



ORIGINAL ARTICLE 

Year : 2021  Volume
: 4
 Issue : 1  Page : 5863 

Fibonacci's mathematical sequence predicts functional and actual lengths of the phalanges of the hand
Andrew S Miller, Rhodri Gwyn, Dhritiman Bhattacharjee, Louisa N Banks
Department of Trauma and Orthopaedic, Glan Clwyd Hospital, Bodelwyddan, North Wales, UK
Date of Submission  10Jun2020 
Date of Decision  07Oct2020 
Date of Acceptance  19Nov2020 
Date of Web Publication  06Jan2021 
Correspondence Address: Louisa N Banks Room 27, Ivor Lewis Building, Glan Clwyd Hospital, Sarn Lane, North Wales, LL18 5UJ UK
Source of Support: None, Conflict of Interest: None
DOI: 10.4103/JNSM.JNSM_65_20
Background: Fibonacci described a mathematical sequence starting 0, 1, 1, 2, 3, 5, 8… where each sub sequential number is the sum of the two preceding numbers. The golden ratio (Phi, Φ = 1.618) is the ratio of two consecutive numbers in the sequence. Here, we investigate whether there is a relationship between these mathematical sequences or ratios and functional or actual lengths of the digits. Methods: Two hundred radiographs of the hand were reviewed by three independent reviewers (interobserver correlation r >0.98), and the actual and functional phalangeal lengths were measured. Results: Both the functional and actual lengths of the phalanges of the little finger followed Fibonacci's sequence. The index, middle, and ring fingers followed a mathematical sequence related to the Fibonacci's sequence. We were not able to demonstrate any direct relationship between phalangeal length and the golden ration (Φ). The sum of lengths of the distal and middle phalanx equals the length of the proximal phalanx with great accuracy. Conclusions: This is useful in many surgical situations (congenital deformities, polytrauma to the hands, and arthroplasty).
Keywords: Fibonacci, functional lengths, golden ratio, phalanges
How to cite this article: Miller AS, Gwyn R, Bhattacharjee D, Banks LN. Fibonacci's mathematical sequence predicts functional and actual lengths of the phalanges of the hand. J Nat Sci Med 2021;4:5863 
How to cite this URL: Miller AS, Gwyn R, Bhattacharjee D, Banks LN. Fibonacci's mathematical sequence predicts functional and actual lengths of the phalanges of the hand. J Nat Sci Med [serial online] 2021 [cited 2021 Mar 3];4:5863. Available from: https://www.jnsmonline.org/text.asp?2021/4/1/58/306263 
Introduction   
Fibonacci (the Italian Leonardo Pisano Bigollo [c. 1170–c. 1250]) has been considered to be; “the most talented Western mathematician of Middle Ages”^{[1]} Fibonacci described the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34..... a specific example of a Lucas sequence where each number in the sequence is the sum of the previous two numbers (x_{n} = x_{(n − 1)} + x_{(n − 2)}).
The golden ratio (phi, Φ) is a proportion of length of two line segments, where the ratio between the longer and shorter lengths equals the ratio of the sum of the two line lengths and the longer line length [Figure 1]. The golden ratio is also known as the golden mean, extreme and mean ration, medial section, divine proportion, divine section, golden proportion, golden cut, and golden number.^{[2],[3],[4],[5],[6],[7]}  Figure 1: The Golden ratio, illustration adapted from Hutchinson, (2010)
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The ratio of each successive pair of numbers in Fibonacci's sequence approximates phi (1.618....) as 5 divided by 3 is 1.66…, 3 divided by 2 is 1.625, and 8 divided by 5 is 1.60 and so on. The ratios of the successive numbers in the Fibonacci sequence quickly converge on phi (Φ). After the 40^{th} number in the sequence, the ratio is accurate to 15 decimal places: 1.618033988749895.^{[8]}
Phi is ubiquitous throughout nature, geometry, and commercially. Examples include the Toyota™ car logo, the spirals of a nautilus shell and the attack pattern of a Harrier Hawk. Even Leonardo Da Vinci's famous “Vetruvian Man” [Figure 2] shows three distinct sets of Golden Rectangles; the head, the torso, and the legs each increasing in width by a factor of Phi.^{[7],[8]}  Figure 2: The Vetruvian Man “(The Man in Action)” by Leonardo Da Vinci. [b. taken from AbuTaieh E, et al., 2012]
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There has been some debate^{[9],[10]} as to whether Fibonacci's sequence, and ultimately, the golden ratio can be applied to the hand, so this paper endeavors to investigate as to whether there is a relationship with regard to both radiographic functional and actual lengths of the digits. Should such a relationship exist it would undoubtedly aid clinical decisionmaking and preoperative planning across many different facets of hand surgery.
