Bertrand Russell expressed his sense of mathematical beauty in these words:
Rota, however, disagrees with unexpectedness as a necessary condition for beauty and proposes a counterexample:
This disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.
In a general Math Circle lesson, students use pattern finding, observation, and exploration to make their own mathematical discoveries. For example, mathematical beauty arises in a Math Circle activity on symmetry designed for 2nd and 3rd graders, where students create their own snowflakes by folding a square piece of paper and cutting out designs of their choice along the edges of the folded paper. When the paper is unfolded, a symmetrical design reveals itself. In a day to day elementary school mathematics class, symmetry can be presented as such in an artistic manner where students see aesthetically pleasing results in mathematics.
Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention, for example:
There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within.William Kingdon Clifford, from a lecture to the Royal Institution titled "Some of the conditions of mental development"