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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Abū’l-Wafāʾ al-Būzjānī (940–77/8 CE) wrote one of the earliest extant treatises dedicated to magic squares, focused on constructions. He repeatedly referred to the aesthetic value of the methods of he described.

For instance, he wrote about a method of constructing a magic square of order 4:

‘It is possible to arrive at the magic arrangement in this square by means of methods without displacement showing regularity and elegance [niẓām wa-tartīb ḥasan نظام وترتيب حسن]’ (trans. Sesiano)

Such a method with ‘regularity and elegance’ was: (1) place the number 1 in a corner, 2 and 3 adjacent to the opposite corner, and 4 diagonally adjacent to 1; (2) place 5 to 8 in reverse order in positions horizontally symmetrically opposite to 1 to 4; (3) place $17 − n$ diagonally two places away from $n$ for $n = 1,\ldots,8$ (see attached image).

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

#MagicSquare #HistMath #MathematicalElegance #elegance #aesthetics

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

Tout juste paru chez College Publications (Londres) après une loooongue période de maturation :

Circulations des mathématiques dans et par les journaux — Histoire, territoires, publics

Cet ouvrage résulte des travaux du collectif international réuni autour du projet ANR « Cirmath » qui a réuni plus de 30 contributrices et contributeurs de près d’une dizaine de pays.

https://cirmath.hypotheses.org/2316

#histmath #histsci

Mathématiques en Chine : à la racine du calcul | Sciences Chrono France Culture

Karine Chemla au micro d'Antoine Beauchamp !

https://www.radiofrance.fr/franceculture/podcasts/sciences-chrono/sciences-chrono-emission-du-vendredi-30-janvier-2026-5278703

#histmath #histsci #ConseilPodcast

Ça intéresserait certainement mes collègues du projet @PatriMaths qui travaillent sur les bibliothèques...

J'ai moi-même travaillé sur la bibliothèque du Genevois Gabriel Cramer (1704-1750) mais je ne dispose pas d'inventaire après décès, et les pièces de cette bibliothèque ont été dispersées dès les semaines suivant son décès :

https://images.math.cnrs.fr/un-mathematicien-et-ses-livres/

#bibliotheques #hitsci #histmath

Le colloque final du projet #Patrimaths se tiendra la semaine prochaine, du 12 au 14 novembre, à Nancy.

Le programme est disponible ici : https://patrimaths.hypotheses.org/719

#histsci #histmath #patrimoine #mathématiques #bibliotheques #encyclopedies

As a several-days-late contribution for the #Mathober Day 25 prompt ‘Wedge’, I would like to point out a little historical curiosity involving ‘wedge’.

Attached is a detail from an Old Babylonian clay tablet of geometrical problems and a reconstruction of the diagram.

The cuneiform text reads: ‘The square-side is 1 cable. ⟨Inside it⟩ I drew 12 wedges and 4 squares. What are their areas?’ (trans. Robson, ‘Mesopotamian mathematics’, p.95)

The term ‘wedge’ translates the Akkadian ‘santakkum’, which names any figure with three (possibly non-straight) sides. (1 ‘cable’ = approximately 360 metres)

The exact symmetry of the configuration is vital to the problem. Without symmetry, which is suggested by the (necessarily approximate) diagram, but which is not made explicit in the question, the ‘wedges’ could be (e.g.) non-isosceles triangles of different sizes, and the problem would be insoluble.

The problems on the tablet [https://www.britishmuseum.org/collection/object/W_1905-0515-1] comprise various geometric configurations in which symmetry is implicitly required for the solution.

#HistMath #HistSci #geometry #symmetry #Mathober2025

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

Karl Berry, the treasurer of the TeX Users Group and one of the editors of its journal ‘TUGboat’, and Frank Plastria, Professor Emeritus at Vrije Universiteit Brussel, have (very kindly!) been reading and commenting on the text. Karl also ran his suite of automated tests on the #LuaLaTeX source files.

As a result, the new version incorporates many corrected typos and small improvements.

Thank you, Frank and 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.)

#HistMath #HistSci #MathArt #aesthetics #InternetArchive

The #Mathober Day 9 prompt is ‘Chi’. In graph theory, \(\chi(G)\) denotes the chromatic number of a graph \(G\): the minimum number of colours required to colour all vertices so that no adjacent vertices have the same colour.

The famous ‘four colour theorem’ says that all planar graphs \(G\) have \(\chi(G) \leq 4\). A graph is planar if it can be drawn in the plane without any edges crossing.

The conjecture originated in 1852 when Francis Guthrie (1831–99) noticed that it was possible to colour a map of England using only four colours so that no neighbouring counties received the same colour, and wondered if this held true for all maps. The question for maps is transformed into one for graphs by replacing each region by a vertex and placing edges between vertices corresponding to neighbouring regions.

