Browse Subject Areas

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

Golden Spirals and Scalp Whorls: Nature’s Own Design for Rapid Expansion

  • Sharad P. Paul

    Affiliations School of Medicine, University of Queensland, 288 Herston Road Herston QLD 4006. Australia, Faculty of Surgery, University of Auckland, 2 Park Road, Grafton, Auckland 1023, New Zealand, Auckland University of Technology, 55 Wellesley St E, Auckland 1010, New Zealand

Golden Spirals and Scalp Whorls: Nature’s Own Design for Rapid Expansion

  • Sharad P. Paul


This paper documents what began as an exercise in curiosity—logarithmic spiral designs abound in nature—in galaxies, flowers, even pinecones, and on human scalps as whorls. Why are humans the only primates to have whorls on the scalp? Is the formation of scalp whorls mechanical or genetic? A mechanical theory has long been postulated– the mechanical theory suggests that hair whorl patterning is determined by the tension on the epidermis during rapid expansion of the cranium while the hair follicle is growing downwards—however, this has never before, to the author's knowledge, been experimentally proven conclusively. We found, that under certain conditions, we were able to experimentally recreate spirals on the scalp to demonstrate that the basis of scalp whorls is indeed mechanical—and that logarithmic spirals may be nature’s own design for rapid expansion of organic tissues. Given our experiments only created whorls when certain conditions were satisfied (and not in others), they have given us great insight into the mechanical formation of skin whorls and the physiology of skin stretch. We believe that these findings will lead to many more advances in understanding skin dynamics and indeed the changes that occur in tissue when confronted by stretch.


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 10th to 12th 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.

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

The Experiment

Twenty freshly slaughtered pig bellies were obtained from an abattoir (supplied by FreshPork Northern, with the animals sourced from Timaru, New Zealand). Two parallel skin lines were marked along hair-follicle directions on pigskin, thereby indicating two parallel skin tension lines, about 2 cm apart. These were marked along ‘Langer’s Lines’ i.e. cleavage lines on the pig and not along wrinkle lines (photographs show that in some cases, wrinkle lines ran across the skin tension lines). Permanent indelible ink markers were used for each of these markings. A 15 ml silicone tissue expander was used in each case (Shanghai Wushan Industrial Co., Ltd, under license from Donasis Bio Labs, Japan). In this experiment, the author deliberately used a rectangular silicone expander to ensure that any spirals formed were not merely due to the shape of the implant used. A 2cm incision was made in the skin, and was deepened to the muscle layer below the ‘dermis’ in the pig. A pocket was created, only wide enough to accommodate the collapsed silicone expander—just wide enough to fit the silicone expander that was firmly compressed and squashed into position. The incision was closed in layers. Care was taken to incorporate deeper layers within the deeper sutures to make sure that the ‘advancing front’ due to this rapid tissue expansion would be in one direction only (away from the suture-line) i.e. one of the two tension lines marked would be fixed to underlying tissues and not capable of expansion. The expanders were filled with saline rapidly (for purposes of this experiment, from previous testing, we defined this as expansion of the silicone expander with 10 ml of saline in under 2 seconds, followed by a further expansion to full capacity after 2 minutes). Tracing paper was placed over the pigskin and the curvature of the advancing line due to tissue expansion was marked out. The process was identical in each case, except we used a different pig belly in each case, and markings were traced using different colored inks (“Fig 1”)

The reason for the two-stage rapid expansion is because such staged protocols have been established in surgery for acute tissue expansion [21], with many authors supporting this method of rapid expansion to achieve additional skin cover during reconstructive surgery [22]. Further, during the testing phase, this study found this technique consistently caused enough dermo-epidermal shear to cause displacement of the fatty dermis and the creation of whorl patterns. During such acute expansion of tissue, biological and mechanical ‘creep’ occurs, and authors have postulated that it is the displacement of water from the collagen network and micro-fragmentation of elastic fibers that makes skin more viscous [23]. To mitigate any variability of technique and ensure accuracy, the same person (in this case the author) was the only person performing all the testing.


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.

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

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.


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.


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.

Author Contributions

  1. Conceptualization: SP.
  2. Data curation: SP.
  3. Formal analysis: SP.
  4. Funding acquisition: SP.
  5. Investigation: SP.
  6. Methodology: SP.
  7. Project administration: SP.
  8. Resources: SP.
  9. Software: SP.
  10. Supervision: SP.
  11. Validation: SP.
  12. Visualization: SP.
  13. Writing – original draft: SP.
  14. Writing – review & editing: SP.