Methods   
Two hundred consecutive, standardized, posteroanterior (PA) hand radiographs (taken at a single institution) were reviewed by three independent reviewers (ASM, RG, DB). Each had the actual (P1, P2, and P3) and functional (F1, F2, and F3) [Figure 3] phalangeal lengths of the index (II), middle (III), ring (IV), and little (V) fingers measured by the three reviewers. All data were tested for Gaussain distribution (normal distribution) with D'Agostino Pearson test before using Pearson's correlation test to demonstrate interobserver r values. Statistical analysis was performed using GraphPad Prism version 9.0.0 Software, Inc., California, USA. Due to different measurement techniques, actual and functional lengths were analyzed independently. Actual lengths were measured from the tip to base of each phalanx: P3, distal phalanx, P2, middle phalanx and P3, proximal phalanx [[Figure 3] to illustrate how these were measured by each reviewer]. We have taken the functional length as the distances between the axes of rotation between the interphalangeal (IP) and metacarpophalangeal (MCP) joints. The functional lengths were measured as previously described by Hamilton and Dunsmuir.^{[9]} F3, tip of distal phalanx to center of rotation of the distal IP (DIP) joint; F2, center of rotation of the DIP joint to center of rotation of the proximal IP (PIP) joint; F1, center of rotation of the PIP joint to the center of rotation of metacarpal head [Figure 3]. All measurements were made in millimeters to one decimal place. Consecutive hand radiographs within the institution were reviewed and radiographs excluded if they: Were on subjects under the age of 18 years had any acute or healed fractures; had arthritis of the digits or MCP joints; or any congenital hand deformities. All radiographs were taken in a standardized way by the radiographers, with the middle (third) MCP joint centred on the image receptor. There was no magnification correction.  Figure 3: Radiograph to demonstrate length ratios of the (a) actual (P1, P2, and P3) and (b) functional (F1, F2, and F3) measurements
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All measurements were performed on a picture archiving and communication system (PACS) software and entered into a passwordprotected database. The PACS system tools allow measurements in millimeters to two decimal places. Each of the reviewers submitted their data to the first author (ASM) who subsequently collated all the data. Age and sex of the patient and side were recorded. Mean lengths, standard deviation, and 95% confidence intervals were produced. In order to demonstrate the presence of a Lucas sequence (P1 = P2 + P3), the two shorter lengths were subtracted from the longest length (P1P2P3 = 0). As the answer to this equation approaches zero, the Lucas sequence is proven more accurate. Ratios of each consecutive measurement were also produced to ascertain if there was a relationship with the golden ratio (Phi, Φ = 1.618). The study was approved by our institutional review board (IRB).
Results   
The age showed a mean of 42.8 years (18–84 years), 125 (62.5%) patients were female, and 119 (59.5%) of the radiographs were of right hands.
