#mathhist — Public Fediverse posts

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

Small update to ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large].

Trivial corrections to the text and index and some minor clarifications and additional citations following comments from others.

Sample pages attached from the two-sided print variant showing a spread about the contribution of François Le Lionnais, a chemist and writer, and one of the founders of the OuLiPo group of writers [https://en.wikipedia.org/wiki/Oulipo]. Le Lionnais wrote some important articles about mathematical beauty in the late 1940s.

I have released the #LuaLaTeX style I used for this book: https://codeberg.org/ajcain/minos (Git repository) https://codeberg.org/ajcain/minos/releases (Documentation and source .zip/.tar.gz)

#MathematicalBeauty #MathHist #MathArt #aesthetics #TeXLaTeX

Following my earlier posts on historical aesthetic judgements of Euler's identity, I just found another one, in Constance Reid's popular mathematics book ‘From Zero to Infinity’ [https://en.wikipedia.org/wiki/From_Zero_to_Infinity], where it is called ‘elegant’. This appears on p.165 of the 1964 third edition. Can anyone confirm if it appears in the first or second editions? (1955, 1960)

I am still interested in other aesthetic judgements of Euler's identity before 1988 (I know of few, although one more than I knew of yesterday), and especially before 1940 (I currently know of none).

Incidentally, as printed in the third edition of Reid's book, the equation would evoke a shudder from any typographer (see attached image).

#EulersIdentity #EulersEquation #MathHist #MathematcalElegance #aesthetics #MathArt #Mathematics #typography

Today's post from @paysmaths @Theoremoftheday was on #EulersIdentity or #EulersEquation \(e^{i\pi} + 1 = 0\) (also given in various equivalent forms), which prompts me to talk about a little historical mystery.

Euler's identity is often held up as as exemplar of mathematical beauty, or called the most beautiful or most elegant equation in mathematics.

But when I was researching my book ‘Form & Number: A History of Mathematical Beauty’ [https://archive.org/details/cain_formandnumber_ebook_large], I was unable to find *any* aesthetic judgement of the equation before the 1940 book ‘Mathematics and the Imagination’ by Kasner & Newman [https://archive.org/details/mathematicsimagi0000edwa_l2s0], where it is called ‘elegant’ (p.103). The earliest explicit judgements of it as *beautiful* that I found are in essays by Le Lionnais in the late 1940s.

Does anyone know of any aesthetic judgements of Euler's equations before 1940? (I know of earlier non-aesthetic judgements like ‘mysterious’ or ‘paradoxical’.)

(The history of aesthetic judgements of Euler's equation is on pp.835–9 of ‘Form & Number’, #OpenAccess at the link above.)

#MathematicalBeauty #MathHist #aesthetics #MathArt #Mathematics

‘Form & Number: A History of Mathematical Beauty’ in print! Kind of. :-)

A correspondent with a bookbinding hobby emailed to say that they had printed and hand-bound the entire work. (They prefer to remain anonymous but have given me permission to post about their work.)

A photo is attached, showing five volumes and a temporary slipcase.

They chose to print the small-format version of ‘F&N’ [https://archive.org/details/cain_formandnumber_ebook_small], which has the same content as the primary large-format version [https://archive.org/details/cain_formandnumber_ebook_large] but uses a smaller page size and replaces sidenotes with footnotes, and so runs to something over 2000 pages.

The binding is ‘veg tanned leather (backed with washi paper) ... sewn with a long stitch and link stitching for the changeovers using .8mm linen thread’.

It is printed on 80 gsm paper, and masses 1430 grams in total.

The first volume contains the front matter; the second, third, and fourth volumes respectively contain parts I (purely historical), II (themes relevant today), and III (mathematical beauty in natural science); the fifth volume contains part IV (reflections) and the bibliography and indexes.

My first reaction on seeing ‘F&N’ printed and bound like this was amazement at the care and craftsmanship involved.

My second reaction was basically ‘Ye gods, did I really write all that?’ :-)

#BookBinding #craft #Mathematics #MathArt #MathHist #MathematicalBeauty #aesthetics

I have updated the annotated edition of G.H. Hardy’s ‘A Mathematician’s Apology’ [https://archive.org/details/hardy_annotated] on archive.org.

I have improved a couple of annotations, and the index now distinguishes references which are only to annotations. Another short contemporaneous review was pointed out to me and I have added it to the included survey of reviews that the ‘Apology’ received on its first publication.

The PDF of the new version is available for download immediately, and archive.org will re-generate the pages for the in-browser reader over the next few hours.

#MathHist #MathPhil #philosophy #GHHardy #MathematicalBeauty