A Retrospective Radiographic Study of Metacarpal and Phalangeal Length Proportions: Testing the Golden Ratio, Fibonacci, and Lucas Models

AimsThis study tested whether the radiographic lengths of the metacarpals and phalanges conform to five previously proposed proportional models based on the Golden Ratio, the Fibonacci sequence, and the Lucas sequence in adult and paediatric hands. MethodsA retrospective observational study was performed of posteroanterior hand radiographs obtained at a district general hospital between January 1 and March 9, 2024.Of 467 radiographic examinations screened, 413 hands from 315 patients met the inclusion criteria: 353 adult hands (patients aged 19-95 years) and 60 paediatric hands (aged 6-18 years).Metacarpal and phalangeal lengths were measured on the picture archiving and communication system to 0.1 mm.Five a priori hypotheses were tested: Golden Ratio relationships between adjacent bones, Lucas sequence proportions (1:1.3:2.3:3.6), a 1:1:2 Fibonacci pattern in the little finger, proportional balance between bone segments, and additive sequence behaviour.Each mean ratio or difference was compared with its predicted value using a two-tailed one-sample z-test and its 95% confidence interval (CI), in the whole sample and six subgroups (336 comparisons).A predicted value lying within the 95% CI was taken as not rejected, not as proof of equivalence. ResultsOf 336 comparisons, 324 differed significantly from the predicted values, and 306 remained significant after adjustment (311 at p < 0.001).No consistent Golden Ratio relationships were identified in adjacent bone segments.Subgroup analysis demonstrated isolated approximations, particularly in the proximal-to-middle phalanx ratio of the middle finger in paediatric and adult male groups.The proposed Lucas sequence pattern was largely absent, with only the paediatric ring finger demonstrating a partial correspondence.The little finger did not demonstrate a consistent Fibonacci relationship.Evidence of proportional balance between phalanges was limited to isolated observations, notably in the adult index finger and paediatric middle finger.Additive sequence relationships were similarly restricted to these subgroups. ConclusionIn this single-centre radiographic sample, metacarpal and phalangeal lengths did not consistently conform to the Golden Ratio, Fibonacci sequence, or Lucas sequence.The few subgroup ratios close to the predicted values were isolated, occurred mostly in the smallest subgroup, and are compatible with chance across multiple comparisons.They should be treated as hypothesis-generating, and confirmation in multicentre and longitudinal samples is required.