#mathematicalbeauty — Public Fediverse posts

A quote from Kiyosi Itô, mathematician and 1998 Kyoto laureate, born #OnThisDay in in 1915:

‘In precisely built mathematical structures, mathematicians find the same sort of beauty others find in enchanting pieces of music, or in magnificent architecture.’

— Itô, 'My Sixty Years along the Path of Probability Theory'. 1998 Kyoto Prize lecture abstract [https://www.kyotoprize.org/en/speech/the-1998-kyoto-prize/]

(One of innumerable testimonies indicating that the subjective experience of #MathematicalBeauty parallels that of visual or auditory beauty.)

#aesthetics

@paysmaths @Theoremoftheday

In his essay ‘The Phenomenology of Mathematical Beauty’, Gian-Carlo Rota said Morley's theorem was surprising but not beautiful. (He was arguing against the claim that beauty depended on unexpectedness, which was how he (perhaps not quite accurately) interpreted G.H. Hardy.)

But every other aesthetic judgement of Morley's theorem that I know of says that it is beautiful:

• ‘[in] a beauty contest involving geometrical theorems [...] one of the chief contenders would be Morley’s Triangle Theorem’ — Bankoff
• ‘[a] gorgeous result’ — Chamberland
• ‘startling, difficult to prove, and utterly beautiful’ — Dunham
• ‘Morley's beautiful theorem’ — Gardner
• ‘absolutely stunning and beautiful’ — Hofstadter
• ‘for sheer beauty it has few rivals’ — Oakley & Baker

(I agree that it is a beautiful result.)

#MathematicalBeauty #aesthetics #GHHardy

Update to ‘An Annotated Mathematician’s Apology’, with a new section on Wittgenstein’s response: https://archive.org/details/hardy_annotated (accessible #TaggedPDF, #OpenAccess)

My thanks go to Wolfgang Kienzler, professor at Friedrich Schiller Universität in Jena, who pointed me to Wittgenstein’s remarks, sent me notes about relevant sources, and critiqued my draft for the new section.

#GHHardy #Wittgenstein #philosophy #mathematics #MathematicalBeauty

Big update to ‘Form & Number: A History of Mathematical Beauty’: the PDFs are now *tagged* (plus various minor improvements).

A #TaggedPDF contains hidden extra semantic information to assist screen-reading software etc., including alt text for diagrams+images (in the case of ‘F&N’, 552 pieces of alt text were required, some quite lengthy).

The new version is available (#OpenAccess as always) at: https://archive.org/details/cain_formandnumber_ebook_large

Feedback from users of screen readers would be very welcome!

Creating this version has been possible because of the ongoing LaTeX tagging project [https://latex3.github.io/tagging-project/] (thank you to the LaTeX team and other contributors!). I have been adapting my own #LuaLaTeX styles [https://codeberg.org/ajcain/minos] to use the new interfaces.

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#accessibility #a11y #MathArt #HistMath #HistSci #MathematicalBeauty #aesthetics #typography #TeXLaTeX

From a Mathematical Curiosity to the Foundation of Internet Security 🧮🧐🔐✨

In 1636, Pierre de Fermat made a remarkable observation: if you take an integer a, raise it to a prime power p, and then subtract a, the result is always divisible by p. In other words, for a prime p,

aᵖ ≡ a (mod p).

For example, when p = 5:
2⁵ − 2 = 30, which is divisible by 5, and
3⁵ − 3 = 240, which is also divisible by 5.

Fermat found this property elegant and tried to spark interest in this curious corner of mathematics. Not everyone was impressed. John Wallis reportedly dismissed such results, remarking, “Big deal; I could find other relationships just as interesting without much effort, and none of them are important.”

About a century later, Leonhard Euler was encouraged by Christian Goldbach to study Fermat’s ideas. Though initially unenthusiastic, Euler soon uncovered deeper structure within them. His work laid much of the foundation for what we now call modern number theory.