Interobserver correlation was excellent for both actual and functional measurements (r >0.98) was a strong correlation between all observers. Mean actual (P) and functional (F) lengths are demonstrated in [Table 1]. Actual and functional lengths of the little finger (V) followed the sequence 1:1:2. Functional lengths of the phalanges of the index (II), middle (III) and ring (IV) fingers all followed the sequence 1:1.3:2.3 as did actual lengths in the index (II) finger. Actual lengths for the middle (III) and ring (IV) followed the sequence 1:1.4:2.4 [Figure 4].  Table 1: Mean actual and functional lengths of the phalanges of each digit in millimeters, standard deviation, and 95% confidence intervals
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 Figure 4: Functional lengths of the phalanges as determined by Hamilton (Journal of Hand Surgery [Edinburgh, Scotland] 27:546, 2002)
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When the two distal measurements were subtracted from the most proximal measurement (P1P2P3 and F1F2F3), the results closely converged on zero. Mean actual measurements; index (II) 0.5 mm, middle (III) – 0.3 mm, ring (IV) – 1.1 mm, and little (V) – 0.9 mm. Standard deviations for actual measurements ranged from 1.6 to 1.8 mm. Functional measurements; index (II) 0.0 mm, middle (III) – 0.1 mm, ring – 0.5, little (V) 0.3 mm. Standard deviation for this group was slightly larger ranging from 1.5 mm to 3.3 mm. Confidence intervals remained narrow across all measurements [Table 1].
No relationship between the golden ratio (Phi, Φ = 1.618) and consecutive phalangeal measurements could be identified. Ratio ranged from 1.00 to 1.99 [Table 2].
Discussion   
Littler published a paper in 1973 proposing that the motion of the tips of the fingers follow an equiangular spiral.^{[11]} He also thought that the lengths of the phalanges followed Fibonacci's series [Figure 5]. This article was revisited in 2010 by Hutchison and Hutchison who claimed that; “the functional lengths of the phalanges of the little finger actually do follow a Fibonacci series and that the functional lengths of the index, long and ring fingers follow a mathematical relative of the Fibonacci series.^{[10]}”  Figure 5: Taken from Littler's article. The measurements from the centers of joint rotation and the equiangular spirals follow the path of the fingertip and are similar to the form of the nautilus shell (Hand 5 (3):188, 1973;^{[12]})
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There are anatomical differences between the IP and MCP articulations of the digits and even between the joints at the same level for each digit. The literature suggests that there are several reasons for the differences between the articulations.^{[10],[12],[13],[14],[15],[16]} These include the shape and orientation of the joints, the synovial insertion, the disposition of the collateral ligaments, “the degree of play in the volar plate” and the surrounding soft tissues.^{[10],[12],[13],[14],[15],[16]}
In Hutchinson and Hutchinson's critique of Littler's paper, they point out that Littler gave no explanation for the lengths that were assigned to the phalanges and as such, it is not possible to know whether these were theoretical, or measured, or indeed whether they were functional or actual lengths. They, therefore, questioned the validity of Littler's 1973 article.^{[10]}
Hamilton looked at the functional length of the bones of the digits (197 participants) and found that the ratios of the DIPtip/PIPDIP/MCPPIP (DIP, PIP) was 1:1:2 for the little finger and 1:1.2:2.3 for the other fingers.^{[9]} Hutchinson claims that when Hamilton disputed Littler's theories, it was because the 1973 paper had been “misinterpreted” and felt that their data did in fact correspond with the Fibonacci series.^{[10]} So even though Hamilton concluded that there was no relationship with the Fibonacci sequence, on reexamining their data, Hutchinson proposed that, when applied to the index, middle, and ring digits, Hamilton's study had a related series of x, 1.3x, 2.3 × 9.10.
Park et al. also looked at 100 PA hand radiographs.^{[17]} They subtracted the length of the proximal phalanx (P1) from the sum of the lengths of the middle phalanx (P2) and the distal phalanx (P3). They felt that there was a large variation in phalangeal relationships in the little finger and concluded that actual bone lengths did not correspond with the Fibonacci sequence, but suggested that the functional lengths might.
