construct an ‘LV’ bag was a four-folded figure with several variations all having the mathematical symbol: 4mm (Fig. 1, inset). This symbol translates into a figure with four-fold repetition and mirror planes (reflection symmetry) every 45 degrees, around the circle. This ‘point symmetry’ is very common in many decorations and even in the architectural design of many courtyards from the 15th century.
A few days ago, on my daily commute to Chicago, the passenger next to me pulled an LV bag to get the train ticket and it was then that I realized that the decoration in the LV bag was in fact a two-dimensional crystal with two 4-folded motives and the ‘LV’ logo in the middle (Fig. 3)

Fig. 3. The repetition (‘crystal’) motif in the design of LV bags and other ‘high fashion’ objects of the ‘LV’ brand.

I was pleasantly surprised for two reasons. First, I was elated to recognize a crystal motif in an ‘haute couture’ object. Second, I was in awe as to the ingenuity of the designers to use a highly symmetrical motif to convey luxury, elegance and distinction. In the end, it is only a repetitive symmetrical pattern.