#Mathematics #Math #NumberTheory #Fermat #Euler #EulerTheorem #FermatsLittleTheorem #ModularArithmetic #RSA #Cryptography #CyberSecurity #HistoryOfMathematics #MathHistory #PrimeNumbers #EulerTotient #AbstractMath #PureMathematics #STEM #ScienceCommunication #MathEducation #EducationalContent #DidYouKnow #MathematicalBeauty #InternetSecurity #Encryption #Curiosity #Innovation #History #Learning #Knowledge

A short #BookReview of ‘Form & Number: A History of Mathematical Beauty’ in ‘Mathematics Magazine’ has been pointed out to me: https://doi.org/10.1080/0025570X.2025.2587000 (not open-access)

I am quite pleased by the reviewer's evaluation :-) ‘a very careful and absolutely thorough survey of the “rich heritage of scholarship” about both beauty in mathematics and the study of that topic’.

There was a more detailed review by Viktor Blåsjö in ‘TUGboat’ last year: https://doi.org/10.47397/tb/46-2/tb143reviews-cain (open-access)

#review #MathematicalBeauty #HistMath #HistSci

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

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

Evangelista Torricelli’s (1608–47) solid is defined by rotating the hyperbola $y = 1/x$ about the $x$ axis and truncating it at $x=1$ (see attached image).

It has infinite length and infinite surface area but finite volume.

This counter-intuitive discovery caused philosophical disturbance, for it seemed to violate the distinction between finite and infinite.

Torricelli, foreseeing the scrutiny to which his work would be subjected, took the precaution of preempting some criticisms by supplying two different proofs, one by ‘indivisibles’, one by exhaustion.

But René Descartes (1596–1650) seems not to have been provoked to any philosophical objections and thought that Torricelli's discovery was beautiful.

Henry Needler (fl. 1690–1718), a perhaps slightly obscure figure who foreshadowed 18th-century discussions of the sublime, seemed to be impressed by the solid's ‘Grandeur and Magnificence’ and thought that it would ‘afford the greatest Delight and Satisfaction to curious Minds’.

(Today, Torricelli's solid is also called ‘Gabriel's horn’ or ‘Torricelli's trumpet’.)

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#infinite #Descartes #HistMath #HistPhil #Torricelli #MathematicalBeauty #sublime #aesthetics

Number theory was the one area of mathematics on which Pierre de Fermat (1607–65) worked throughout his life, and he found it ‘very beautiful and very subtle’.

Among other results, he said that the Polygonal Number Theorem (which asserts that every natural number is the sum of at most $n$ $n$-gonal numbers) was ‘a most beautiful and wholly general proposition […] this marvellous proposition’.

(He offered no proof of this result, but claimed to have one in a marginal note to Diophantus' Arithmetica; this was the same book in which he noted what became known as Fermat's Last Theorem.)

Fermat also seems to have counted magic squares and analogous configurations as part of number theory, and wrote that: ‘I know hardly anything more beautiful in arithmetic than these numbers that some call planetary and others magic’. (The term ‘planetary’ is derived from certain treatises linking the magic squares to planets used in talismans.)

He said he had found a rule to find magic cubes (one of his examples is in the attached image) and also determined how many different ways each such cube can be arranged, which he called ‘one of the most beautiful things in arithmetic’.

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[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

#Fermat #NumberTheory #HistMath #MathematicalBeauty #MagicSquare

Book XVI of Girolamo Cardano's (1501–76) ‘De Subtilitate’, a compendium of natural philosophy, begins by presenting sixty useful properties of geometrical figures, and says:

‘These are the sixty properties, outstanding in distinction and beauty and regard [nobilitate, & pulchritudine, & admiratione præstantiores], of the geometrical figures both surface and solid. Yet it does not escape me that properties exist that are practically numberless, but cannot be compared with these for elegance’

Some of the properties Cardano so admired seem to lapse into triviality, but others are important results from Euclid, Archimedes, Apollonius, and Ptolemy. For example:

Let $A_i$ be the point at the end of the $i$-th turn of an archimedean spiral. Consider the line through the $A_i$ and the perpendicular to this line through the origin $O$. Suppose that the tangent to the spiral at $A_i$ intersects the perpendicular at $B_i$ . Then $OB_i$ equals the circumference of the circle with centre $O$ and radius $OA_i$ (see attached image).

