A week or so of exhaustive LatticeReducing (FindIntegerNullVector cr*ps out around dimension 50) "factorizes" the big surd in eta(e^(-80 pi)) (just south of Monterey) into binomial surds: DedekindEta[ 40 I] == ((3/(5 (1 + 2 Sqrt[2/5])))^(7/8) Sqrt[ 7/((1 + 3/Sqrt[2]) (1 + Sqrt[2]/5^( 1/4)))] (((1 + 3 Sqrt[2/5]) (1 + 1/5^(3/4)) (1 + 2/5^( 3/4)) (1 + 3/5^(3/4)) (1 + Sqrt[2]/5^(3/4)))/((1 + ( 2 2^(1/4))/Sqrt[5]) (3 + 5^(1/4)) (1 + 5^(1/4)/Sqrt[ 2]) (1 + 2^(3/4) 5^(1/4)) (1 + 2^(1/4) Sqrt[5]) (4 + 2^(3/4) Sqrt[5]) (1 + 5^(3/4)) (2 + 5^(3/4)) (1 + 5^(3/4)/ Sqrt[2]) (2 + 10^(1/4))))^( 1/4) (((1 + (3 Sqrt[5/2])/2) (1 + 8/5^(3/4)) (1 + 2^(1/4)/Sqrt[ 5]) (4 + 5^(1/4)) (4 + Sqrt[2] 5^(1/4)) (4 + Sqrt[5]) (4 + 2^(1/4) Sqrt[5]))/((1 + 2 (2/5)^(1/4)) (1 + 4/Sqrt[ 5]) (1 + (2 2^(3/4))/Sqrt[5]) (1 + 4/5^(1/4)) (1 + ( 2 Sqrt[2])/5^(1/4)) (1 + (2 Sqrt[5])/3) (1 + Sqrt[5]/2^( 1/4)) (3 + 5^(3/4)) (1 + 1/3 2^(1/4) 5^(3/4)) (5 + 3 10^(1/4)) (4 + 5 10^(1/4))))^(1/8) Gamma[1/4])/(4 (1 + 1/Sqrt[2])^( 5/4) ((1 + (2/5)^(1/4)) (2 + Sqrt[5]) (3 + Sqrt[5]))^( 5/8) ((1 + (2 5^(1/4))/3) (1 + 5^(1/4)) (1 + Sqrt[5]) (1 + 10^( 1/4)))^(3/8) ((1 + Sqrt[2/5]) (1 + 1/2^(1/4)) \[Pi])^(3/4)) For what algebraic numbers are such factorizations possible? (Massively nonunique, in this case.) (Apropos Allan's roadogeny (pathogenesis?), I could swear seeing a pointer here to a website featuring a cellular type (celluloid?) algorithm for synthesizing diagrams strongly resembling maps of towns.) --rwg