The Meta-golden Ratio Chi

Dirk Huylebrouck
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014)
Pages 151–158 Regular Papers


Based on artistic interpretations, art professor Christopher Bartlett (Towson University, USA) independently rediscovered a mathematical constant called the ‘meta-golden section’, which had been very succinctly described 2 years earlier by Clark Kimberling. Bartlett called it ‘the chi ratio’ and denoted it by χ (the letter following φ, the golden section, in the Greek alphabet). In contrast to mathematician Kimberling, Bartlett motivated his finding on artistic considerations. They may be subject to criticism similar to the ‘golden ratio debunking’, but here we focus on showing that his chi ratio is interesting as a number as such, with pleasant geometric properties, just as the golden ratio. Moreover, Bartlett's construction of proportional rectangles using perpendicular diagonals, which is at the basis of his chi ratio, has interesting references in architecture and in art.