Introduction

Logarithmic spirals such as golden spirals (with a growth factor of φ = 1.6180339887, the golden ratio) are ubiquitous in nature and are seen in arrangements of leaves, seeds, pinecones and many different arrangements in nature [1]. Such spiral patterns are frequently observed and utilized in a variety of phenomena including galaxies, biological organisms, as well as turbulent flows [2]. Many have attributed mysticism to the presence of origins—the golden ratio was called τ (tau), the symbol of life, and was also known to ancient Egyptians (The ankh) and Hindus[3]. One of the viewpoints for the occurrence in plants was inherently protoplasmic i.e. “spirals” exist in nature through constitutional rather than external mechanical influences [4]. However, a team noted that metallic nanoparticles could be made to arrange themselves with floral spiral design and commented that the shape of spirals depended on size of particles, thickness of shell and rate of cooling [5]. Others have stimulated spiral wave forms in neuronal circuits of the human body [6]. While Fibonacci spirals are everywhere in nature, and indeed common in art and architecture—in a laboratory setting, spontaneous assembly of such patterns has rarely been realized in mechanical tests, and never before on skin [6]. Some researchers demonstrated Fibonacci spiral patterns could be reproduced through stress manipulation on the metal-core based shell microstructures and commented on the possible role of mechanical stress in influencing these spiral patterns [7]. And, in a study of rocks and tectonic plates, researchers noted spiral patterns of cracks—where the cracking arises not from twisting forces but from a progressing stress front [8].

As an animal biologist and skin cancer academic, the author's fascination has been with the fact that humans are the only animals to have scalp whorls on the top of their heads that follow spiral patterns seen in nature. Others have noted that of all mammals, only humans have hair whorls on the vertex of the scalp, and that each human individual must have a hair whorl [9]. Animals like horses do have whorls on the body or faces, but not on the top of their heads—and this is not universal in all horses—however, such patterns demonstrate less variability, and show more Mendelian forms of inheritance than those seen in humans [10]. In keeping with such genetic theories, the number of whorls on a horse’s face and the direction of such patterns are used to gauge equine temperament such as calmness, enthusiasm or wariness [11]. Notably, other primates like chimpanzees, that are evolutionarily closer to humans, do not have whorls [12]. In this experiment we re-created skin whorls by causing shearing forces due to rapid skin expansion, along a progressing front.

Materials and Methods

The Medium

The author spent the past year in testing and studying the properties of pigskin, and its suitability as an equivalent and comparable testing medium to human skin, before formally testing the mechanical theory of spiral formations in organic tissue. This involved considerable time taken to perfect the technique of create rapid dermo-epidermal shear by inflating a saline expander rapidly.

Pigskin has been noted to be the closest medium to human skin in several studies—Herron noted that pigs were the ideal dermatological models in skin surgery, wound healing, burns and skin therapies such as lasers [12]. Schmook and others compared human and animal skin and concluded that “in agreement with published data, pig skin appeared as the most suitable model for human skin: the fluxes through the skin and concentrations in the skin were of the same order of magnitude for both tissues” [13].

Before one can study skin bio-dynamics or surgical closure techniques, one needs a good understanding of the directions of skin tension lines of normal skin. Karl Langer (1819–1887), Professor of Anatomy in Vienna, was one of the earliest people to undertake such studies, when he undertook experiments to study physical and mechanical properties of human skin [14] in cadavers by making small circular incisions on skin—and then noting the directions they were distorted in—these ‘skin tension lines’ he termed as ‘cleavage lines’ [15]. Since Langer’s original study, many other variations of skin tension lines have been mapped out—and people have noted that in certain parts of the face, Langer’s lines deviate from relaxed skin tension lines—nevertheless Langer’s work remains the most important in any discussion regarding skin tension lines [16]. More recently, investigators studied porcine skin to assess the presence of ‘Langer’s Lines’ and concluded that, as in humans, Langer’s Lines do exist, and have specific patterns in pigskin [17] and are equally dynamic with movement—with the authors noting that in pigskin the ‘tension lines were oblique to the vertebra in the cephalic area, perpendicular to the vertebra in the middle torso, and parallel to the vertebra in the caudal area’ [17] One of the other factors that was considered prior to this testing was the relationship between Langer’s Lines and hair follicles—experimental studies have shown that there is a good correlation between Langer’s Lines and direction of hair streams [17] and this knowledge, hitherto helpful in studying wounds, also helped in the design of this experiment—especially in the planning of skin markings and incisions for this study.

