Alan J. Cain

Each day of February I posted a fact/image/anecdote about the aesthetics of mathematics, which seemed to provoke a certain amount of interest.

The posts are all collected on my personal website, with minor fixes and improvements (including vector versions of diagrams).

The index is here: https://ajcain.codeberg.page/posts/2026-03-01-aesthetics-of-mathematics.html

#aesthetics #MathematicalBeauty #HistMath #elegance #beauty

Maxwell's equations seem to be universally and consistently held up as exemplars of mathematical beauty in physical law. Expressed in modern notation as differential equations, they are as shown in the first attached image.

Even someone unaware of the physical interpretation of the symbols can see clear symmetries in the equations.

Henri Poincaré (1854–1912) thought James Clerk Maxwell (1831–79) was able to reformulate electromagnetic theory in part due to seeing how the equations would become more symmetrical:

‘It was because Maxwell was profoundly steeped in the sense of mathematical symmetry; would he have been so, if others before him had not studied this symmetry for its own beauty?’

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#MaxwellsEquations #MathematicalBeauty #symmetry #HistSci

In 1948, François Le Lionnais (1901–84) published an essay in which he distinguished two types of beauty in mathematics:

• ‘Classical’ mathematical beauty, which impressed by its control and austerity.

• ‘Romantic’ mathematical beauty, which manifested in wildness, non-conformity, and strangeness.

Classical beauty was found where there was unification, such as in the 9-point circle of a triangle (see 1st attached image), or how the circle, ellipse, hyperbola, and parabola all arise from the focus–directrix construction (see 2nd attached image) and from conic sections, and can transformed into one another by projective transformations.

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#MathematicalBeauty #ClassicalBeauty #Classicism #RomanticBeauty #Romanticism #ClassicalVsRomantic #aesthetics

The philosopher, biologist, and political theorist Herbert Spencer (1820–1903) has a minor but curious role in the history of mathematical beauty, because of comments he made about Monge’s theorem, which states:

For any three circles in a plane, none contained within another, the intersections of the outside tangents of the three pairs of circles are collinear. (See attached image.)

Spencer said that when he thought of it he was

‘struck by its beauty at the same time that it excites feelings of wonder and of awe: the fact that apparently unrelated circles should in every case be held together by this plexus of relations, seeming so utterly incomprehensible.’

However, Spencer’s reaction of wonder and of awe may ultimately have been born of his limited mathematical ability.

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#geometry #HerbertSpencer #MathematicalBeauty #HistMath