  1. 1. Naylor M. (2002) Golden, “√2, and φ flowers: A spiral story” Mathematics Magazine. 75(3), pp. 163–172
  2. 2. Catrakis H.J. (2011), "The Logarithmic Spiral: Mathematical Aspects and Modeling in Turbulence", Journal of Mathematics Research, 3 (3) pp. 3–11.
  3. 3. Ingrouille M. (1994)"Steps Up the Botanic Spirals", Nature, Book Reviews. 369 (7052) pp. 718
  4. 4. “Spirals in nature” (1933) Journal of the Franklin Institute. 215(3) pp. 352
  5. 5. "Natural spirals", (2005), Nature, 436 (7052) pp. 757
  6. 6. Ma J, Hu B, Wang C, Jin W. “Simulating the formation of spiral wave in the neuronal system.” Nonlinear Dynamics. 73, (1, 2) pp. 73–83
  7. 7. Li Chaorong and Ji Ailing and Cao Zexian. (2007) Stressed Fibonacci spiral patterns of definite chirality, Applied Physics Letters, 90 (16) pp. 164102
  8. 8. Leung KT, Józsa L., Ravasz M., Néda Z. (2001), "Pattern formation: Spiral cracks without twisting", Nature, 410 (6825) pp.166.
  9. 9. Wunderlich RC, Heerema NA. (1975) “Hair crown patterns of human newborns. Studies on parietal hair whorl locations and their directions.” Clinical Pediatrics 14 (11) pp. 1045–1049 pmid:1183129
  10. 10. Samlaska CP, James WD, Sperling LC. (1989) “Scalp Whorls.” Journal of the American Academy of Dermatology 21(3, 1) pp. 553–6.
  11. 11. Randle H., Webb T.G., Gill L.J., (2003). “The relationship between facial hair whorls and temperament in Lundy ponies.” Annual report of the Lundy Field Society (52) pp. 67–83
  12. 12. Herron AJ. Pigs as Dermatologic Models of Human Skin Disease. Proceeding of the ACVP/ASVCP Concurrent Annual Meetings December 5–9, 2009 Monterey, California, USA
  13. 13. Schmook FP, Meingassner JG, Billich A. Comparison of human skin or epidermis models with human and animal skin in in-vitro percutaneous absorption. Int J Pharm. 2001 Mar 14; 215(1–2): pp. 51–6 pmid:11250091
  14. 14. Gibson T. Editorial. “Karl Langer (1819–1887) and his lines.” British Journal of Plastic Surgery 1978; 3 (1): 1–2
  15. 15. Langer K. “On the anatomy and physiology of the skin I. The cleavability of the cutis”. British Journal of Plastic Surgery 1978; 31 (1) pp. 3–8
  16. 16. Borges A.F., (1984) “Relaxed skin tension lines (RSTL) versus other skin lines,” Plastic and Reconstructive Surgery, 73 (1) pp. 144–150 pmid:6691065
  17. 17. Kwak M, Son D, Kim J, and Han K, (2014) “Static Langer's line and wound contraction rates according to anatomical regions in a porcine model,” Wound Repair and Regeneration 22(5) pp. 678–682 pmid:24942927
  18. 18. Samlaska CP, James WD, Sperling LC. (1989) “Scalp Whorls.” Journal of the American Academy of Dermatology 21(3, 1) pp. 553–6
  19. 19. Anderson DB, Kauffman RG, & Kastenschmidt LL (1972) “Lipogenic enzyme activities and cellularity of porcine adipose tissue from various anatomical locations.” Journal of Lipid Research 13(5) pp. 593–599 pmid:5075506
  20. 20. Driskell RR, Jahoda CA, Chuong CM, Watt FM, Horsley V. (2014). “Defining dermal adipose tissue.” Experimental Dermatology 23(9) pp. 629–631 pmid:24841073
  21. 21. Chandawarkar RY, Cervino AL, Pennington GA. (2003) “Intraoperative acute tissue expansion revisited: a valuable tool for challenging skin defects.” Dermatologic Surgery Aug; 29(8) pp. 834–8
  22. 22. Sasaki GH. (1987) “Intraoperative sustained limited expansion (ISLE) as an immediate reconstructive technique.” Clinics in Plastic Surgery 14 (3) pp. 563–73 pmid:3608368
  23. 23. Wilhelmi BJ, Blackwell SJ, Mancoll JS, Phillips LG. (1998) “Creep vs. stretch: a review of the viscoelastic properties of skin.” Annals of Plastic Surgery 41 (2) pp. 215–9. pmid:9718160
  24. 24. Van Doorn WG, Van Meeteren U.(2003) “Flower opening and closure: a review.” Journal of Experimental Botany 54 (389) pp. 1801–1812. pmid:12869518
  25. 25. Zhu Y, Luo J, Barker J, Hochberg J, Cilento E, Reilly F. (2002) “Identification of genes induced by rapid intraoperative tissue expansion in mouse skin.” Archives of Dermatologic Research. 293(11), pp. 560–568
  26. 26. Yang M, Liang Y, Sheng L, Shen G, Liu K, Gu B et al. (2011) “A preliminary study of differentially expressed genes in expanded skin and normal skin: implications for adult skin regeneration.” Archives of Dermatologic Research. 303(2), pp. 125–133
  27. 27. Paine ML, Paine CT, Machin GA. (2004) “Hair Whorls and Monozygosity.” Journal of Investigative Dermatology. 122 (4) pp. 1057 pmid:15102099
  28. 28. Hakim V and Karma A. Spiral Wave Meander in Excitable Media: The Large Core Limit. Phys. Rev. Lett. 79, pp. 665