Markley commented on Park's hypothesis the same year and agreed that functional lengths were more likely to correlate with a Fibonacci or Lucas series. He commented that; “it is likely that the radius vectors determining the equiangular sweep of digital flexion are a reflection of functional interaxial lengths.”^{[18]}
Gupta et al. looked at eight normal subjects and experimentally used motion analysis to look at the motion path of the fingertips of the digits.^{[19]} They placed retroflective markers at the proximal and distal aspects of the index (II) to little (V) metacarpals, the PIP and DIP joint lines and in the center of the nail plate 2 mm from the tip of the digit. They confirmed that the motion path of the finger does follow the equiangular spiral during flexion and extension as proposed by Littler (1973).^{[19]} However, as Markley pointed out that they did not report the tangent angles or proportionality ratios.^{[18]}
Choo et al. looked at 100 male Chinese subjects and felt that “functional human hand proportion, as defined by flexion creases” was approximated by the Fibonacci series (1:1:2).^{[20]} They also concluded that their results needed; “to be interpreted with caution” given the fact that hand proportion varies with age, gender, and ethnicity. Hamilton, however, felt that after looking at three cadaver specimens and 197 AP hand radiographs, that the relationships that they observed between the phalanges would be “constant for all human hands.”^{[9]}
Our results support the conclusions made by Hamilton and Choo et al. that the functional lengths of the little finger do indeed follow the sequence 1:1:2.^{[9],[20]} We can also corroborate with Hamilton's conclusion that functional lengths of the index to ring finger follow a specific Lucas sequence 1:1.3:2.3 [Figure 4].
Previous authors^{[17]} have suggested that there is no correlation between any Lucas sequence and actual phalangeal lengths. Our study, however, assessed more hand radiographs than any previous paper (n = 200). We have demonstrated that actual phalangeal measurements for the index (II) and little (V) finger follow the same sequence as the functional measurements. The little (V) finger following the sequence 1:1:2 and index following the sequence 1:1.3:2.3. The sequences for the actual lengths of the middle (III) and ring (IV) finger (1:1.4:2.4) do not correlate exactly with the sequences for the functional lengths (1:1.3:2.3).
Of greatest interest however is the fact that our results have shown that the lengths of P2 plus P3 equal the length of the corresponding P1 with an average accuracy of around 1 mm (−1.10.5 mm). This is also true of the functional lengths where the measurements of F2 plus F3 equal F1 (–0.50.0 mm). This has been observed in the literature when looking at the A1 pulley and surface landmarks.^{[21]}
Study limitations
The mean actual lengths of the middle and ring finger followed a Lucas sequence very close to Littler's sequence 1:1.4:2.4. This inconsistency may be due to a radiographic anomaly causing relative shortening of the distal phalanx in relation to the proximal of the longest fingers. Potential explanations include the parabolic effect of the radiograph, which is centred on the middle of the hand; or the tendency for the DIP joints of the middle and ring finger to be held in slight hyperextension as the patient's hand is placed on the radiographic cassette. It is clear that the sequence predicting the lengths of the phalanges of the little finger differs from that of the remaining fingers. What is not clear is why this is the case and is certainly an area of interest for further study. Larger patient numbers and pressure transducers placed on the fingers during radiographs or landmark measurements of patient's hands may help to answer this question in the future.
Conclusions   
There has been much debate since Littler's original paper in 1973^{[11]} as to whether any relationship does in fact exist between Fibonacci's sequence, the golden ratio and the hand.
The finding of greatest clinical significance is the fact that lengths of the distal and intermediate phalanx equal that of the proximal phalanx extremely accurately. This is the case for both actual and functional measurements of all the fingers. This is useful for the planning of congenital deformity surgery, such as reconstruction of failure of formation, undergrowth or overgrowth of the digits, where this information can be used to predict the length of the reconstructed digit. It is also useful where there has been polytrauma to the hand(s) so that this algorithm can predict the actual lengths of each of the digits when planning reconstruction or fixation where there has been bone loss. Knowing the predicted actual lengths of the digits is also useful in planning arthroplasty surgery such as PIP or metacarpal phalangeal joint replacements so that accurate placement of prostheses can be achieved. These can be difficult to appreciate if the arthritic process has deformed multiple joints. This is the first study, to our knowledge; to study both actual lengths and functional phalangeal lengths in the same hand to see if any relationship does exists.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5]
[Table 1], [Table 2]
