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Introduction to Inorganic Chemistry
Published in Caroline Desgranges, Jerome Delhommelle, A Mole of Chemistry, 2020
Caroline Desgranges, Jerome Delhommelle
Yet the power of group theory is far more impressive! Indeed, the question of solving quintic equations (polynomial equations of degree 5) still remains a very challenging problem at the end of the 18th century. Lagrange in “Sur la résolution algébrique des équations” (1771) understands that new methods need to be developed to solve these equations. Furthermore, Abel (1802–1829) confirms: “Au lieu de demander une relation dont on ne sait pas si elle existe ou non, il faut demander si une telle relation est en effet possible […] de telle ou telle manière”. Lagrange understands that symmetry and permutation play a major role in finding the solutions of polynomial equations. A young French mathematician Galois (1811–1832) (see Figure 6.3) finally finds the solution in 1831 but dies one year later during a duel. His idea is to define a new method based on “groups”: “On peut se donner arbitrairement une première permutation, pourvu que les autres permutations s’en déduisent toujours par les mêmes substitutions de lettres. Le nouveau groupe ainsi forme jouira évidement des mêmes propriétés que le premier.” (One may take an arbitrary first permutation, if the others are deduced from the same substitutions of letters. The newly obtained group will have the same properties as the first one.) For instance, if we take the sequence “A B C D” then the permutations give “B A D C”, “C D A B” and “D C B A” giving us a group of four elements. Regarding the quintic problem, Galois theorizes that the group keeping all the qth roots p1/q, in which q is the prime of any polynomial p, invariant has at most 1/(q(q – 1)) times as many elements as the group leaving p invariant. Other mathematicians develop group theory, including Cauchy (1789–1857), Cayley (1821–1895) and Lie (1842–1899). In particular, Cauchy introduces the theory of permutation groups, as well as the theory of matrices. Klein (1849–1925) in his 1872 Erlangen program classifies geometries by their symmetry groups, linking geometry and group theory. But the most impressive work comes from Frobenius (1849–1917) and his idea of representation theory. Thanks to this new formalism, it is possible to simplify complex problems in terms of group representations and their associated characters. More specifically, the elements of a group can be described as matrices and the characters as the traces of matrices. During the 20th century, Weyl (1885–1955) and Wigner (1902–1995) make crucial contributions by linking group theory and quantum mechanics! According to Wigner, symmetry, and especially group theory, is the key: “I have come to agree […] that the recognition that almost all rules of spectroscopy follow from the symmetry of the problem is the most remarkable result”. In 1963, Wigner receives the Nobel Prize in physics “for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles”. Bethe (1906–2005), another Nobel Prize awardee, shows that group theory can be applied to problems concerning the nature of crystals.
Topological perception on attention to product shape
Published in International Journal of Design Creativity and Innovation, 2020
The shape of a product can be simplified to its basic geometry from a global viewpoint. In the product iteration process, shapes would be altered for competing with other brands, while the shapes of some products remain invariant, especially the identity of brand properties. There is a way to clarify the relationship between the invariants and their levels of structural stability. According to German mathematician Klein’s Erlangen program, any geometry is dealing with invariant properties under a certain transitive group action. Using this principle, he built a hierarchy of geometries, stratified in ascending order of stability: Euclidean geometry, affine geometry, projective geometry, and topology, which provides the highest stability (Kisil, 2012). Various studies on perceptual organization have demonstrated that the psychological reality (i.e. psychological representation of knowledge) of Klein’s development reveals a functional hierarchy of shape perception. Perceptual organization is the grouping of parts or regions of the field with one another, and the differentiation of the figure from the ground (Rock, 1986). This functional hierarchy is remarkably consistent with the stratification of geometries. For instance, the reaction times for perceiving differences in geometry are closely related to the geometrical stratification related to form stability (Chen, 2005). Specifically, the highest stability of topology provides the shortest reaction times among the gradations of geometry. In other words, people perceive topological features before any other geometrical variation. Therefore, we addressed customer attention to topological differences in the shape of products to foster innovative processes. Under normal circumstances, customers integrate information from stimuli with previous knowledge (Kruglanski & Gigerenzer, 2011). Especially for consumer products, consumers may be familiar with product shapes by recalling related memories. When a product with a novel shape is presented to consumers as a marketing stimulus, they integrate information from that stimulus with the previously known shape, and greater differences are likely to elicit more consumer attention.