The mechanical theory of scalp whorl formation suggests that during the 10^th to 12^th week of fetal life, the brain expansion is so rapid that it creates a shearing force between the two layers of skin, at the dermo-epidermal junction, which in turn causes the hair follicles to curve [18].While planning for this study, the author conducted expansion studies using saline expanders to understand the pressure required to create such shearing forces and the technique needed to replicate dermo-epidermal shear, and thereby the slanting of hair follicles. In a cross section of pigskin, as in Fig 1 one is able to observe the epidermis and a layer of adipose tissue that serves as a fatty ‘dermis’ layer. Using saline expanders at the margins of pigskin cross sections allowed us to observe dermo-epidermal shear. In this experiment, we found rapid expansion resulted in enough shearing force to separate the layers of skin. Slow expansion over hours, days or weeks did not result in creating enough ‘shear’ in pigskin to produce spirals. Pigskin has three separate fat layers, separated by two layers of fascia [19]. The uppermost layer of fat in pigskin corresponds to the dermis in human skin—and authors have coined the term ‘intradermal adipocytes’ to describe these cells, as the term accurately reflects both their developmental origin and anatomical location [20]. Therefore, for the purposes of this experiment, the author decided to see if the causation of shearing forces between the uppermost two layers of pigskin (the epidermis and fatty dermis) would result in the formation of spirals along an advancing front.

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Fig 1. Image shows saline expander, pigskin anatomy and curves formed by the advancing front of rapid expansion.

https://doi.org/10.1371/journal.pone.0162026.g001

Discussion

In our experiment, skin tracings of the advancing front (i.e. the non-sutured skin tension line that becomes deformed into curves under the force of rapid expansion) were marked out as detailed above (“Fig 1”). To the naked eye, we could see that we had achieved some degree of displacement of the original straight-line markings into curves (“Fig 2”). Interestingly, when we had traced out all the patterns and then superimposed all our tracings, one on top of each other to create a composite image, we noted a distinctive logarithmic spiral pattern. The results are presented here—the composite image created by overlaying all our tracings show a clear logarithmic spiral pattern, in our case very close to a golden spiral pattern seen in nature. “Fig 2” shows each individual tracing, followed by a composite image, finally overlaid by a golden spiral. Each tracing, when analyzed retrospectively did indeed conform to a portion of a golden spiral.

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Fig 2. Rapid expansion and skin line tracings; composite image created by overlaying tracings.

https://doi.org/10.1371/journal.pone.0162026.g002

This experiment demonstrates clearly that under certain conditions, rapid tissue expansion causes dermo-epidermal shear, which can result in the formation of spirals—in our case, nature’s own design: the golden spiral, perhaps nature’s own design for rapid organic expansion.

As discussed earlier, spiral formations abound in nature and in tissues; they may represent a feature of rapid expansion along an advancing front. After all, the opening of flowers is often rapid –for example, Hedera helix, the English Ivy, opens in about 5 minutes [24], and many flowers demonstrate spiral patterns around their inflorescence axis. The opening of flowers is already known to be due to cellular expansion—while floral openings can be classified as nocturnal, diurnal, single or repetitive, previously published papers have demonstrated that the opening of flowers is generally due to cell expansion [24]. As to the genetics of such rapid expansion, researchers have studied rapid intraoperative tissue expansion in mouse skin and found that in response to stretch caused by a balloon, similar to that used in our experiment, the following genes are induced—L1 (truncated long interspersed nucleotide element 1), myotubularin and insulin [25]. In another study on adult skin, researchers found a significant difference in 77 genes after expansion—with a significant finding being the expression of regeneration related genes, such as HOXA5, HOXB2 and AP1, after tissue expansion, with implications for further research into skin regeneration [26]. Indeed studies in monozygotic twins have shown individual variations of scalp whorls, and therefore both skin stretch and the subsequent gene expression can be considered responsible for the formation of spirals [27].

Hakim and colleagues have studied spiral wave meander in excitable media and suggest that the dynamics of spiral waves can be reduced to a nonlinear equation of motion for the wave tip that is asymptotically exact in a parameter range where the motion takes place around a large central core region [28]. Conceptually, that is not unlike our findings in this experiment. Knowledge gained from this experimental study has helped the author design a new technique for closing scalp defects based on the golden spiral where the motion takes place around a central core (measurements show cutaneous flaps raised in this way can stretch more than when raised in other shapes, and therefore can help in reconstruction of larger areas of scalp soft tissue), and this will be the subject of a separate paper in a cutaneous surgery journal.

Conclusion

The mechanical theory of scalp expansion causing whorls, while hypothesized, has never been re-created, to our knowledge, in a controlled experiment on skin. This study therefore represents a substantial advance in the understanding of both the formation of scalp whorls, skin biomechanics and specifically, the golden spiral pattern—which is widely found in galaxies and plants—and is probably one of nature’s own designs for rapid expansion of organic matter, along an advancing front. Given this this is an interesting and exciting new discovery, and may lead to further research in other fields of scientific endeavor, I am presenting this to the wider scientific community.

Acknowledgments

I would like to thank Jan Gardner at Auckland University’s Advanced Clinical Skills Centre for helping obtain pigskin and use of the Centre’s facilities for my experiments. This paper resulted from a study that forms part of my Ph. D research project at the University of Queensland’s School of Medicine and I would like to acknowledge my supervisor, Assoc. Professor Cliff Rosendahl. And finally, I would like to thank Ryan Butler, Auckland University of Technology, for his help with photography to document this experiment.