Fundamental ratios and logarithmic periodicity in human limb bones

Abstract

Fundamental mathematical relationships are widespread in biology yet there is little information on this topic with regard to human limb bone lengths and none related to human limb bone volumes. Forty-six sets of ipsilateral upper and lower limb long bones and third digit short bones were imaged by computed tomography. Maximum bone lengths were measured manually and individual bone volumes calculated from computed tomography images using a stereologic method. Length ratios of femur : tibia and humerus : ulna were remarkably similar (1.21 and 1.22, respectively) and varied little (<7%) between individuals. The volume ratio of femur : tibia was approximately half that of humerus : ulna (1.58 and 3.28, respectively; P < 0.0001). Lower limb bone volume ratios varied much more than upper limb ratios. The relationship between bone length and volume was found to be well described by power laws, with R 2 values ranging from 0.983 to 0.995.The most striking finding was a logarithmic periodicity in bone length moving from distal to proximal up the limb (upper limb k = 0.72, lower limb k = 0.93). These novel data suggest that human limb bone lengths and volumes follow fundamental and highly conserved mathematical relationships, which may contribute to our understanding of normal and disordered growth, stature estimation, and biomechanics.

Exploratory Evidence of Sex-Dimorphic Associations of the Ulna-to-Fibula Ratio, a Potential Marker of Pubertal Sex Steroid Exposure, with the Implicit Need for Power

“…In our view, our results suggest that long bone length, particularly the differential growth effect reflected in UFR, is a potential marker of pubertal organizing hormone effects. Research pertaining to forming limb ratios is rare, which is surprising given the very conserved mathematical relationships they yield (Pietak et al 2013). And if such ratios are calculated, researchers often seem to focus on within-extremity ratios like between ulna and radius (or humerus) or between tibia and fibula (Brons et al 1990;Pietak et al 2013).…”

Is there a relationship between the lengths of the ipsilateral clavicle and the ulna? An anthropometric and statistical study

“…The anthropometric relations between the clavicle and the ulna is an unknown issue and there are very limited studies focusing on this issue [1,2,5]. Our study revealed that there are fixed proportions between ulna and the clavicle lengths.…”

“…Mathematical relationships between bones lengths and volumes have been evaluated to improve our understanding of normal and disordered growth, stature estimation, and biomechanics [1].…”

“…This approach known as "process structualism" advocates exploring regularities in biology as, this regularities may be very important [1,2]. Dogan and Aydınlioglu et al [3,4] had found evidence of fixed proportions between the length and width of metacarpals and phalanges in the foot.…”

Michelangelo, the Last Judgment fresco, Saint Bartholomew and the Golden Ratio

“…But fascination with the GR is not confined to mathematicians. Biologists, artists, musicians, architects, psychologists, historians, and even mystics have pondered and debated the basis of its ubiquity and appeal (Livio, 2002;Lu and Steinhardt, 2007;Newell et al, 2008;Pietak et al, 2013;De Campos, 2014).…”