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
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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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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The mathematical theory of linkages was once found beautiful, but is now comparatively unknown.
One of its highlights was what Florian Cajori (1859–1930) called the ‘beautiful discovery’ of the Peaucellier–Lipkin linkage (found independently in 1864 and 1871), a simple mechanism that transforms circular motion into linear motion (see attached image).
The importance of such a mechanism is that it can produce straight-line motion without using guide-rails and thus reducing friction. Previous linkages such as James Watt's (1736–1819) only *approximated* straight-line motion.
J.J. Sylvester, (1814–97) (who characterized his mathematical work as ‘the worship of the True & Beautiful’) admired a pump based on the linkage for ‘[i]ts elegance, and the frictionless ease with which it can be worked (beauty as usual the stamp and seal of perfection)’.
When the physicist William Thomson (later Baron Kelvin; 1824–1907) was able to work a model of the linkage, he was reluctant to hand it back, saying: ‘No! I have not had nearly enough of it — it is the most beautiful thing I have seen in my life’.
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#linkage #mechanism #mechanics #Kelvin #Sylvester #MathematicalBeauty
Joseph-Louis Lagrange (1736–1813) found mathematical beauty in many of the fields in which he worked. Here I present some examples from solid geometry.
He considered beautiful Albert Girard’s (1595–1632) theorem relating the area and the angles of a spherical triangle:
The area of a triangle ABC on the surface of a unit sphere is A + B + C − π.
Jean-Étienne Montucla (1725–99) had earlier called the result ‘very elegant’, but had complained that Girard had proved it in a ‘quite laborious and obscure’ fashion. Lagrange thought John Wallis’s (1616–1703) proof was beautiful.
Another result that Lagrange admired was the following:
In any stereographic projection of a sphere onto a plane, any circle on the sphere that does not pass through the point of projection is projected to a circle on the plane (see attached image).
Hence to find the image of such a circle under projection it suffices to find the images of three distinct points on the circle. This fact Lagrange thought a ‘beautiful property of the stereographic projection’.
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#Lagrange #geometry #SphericalGeometry #SolidGeometry #Montucla #MathematicalBeauty #elegance
In retrospect, modern aesthetics is seen to have emerged at the end of the 17th and in the 18th centuries, with the term ‘aesthetic’ being coined by Alexander Gottlieb Baumgarten (1714–62) in 1735 from the Greek aisthētikos [αἰσθητικός].
Many of the early thinkers considered mathematical beauty to be an archetypical form of beauty and integrated it into their theories.
For example, Jean-Pierre de Crousaz (1663–1750) and Francis Hutcheson (1694–1746) both analysed beauty in terms of ‘unity (or uniformity) amidst variety’. Hutcheson thought that this explained why regular polyhedra were more beautiful than irregular ones, and that Archimedes' celebrated theorem
‘The ratios of volumes of a cylinder, its inscribed sphere, and a cone of equal base and height are 3 ∶ 2 ∶ 1’
was more beautiful than the less precise
‘A cylinder has greater volume than an inscribed sphere, which in turn has greater volume than a cone of equal base and height’
because they had equal variety (since they applied to the same objects), but the first theorem had greater unity.
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#aesthetics #HistPhil #Baumgarten #Crousaz #Hutcheson #UnityAmidstVariety #UniformityAmidstVariety #MathematicalBeauty
Results ranging from visualizable theorems of solid geometry to abstract propositions of analysis were called beautiful by Leonhard Euler (1707–83). For instance, he thought beautiful the following result:
If an elliptical cylinder is cut by any plane at an angle θ, then the ratio of the product of the principal axes of the section and of the product of the principal axes of the base is 1:cos θ (see attached image).
Aesthetic concerns seem to have been part of what drew Euler to number theory. Christian Goldbach (1690–1764) persuaded him to take an interest in the subject and to make a serious study of Fermat's work. His attention was drawn by the theorem:
Every natural number can be expressed as a sum of four squares.
With presumably deliberate understatement, Euler described it as a ‘not inelegant theorem’. The result remained unproven in Euler's time, and the first proof was given by Joseph-Louis Lagrange (1736–1813), becoming known as ‘Lagrange’s four-square theorem’.
Thus, for Euler, *unproven* conjectures could have aesthetic value. And so he judged another well-known then-unproven result of Fermat:
‘In Fermat there is another very beautiful theorem for which he claims to have found a proof. […] the formula $a^n + b^n = c^n$ is impossible whenever $n > 2$’
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[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]
#Euler #Fermat #Goldbach #Lagrange #FermatsLastTheorem #MathematicalBeauty
Gottfried Wilhelm Leibniz's (1646–1716) first great mathematical achievement was the ‘arithmetic quadrature’ of the circle through his alternating series: π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
He communicated the result to his mathematical mentor Christiaan Huygens (1629–95), who thought it ‘very beautiful and very pleasing’. Isaac Newton (1642–1726) welcomed Leibniz’s work as ‘very elegant’. Leibniz himself wrote that there was no simpler or more beautiful way of expressing the area of a circle using rational numbers.
In a short note concerning the beauty of theorems, Leibniz wrote:
‘Theorems are not intelligible except by their signs or characters. […] The beauty of theorems consists in the beautiful arrangement of their characters.’
To illustrate ‘beautiful arrangement of characters’, Leibniz gave the example of a theorem concerning Berthet’s curve (shown in red in 1st attached image). The detail of its definition is not important here, but it is defined with reference to an arc AC centred at B.
Leibniz's result was a way of finding the tangent to the curve at E: take the tangent to the arc at its intersection with BE (i.e., at D), and find the point F such that FD ∶ DE ∶∶ EB ∶ BD. Then EF is the desired tangent (see 1st attached image).
Why is there a ‘beautiful arrangement of characters’? Because the proportion FD ∶ DE ∶∶ EB ∶ BD is easily remembered via a mnemonic: one can draw the path FD ⋅ DE ⋅ EB ⋅ BD without raising one's pen (2nd attached image).
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#geometry #Leibniz #Huygens #Newton #MathematicalBeauty #aesthetics #HistMath
An enduring locus of mathematical beauty in the seventeenth century concerned curves like the cycloid and the catenary.
A cycloid is the path followed by a point on the circumference of a circle rolling along a straight line (see attached image).
Christopher Wren (1632–1723) proved that the arc length of the cycloid is four times the diameter of its generating circle.
Christiaan Huygens (1629–95) thought Wren's work ‘really beautiful’. Blaise Pascal (1623–62) also called it ‘beautiful’ (even though he also seemed to repudiate any true notion of mathematical beauty in his ‘Pensées’.
Huygens proved that an inverted cycloid was the ‘tautochrone’: the curve along which a body starting from rest and freely accelerated by uniform gravity reaches the lowest point in the same time, independently of its starting point.
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#cycloid #tautochrone #Huygens #HistMath #MathematicalBeauty #Pascal
Alan J. Cain