This result effectively comprises Propositions 18+19 of Archimedes' ‘On Spirals’.

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#MathematicalBeauty #elegance #Cardano #Archimedes #spiral

Luca Pacioli’s (c.1445–1517) book ‘Divina proportione’ (written 1496–8, published 1509) is famous in the history of mathematical beauty, but mostly for the wrong reasons.

The term ‘divine proportion’ refers to Euclid's ‘extreme and mean ratio’, known since the late 18th century as the ‘golden ratio’: $1.61803\ldots:1$.

Pacioli's use of the term ‘divine’ **was not based upon aesthetic appreciation**.

Rather, he made a mystical identification of certain properties of the ratio with attributes of God. E.g., the incommensurability of the ratio = the indefinability and ineffability of God.

But Pacioli aesthetically admired the five regular polyhedra — the platonic solids — and the archimedean solids that he knew. In the dedication of ‘Divina proportione’ he wrote that hoped that his patron would see ‘their most sweet harmony’. He linked the aesthetic value of the solids to that of the sphere, from which he saw them as deriving. He seems to have placed special value on the ‘most noble’ dodecahedron.

In his portrait (attached), a dodecahedron sits on top of one of his books as a symbol of mathematical success. His diagram is part of the construction of the tetrahedron. A glass rhombicuboctahedron hangs behind him.

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[Each day of February, I am posting a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

#GoldenRatio #DivineProportion #MathematicalBeauty #MathArt #polyhedron #RegularSolid #PlatonicSolid

Islamic geometrical art is justly famous for its mathematical beauty. Here I will point to some documented connections between the artisans who created it and mathematicians.

The mathematician Abū’l-Wafāʾ al-Būzjānī (940–77/8 CE) wrote in a book on geometry useful for craftsmen (more on this later) that he had been present at discussions between artisans and geometers. Al-Khayyāmī (Omar Khayyam; c.1048–c.1131), the father of algebra, wrote of attending such a meeting. Caʿfer Efendi (fl. early 17th century) wrote of discussions between ‘learned men’ and architects on the subject of geometry.

Another connection is in the person Abu’l-Faḍl ibn ʿAbd al-Karīm (d. 1202/3), whom Ibn abi Uṣaybiʿah recorded as having been called ‘the geometrician’ for his famed knowledge of geometry. He was by profession a carpenter and stonemason. He studied Euclid to improve his carpentry; he then proceeded to Ptolemy’s ‘Almagest’ and became an authority in geometry. His skill is evident in the doors he made for the Bīmaristān Nūr al-Dīn in Damascus, which have upon them a geometrical pattern of hexagons and pentagrams made up of brass nails (the design is shown in the attached image).

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#geometry #MathArt #aesthetics #MathematicalBeauty #tiling #IslamicArt

Abū Sahl al-Kūhī (or al-Qūhī; fl. c.970–c.1000), who was regarded by contemporaries as the ‘Master of his age in the art of geometry’, once wrote of his motivation for considering a problem:

‘Having completed the construction of a regular heptagon in a circle, we set out to investigate another proposition, one more beautiful [ʾaḥsan أحسن], deeper, more opaque and more difficult to find out than the construction of the heptagon […]. This is the construction of an equilateral pentagon in a known square.’

Al-Kūhī's construction is shown in the attached diagrams. The red and blue curves are hyperbolae, both with latus rectum equal to 2AG, and with major axes NS and PI. The segments AE and DK (found using the the intersection of the hyperbolae) are equal to the required sides of the pentagon and a simple translation moves them into position.

(Note that the pentagon is only *equilateral*, not *equiangular*, and so is not regular.)

#MathematicalBeauty #geometry #HistMath #conics

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Leonardo Pisano (c.1170–after 1240), dubbed ‘Fibonacci’, thought that Archimedes' proof that π was between $3\frac{10}{71}$ and $3\frac{1}{7}$ was beautiful [pulcra].

Archimedes' proof proceeds by calculating approximate ratios of the perimeters of 96-gons circumscribed about and inscribed in a circle to the diameter of that circle, implicitly starting with dodecagons and repeatedly bisecting edges to obtain 24-, 48-, and then 96-gons (see attached image).

