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• Post #2298919

  I have uploaded a new version of my book ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large] to archive.org. The PDF of the new version is available immediately, and archive.org will re-generate the pages for the in-browser reader over the next few hours. Changes: • Improvements to several figures. • Various small improvements to the text. • Several minor corrections, including a number of reader-submitted ones. • All end-of-l...

• Post #2298904

  There seems to have been a long-standing connection in mediaeval Islamic thought between calligraphy and geometry. The scholar Abū Ḥayyān al-Tawḥīdī (c.930–1023) wrote that ‘handwriting is spiritual geometry by means of a corporeal instrument’. Much later, Aḥmad Ibrāhīmī Ḥusaynī Qummī (b. 1546) wrote in his treatise ‘Calligraphers and Painters’ (c.1606) that ‘writing is the geometry of the soul’. Al-Tawḥīdī attributed the idea to Euclid; Qummī pointed to Plato, so both undoubtedly thought that...

• Post #1733538

  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/1...

• Post #1670156

  Yves-Marie André's (1675–1764) [https://en.wikipedia.org/wiki/Yves_Marie_André] ‘Essay on Beauty’ (‘Essai sur le beau’, 1741) was an important work of 18th-century aesthetics. In the article on beauty in the 2nd volume (1752) of the ‘Encylopédie’, Denis Diderot praised André's system as the best he knew, elevating it above the work of Plato, Augustine, Wolff, Crousaz, Shaftesbury. In 2010, I released a #CreativeCommons-licensed annotated English translation [https://archive.org/...

• Post #1666326

  Attached is a typographical failure which I encountered in a book from a respected academic publisher. Less-than and greater-than symbols < > have been used instead of angle brackets ⟨ ⟩ (\langle and \rangle). This error should not have survived editorial scrutiny. Even if the author preferred the *appearance* of less-than and greater-than symbols, the real problem is that TeX spaces them as binary relations rather than delimiters. Thus there is no space between = and <,...

• Post #830942

  My new #LuaLaTeX package: lua-simple-wrap This package allows easy wrapping of text around small figures, pull quotes, etc. (See attached image.) Unlike (I think) all other LaTeX packages for wrapping text around things, the wrapped object can be specified at its natural place in the text, and the package works out where it should appear. Virtually all the processing happens on the Lua side. The package is semi-experimental, and definitely requires more work before I submit it to CTAN, but th...

• Post #751166

  Just received an email asking me to peer-review a paper, and asking me to decide whether to review it based on an abstract in which every piece of LaTeX code has been replaced with ‘[[EQUATION]]’, so that it reads like: ‘For [[EQUATION]], let [[EQUATION]] be the semigroup of partial bijections on the lattice [[EQUATION]]. For a subset [[EQUATION]] of [[EQUATION]] with size at most [[EQUATION]], we show that [[EQUATION]] and [[EQUATION]] are the unique possible...’ I am not quoting the actual t...

• Post #557474

  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...

• Post #557473

  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 f...

• Post #557471

  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 woul...

• Post #490056

  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