Mathematical beauty
For example, Gauss's Theorema Egregium is a deep theorem that states that the gaussian curvature is invariant under isometry of the surface.
In his 1940 essay A Mathematician's Apology, G. H. Hardy suggested that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy".
[9] In 1997, Gian-Carlo Rota, disagreed with unexpectedness as a sufficient condition for beauty and proposed a counterexample: A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.
[10] In contrast, Monastyrsky wrote in 2001: It is very difficult to find an analogous invention in the past to Milnor's beautiful construction of the different differential structures on the seven-dimensional sphere...
The original proof of Milnor was not very constructive, but later E. Brieskorn showed that these differential structures can be described in an extremely explicit and beautiful form.
Group theory, developed in the early 1800s for the sole purpose of solving polynomial equations, became a fruitful way of categorizing elementary particles—the building blocks of matter.
In a general Math Circle lesson, students use pattern finding, observation, and exploration to make their own mathematical discoveries.
Some of the topics and objects seen in combinatorics courses with visual representations include, among others Four color theorem, Young tableau, Permutohedron, Graph theory, Partition of a set.
[17] Brain imaging experiments conducted by Semir Zeki and his colleagues[18] show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial orbito-frontal cortex (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty.
Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.
These mathematicians believe that the detailed and precise results of mathematics may be reasonably taken to be true without any dependence on the universe in which we live.
[20] Hungarian mathematician Paul Erdős[21] spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs.
As there are exactly five Platonic solids, Kepler's hypothesis could only accommodate six planetary orbits and was disproved by the subsequent discovery of Uranus.
G. H. Hardy[23] analysed the beauty of mathematical proofs into these six dimensions: general, serious, deep, unexpected, inevitable, economical (simple).
Paul Ernest[24] proposes seven dimensions for any mathematical objects, including concepts, theorems, proofs and theories.
[31][32] Examples of the use of mathematics in music include the stochastic music of Iannis Xenakis, the Fibonacci sequence in Tool's Lateralus, counterpoint of Johann Sebastian Bach, polyrhythmic structures (as in Igor Stravinsky's The Rite of Spring), the Metric modulation of Elliott Carter, permutation theory in serialism beginning with Arnold Schoenberg, and application of Shepard tones in Karlheinz Stockhausen's Hymnen.
Sacred geometry is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history in Islamic architecture.
It also provides a means of meditation and comtemplation, for example the study of the Kaballah Sefirot (Tree Of Life) and Metatron's Cube; and also the act of drawing itself.
British constructionist artist John Ernest created reliefs and paintings inspired by group theory.
[33] A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, including Anthony Hill and Peter Lowe.
Bertrand Russell expressed his sense of mathematical beauty in these words: Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.