Fibonacci’s judgement seems to be the earliest extant description of a *proof* as beautiful in the European tradition. [Al-Nasawī (fl. 1029–44) had earlier described a proof as beautiful.]

But there is a twist in the story...

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#MathematicalBeauty #BeautifulProof #HistMath #Fibonacci #Archimedes #Pi

‘Form & Number: A History of Mathematical Beauty’ has been updated: https://archive.org/details/cain_formandnumber_ebook_large

Quite a few typos have been fixed, and there are *many* small but important typographical improvements, mostly to kerning and spacing.

Karl Berry, the treasurer of the TeX Users Group and one of the editors of its journal ‘TUGboat’, has been doing a frankly heroic job of proof-reading the book with a typographer's eye — thank you Karl!

(The new PDF is available for download immediately; archive.org will re-generate images for the in-browser reader during the next few hours.)

Attached is a preview of a two-page spread from the print variant of ‘Form & Number’, showing the start of a section on the Ikhwān al-Ṣafāʾ and a diagram of the numerical structure of their cosmology.

#MathematicalBeauty #HistMath #MathArt #aesthetics #InternetArchive

According to the biography by Diogenes Laertius, Pythagoras (c.570–c.490 BCE) ‘held that the most beautiful figure is the sphere among solids, and the circle among plane figures’.

This aesthetic preference for the circle and sphere can be traced through thinkers like Plato (who, according to later writers, set the problem of describing the movements of the heavens using uniform circular motions), Cicero (106–43 BCE), and Proclus (410/12–485 CE), and into the middle ages.

Thomas Bradwardine (1290/1300–1349), one of the mediaeval ‘Oxford calculators’, was obviously influenced by this tradition when he wrote that the circle ‘is the first and most perfect of figures, the simplest and most regular, the most capacious and the most beautiful of figures’.

But Bradwardine then presented evidence that he saw as attesting to the beauty and perfection of the circle: (1) the construction to find the centre of a circle by bisecting a diameter found as the perpendicular bisector of a chord; (2) that the intersections of six equally-spaced radii with the circumference define a regular hexagon; (3) that exactly six circles of equal size can touch a given circle (see attached image).

For Bradwardine, the perfection of the circle was thus linked to the perfection of the number 6 = 1+2+3: the construction involves six intersections with the circle; the hexagon is made up of six lines; the third result involves six outer circles.

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#MathematicalBeauty #HistMath #Pythagoras #Bradwardine #geometry #aesthetics #PerfectNumber

As noted in a previous post, Archimedes thought highly of the result that the ratio of either the volumes or surface areas of a cone, a sphere, and a cylinder exactly circumscribing them is $1:2:3$.

So did others: three centuries later, the architect Nicon (d.149/50 CE), father of the philosopher and physician Galen (129–c.210/217 CE), thought it fitting to point out the ratio of the configuration in a public inscription in his city, Pergamon:

‘the cone, the sphere, the cylinder.
If a cylinder encloses the other two shapes,
[...]
Competition the principle and in solids
the progression $1 ∶ 2 ∶ 3$,
a noble, divine equalization,
but also mutual interdependence
of the solids, always in the ratio $1 ∶ 2 ∶ 3$.
They should be beautiful and wonderful,
the three solid shapes’

Nicon doubtless admired these ratios as an architect: a sphere inside a cylinder brings to mind the Pantheon at Rome, of which the Temple of Zeus Asclepius Soter in Pergamon was a half-scale copy. These buildings were designed so that a basically cylindrical rotunda was crowned with a hemispherical dome under which a sphere would fit (see attached image).

[Each day of February, I am posting a story/image/fact/anecdote related to the aesthetics of mathematics.]

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#MathematicalBeauty #HistMath #Archimedes #geometry #architecture #aesthetics

Max Dehn (1878–1952) said that Archimedes’ (c.287–212 BCE) discovery that the surface area of a sphere was four times its great circle was the one of the most beautiful results of Greek mathematics.

