Alan J. Cain

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

Feb 3 at 8:38amWeb
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References

• Archimedes. ‘The Works of Archimedes’, vol.I: ‘The Two Books on the Sphere and the Cylinder’. Translated into English, together with Eutocius' commentaries, with commentary, and critical edition of the diagrams. Cambridge University Press, 2004. ISBN: 978-0-521-66160-7

• A.J. Cain. ‘Form & Number: A History of Mathematical Beauty’. Lisbon, 2024. URL: https://archive.org/details/cain_formandnumber_ebook_large
pp.106–7.

• Cicero. ‘Tusculan Disputations’. Loeb Classical Library, no.141. Cambridge, MA: Harvard University Press, 1927. DOI: 10.4159/DLCL.marcus_tullius_cicero-tusculan_disputations.1927 § V.xxv.

• M. Dehn. 'The Mentality of the Mathematician: A Characterization'. In: The Mathematical Intelligencer. 5, no.2 (June 1983), pp.21–2. DOI: 10.1007/bf03023621

• Plutarch. ‘Marcellus’. In: ‘Lives’, vol.V. Loeb Classical Library, no.87. Cambridge, MA: Harvard University Press, 1917. ISBN: 978-0-674-99097-5. DOI: 10.4159/DLCL.plutarch-lives_marcellus.1917 Stephanus p.307; § XVII.7.

• I. Schneider. ‘Archimedes: Ingenieur, Naturwissenschaftler, Mathematiker’. 2nd edition. Mathematik im Kontext. Springer Spektrum. ISBN: 978-3-662-47129-6. pp.89-90

Image credits

• Cone/sphere/cylinder from ‘Form & Number’, figure 3.9.

• Paolo Barbotti, ‘Cicero discovering the tomb of Archimedes’. URL: https://commons.wikimedia.org/wiki/File:Cicero_discovering_tomb_of_Archimedes_(_Paolo_Barbotti_).jpeg [Public domain]

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Form & Number: A History of Mathematical Beauty (Ebook, large format) : Alan J. Cain : Free Download, Borrow, and Streaming : Internet Archive

This book offers a history of beauty in mathematics and of the study of beauty in mathematics. Its intention is to examine the historical development of the...

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