Fibonacci Sequence in Flexion-Extension Complex of Index Digit

Introduction

The human hand is a biomechanically efficient functional unit. The manus is a complex made up of a score of intrinsic muscles, twenty seven individual osseous elements articulating through a series of joints. With inputs by ligaments, tendons, bursae assisting the movements, the hand is indeed a marvel in kinetic engineering, Gifted through evolution to perform the ‘opposition’ exercise, the phalangeal, metacarpal, and carpal condylar, saddle and gliding plane joints of the palm flex, extend, abduct and adduct¹.

Prone to injury and malfunction, the loss of ‘man days’ due to hand lesions, cause serious economic loss to industry and economy in many agrarian and industrial nations. Much analytical work has been done by kinesiologists on the function and disabilities of this often used component of the upper limb. In this brief paper we present yet another interesting observation on the biomechanics of the digit.

Materials & Methods

To scale measurements were made of the shadow of a supine hand during performance of simple flexion and extension of the digits, especially, the index. The shadow of the arc described by the exercise in the form of excursion of the flexing index tip projected on a wall mounted white screen board was traced. The resultant arc was analyzed and linear measurements (in mms) of each individual bone element were recorded, from distal phalanx to base of second metacarpal for index digit.

Mathematical models from values obtained were graphically reproduced through Computerized Assisted Designing (CAD).

Observations

Measurements of individual bones show that, despite the soft tissue elements that bar precise and accurate measures, the interarticular distances between the distal, intermediate, proximal phalanges and the metacarpal ends, repeat the Fibonacci sequence. That is the lengths were 17 mm, 28 mm, 45 mm and 72 mm respectively. Put in Fibonacci sequence series, 17 +28+42+73, or at every joint an increment of 1.6 to 1 was evident (Figures 1 & 2).

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Figure 1

The geometrical proportions of individual bones of the hand

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Figure 2

The cochlear spiral configuration of index tip during extension-flexion

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Up to 120 mms., the arc described by the index tip shadow forms a cochlear spiral pattern, beyond which it becomes erratic (chaotic). Flexion at index interphalangeal, metacarpopahalangeal joints spiraled across a hub or axis, which was horizontally mobile for about 10 mms in a complete range of flexion-extension.

Taking the midpoint of the axis, as referral point, when the arc was plotted through CAD, the spiral created displayed a splayed-out form, akin to the molluscan shell.

Measurements at fixed reference points along the spiral show a progressive increment at angulation or its reverse, in extension and flexion of index. Approximately, the increment followed the ‘golden mean’ ratio first enunciated by the European mathematician, Fibonacci, centuries ago, which is around 1: 1.63 (Figures 3 & 4).

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Figure 3

CAD plot of arc produced by flexion- extension of index digit

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Discussion

The need for better utilization of biomedical devices and mathematical models, coupled with a more comprehensive interdisciplinary input, could lead to more understanding, and there-from, a better rehabilitation, of the kinetics of the human hand^{2, 3}. The two measurements presented here demonstrate that function follows precise mathematical formulae, and that such mathematically sound base for movement is probably directly related to the ratios, in the very construction of the skeletal components.

Comparing such recreated spirals drawn from disabled or treated hands, to normal ones, could aid the rehabilitation physiotherapy procedures^{5, 6}. It is quite evident that a normal hand functions in a synchronized and predictable mathematical model⁴, a fact of much significance to biomedical engineers.