Archimedes himself had a high opinion of this result and two others in his two books ‘On the Sphere and the Cylinder’: that the volume and surface area of a sphere and a cylinder exactly circumscribing it are in the ratio $2 : 3$. One can add a cone fitting inside the cylinder to have ratios $1 : 2 : 3$ (see 1st attached image).

It has been suggested that Archimedes’ conjectures for these ratios may have been guided by a conscious or unconscious search for beautiful integer ratios between geometric configurations. There is no direct evidence for this motivation, but Archimedes’ work seems to exhibit a preference for small integer ratios.

According to Plutarch, Archimedes desired that his tomb should be marked by a cylinder enclosing a sphere and an inscription of the ratio of the one to the other; Cicero related how he had sought out Archimedes’ tomb and found a column just so inscribed (see 2nd attached image).

[Each day of February, I intend to post an interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

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#MathematicalBeauty #HistMath #Archimedes #Plutarch #Cicero #geometry #aesthetics

The ‘Conics’ of Apollonius of Perga (c.260–c.190 BCE) became the standard text for ‘conic sections’ — the curves formed by the intersection of a plane and a cone, namely an ellipse, parabola, or hyperbola, depending on the angle of the plane relative to the slope of the cone (see attached image).

In the preface to the ‘Conics’, Apollonius wrote:

‘The third book contains many incredible theorems of use for the construction of solid loci and for limits of possibility of which the greatest part and the most beautiful [kallista κάλλιστα] are new.’

This quotation is triply important in the historiography of mathematical beauty: (1) it is the earliest extant description of a mathematical theorem as ‘beautiful’; (2) it is the earliest extant application of the term ‘beautiful’ to mathematics by a mathematician; and (3) it is the unique extant use of the term ‘beautiful’ to describe theorems by an ancient Greek mathematician.

(There is much discussion of the beauty of mathematics in ancient Greek thought, but it normally applies to the objects or concepts of mathematics.)

[Each day of February, I intend to post an interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

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#MathematicalBeauty #HistMath #Conic #ConicSection #geometry #aesthetics

For each day of February, I intend to post a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.

February has 28 days, and 28 is the second perfect number, so let's start there.

Definitions first: a ‘perfect’ number is equal to the sum of its own proper divisors (28 = 1+2+4+7+14). If the sum of proper divisors is greater than the number itself, the number is ‘abundant’; if less, it is ‘deficient’.

Philo of Alexandria (fl. early 1st century CE) seems to have connected perfect numbers with beauty, at least via the order emplaced in the world during the Biblical 6 (= 1+2+3) days of creation. (The perfect number of days of the hexaëmeron was emphasized by later writers like Methodius of Olympus (d. c.310 CE) and Augustine of Hippo (354–430 CE), though not in explicitly aesthetic terms.)

Nicomachus of Gerasa (fl. 100 CE) seems to have thought that the (rare) perfect numbers were beautiful, and that the much more common abundant and deficient numbers were ugly, likening them to monstrous creatures with too many or too few limbs, mouths, eyes.

Boethius (c.480–c.524 CE) agreed with Nicomachus and was even more specific about the parallel to monsters: deficient numbers were like the one-eyed Cyclopes; abundant numbers were like the triple-headed or -bodied Geryon (image attached).

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#MathematicalBeauty #MathHist #PerfectNumber #aesthetics

@marick An exception is perhaps the late Edsger Dijkstra. He wrote often about the important of beauty in programming and computer science generally.

In particular, he noted that beauty and elegance are the greatest virtues that a computer program could have and that

‘when we recognize the battle against chaos, mess, and unmastered complexity as one of computing science's major callings, we must admit that “Beauty is our Business”’,

and

‘in the practice of computing, where we have so much latitude for making a mess of it, mathematical elegance is not a dispensable luxury, but a matter of life and death’.

(He seems not to have distinguished elegance and beauty, and he subsumed the aesthetics of computer science within the aesthetics of mathematics. )

At risk of making this an advertisement, for the citations for these quotations and other sources for Dijkstra's views on beauty, I will point to pp.697–699 of my open-access book ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large].

#MathematicalBeauty #MathematicalElegance #aesthetics