The Double-Face of Symmetry: A Conceptual History
Giora HonSymmetry is commonly perceived as a concept that expresses bilateral or radial relations, which effectively describes spatial arrangements that most people think is in some sense innate to the human mind. So, does the concept have a history? Has it evolved? Was there a revolution? The long history of the concept of symmetry began in classical Antiquity as a single concept with a range of applications, expressing proportionality with a specific constraint. In fact, symmetry was used in two different contexts: in mathematics it had the technical meaning of commensurable, while generally it meant suitable or well proportioned. The latter usage involves an aesthetic judgment arrived at by comparison with an ideal in the relevant domain, in an attempt to establish a certain property of the object, e.g., that it is beautiful or that it functions efficiently. We offer historical evidence that, despite the variety of usages in many different domains, there is a conceptual unity underlying the invocation of symmetry in the period from Antiquity to the 1790s which is distinct from the scientific usages of this term that first emerged in France at the end of the 18th century. We examine the trajectory of the concept in the mathematical and scientific disciplines as well as its trajectory in art and architecture. The changes in the meaning of symmetry from Antiquity to the eighteenth century can be explained by appealing to evolution—nobody in that period claimed to be doing anything new. The philosopher Immanuel Kant is probably the first thinker to indicate that something is fundamentally missing in the traditional account. In 1768 he introduced the concept of incongruent counterparts to indicate a reversal of ordering in entities that are equal and similar but cannot be superposed. However, the key figure in revolutionizing the concept of symmetry was the mathematician Adrien-Marie Legendre who, in 1794, claimed to be doing something new. Indeed, by introducing a principle of ordering he revolutionized the concept, and laid the groundwork for its modern usages.
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Figures (5)
![Fig. 3. Two spherical triangles (ABC and A’B’C’) that are equal and similar but cannot be made to coincide. ULLE VELOSO ULUEL Tit Malu al AU Lidia UUIUUto. With his concern for the possible directionality of space, KANT proposed a distinction, hinting at the principle of ordering without presenting it in mathematical terms: “The dis- tinctive characteristic in question consists in the particular direction in which the order of the parts is turned.”*’ And he made the following claim without proof and without a figure (see Fig. 3): “A spherical triangle can be exactly equal and similar [vdllig gleich und Ghn- lich] to another such triangle, and yet still not coincide [decken] with it.”](https://figures.academia-assets.com/55881155/figure_004.jpg)
