The Rogers–Ramanujan Continued Fraction and the Icosahedron

Bruce Bartlett ()
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Bruce Bartlett: Stellenbosch University

A chapter in Handbook of Visual, Experimental and Computational Mathematics, 2026, pp 367-389 from Springer

Abstract: Abstract In his celebrated letter to Hardy in 1913, Ramanujan wrote down a continued fraction (independently discovered by Rogers twenty years earlier) and stated some remarkable evaluations of it for special values of τ ∈ ℍ $$\tau \in \mathbb {H}$$ . Duke has explained Duke (2005) that these special evaluations arise because the Rogers-Ramanujan continued fraction is equivariant under the group G of rotations of the icosahedron. We show that this equivariance can be understood as an explicit bijection between the 6 vertex axes of the icosahedron and the 6 points in the projective line over the field with 5 elements. Moreover we show that just as rotations in G preserve the angles between the icosahedron axes, the transformations in PSL ( 2 , 5 ) $$PSL(2,5)$$ preserve the cross ratios of the duad synthemes in the projective line.

Keywords: Rogers-Ramanujan; Icosahedron; Hauptmodul; Modular curve (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16368-4_60

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DOI: 10.1007/978-3-032-16368-4_60

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