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Concept
Mathematical beauty
Summary
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, (a position taken by G. H. Hardy) or, at a minimum, as a creative activity.
Comparisons are made with music and poetry.
Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean:
A proof that uses a minimum of additional assumptions or previous results.
A proof that is unusually succinct.
A proof that derives a result in a surprising way (e.g., from an apparently unrelated theorem or a collection of theorems).
A proof that is based on new and original insights.
A method of proof that can be easily generalized to solve a family of similar problems.
In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs being published up to date. Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity. In fact, Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published.
Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, highly conventional approaches or a large number of powerful axioms or previous results are usually not considered to be elegant, and may be even referred to as ugly or clumsy.
Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated. These results are often described as deep. While it is difficult to find universal agreement on whether a result is deep, some examples are more commonly cited than others.
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Theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations.
Mathematics and art
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts. Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions conjectured to have been based on the ratio 1: for the ideal male nude.
Pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.