bibkey: joyce1997euclidpentagon authors: Euclid; D. E. Joyce (ed.) year: 1997 title: Elements, Book XIII, Proposition 8 doi: null url: https://mathcs.clarku.edu/~djoyce/elements/bookXIII/propXIII8.html claim: A regular pentagon’s crossing diagonals are cut in extreme and mean ratio, with the greater segment equal to a side. strata_touched: [] license: citation-only triage: anchor

Elements, Book XIII, Proposition 8

The primary proposition in David Joyce’s edition states and proves the extreme and mean division of crossing diagonals in an equilateral and equiangular pentagon. Its greater segment is a pentagon side. Consequently the diagonal-to-side ratio $r>0$ obeys $r^2=r+1$, hence $r=(1+\sqrt{5})/2$.

The proposition supplies the classical pentagon ratio. Connecting that quadratic to the spectrum of the specific FIB substitution matrix is an algebraic identification; it does not transport polygon symmetries to the FIB syntax or define physical lengths.