Thus, for the Mathober prompt ‘Chi’, I offer a map of the counties of England as they were in Guthrie's time and the corresponding graph, coloured in the same way with four colours.

Vector version (PDF format) of the map: https://ajcain.codeberg.page/posts/files/england-counties-map-4coloured.pdf

Vector version (PDF format) of the graph: https://ajcain.codeberg.page/posts/files/england-counties-graph-4coloured.pdf

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#Mathober2025 #GraphTheory #HistMath

The #Mathober day 3 prompt is ‘#Polyhedron’. I have no art to offer, but I thought I would use the occasion to draw attention to a lesser-known role of polyhedra in Johannes Kepler's (1571–1630) thought.

Start with the famous part: Kepler argued that the five regular (or Platonic) solids fitted between the orbs of the then-known six planets [https://commons.wikimedia.org/wiki/File:Kepler-solar-system-1.png].

When Galileo discovered four moons of Jupiter, Kepler sought a similar polyhedral structure for the Jovian system, and suggested that ‘semiregular’ solids provided the key. ‘Semiregular’ for Kepler meant that the polyhedra had rhombic faces and satisfied certain technical criteria. (The concept differs from today's notion of semiregular polyhedra.)

There were exactly three such ‘semiregular’ polyhedra:

• the cube (the square being a special kind of rhombus).

• the rhombic dodecahedron (12 faces; diagram attached).

• the rhombic triacontahedron (30 faces; diagram attached).

Kepler placed the rhombic dodecahedron between Io and Europa, the rhombic triacontahedron between Europa and Ganymede, and the cube between Ganymede and Callisto.

Kepler thought that there were six planets because God had shaped the cosmos around the five platonic solids. The three ‘semiregular’ solids would similarly explain why there were four moons of Jupiter.

Perhaps someone who is a better artist than I am could draw a diagram of the three rhombic solids and the orbs of the four Galilean moons in the style of Kepler's famous diagram of the Platonic solids between the orbs of the planets.

#Mathober2025 #HistMath #HistSci

The new issue of ‘TUGboat’ also contains an interview with yours truly, conducted by Jim Hefferon [https://hefferon.net/], a mathematician at the University of Vermont and the author of several free undergraduate mathematics textbooks (‘Linear Algebra’, ‘Theory of Computation’, ‘Introduction to Proofs’ — see his website).

The interview ranges over mathematical beauty, history, typography in general and TeX/LaTeX in particular, and the process of writing ‘Form & Number’.

The interview is #OpenAccess: https://www.tug.org/TUGboat/tb46-2/tb143cain-hefferon.pdf

#typography #TeXLaTeX #MathematicalBeauty #MathArt #HistMath

The new issue (vol.46, no.2) of ‘TUGboat: Communications of the TeX Users Group’ contains a (positive!) review by Viktor Blåsjö of my book ‘Form & Number: A History of Mathematical Beauty’. The review is #OpenAccess: https://www.tug.org/TUGboat/tb46-2/tb143reviews-cain.pdf

Viktor Blåsjö is a historian of mathematics at Utrecht University, and I am delighted (and more than a little surprised) that a professional historian praised the work of a mere dilettante like me! Although his review disagrees with me on some individual points, I cannot imagine a better endorsement than this:

‘Cain’s book means that a systematic starting point for all future work on this topic is now available [...] what I wish most of all is that mathematicians and philosophers who set out to opine on this topic in the future make sure to read this book first, as this is bound to elevate the academic discourse on this subject to a more critically aware and informed level’

Thank you, Viktor!

‘Form & Number’ itself is of course freely available under a #CreativeCommons licence: https://archive.org/details/cain_formandnumber_ebook_large

#MathematicalBeauty #MathArt #HistMath #HistSci

Un autre CDD d'IE humanités numériques dans un projet d'histoire des maths, cette fois pour entraîner des modèles HTR sur des textes de maths du 17e siècle (toujours dans une super équipe) :
https://emploi.cnrs.fr/Offres/CDD/UMR7219-VIRMAO-020/Default.aspx

#histsci #histmath #jobalert #Esr #ResearchJobs #digitalhumanities @histodons

Mercredi, journée d'étude @PatriMaths pour réfléchir à la question du patrimoine mathématique dans les encyclopédies du XVIIIe au XXe siècle.

J'y présenterai une étude de cas sur les fortunes et infortunes du parallélogramme analytique de Newton (une méthode utilisée pour initier des développements en série, notamment en appui d'études de courbes algébriques) dans les encyclopédies de langue anglaise et française entre 1750 et 1850.

https://patrimaths.hypotheses.org/704

#histsci #histmath #patrimaths