THE GOLDEN ANGLE: PHYLLOTAXIS &
THE EPIDERMAL CELL REDUCTION RATIO
In SB Hoath and DG Leahy, "The Functional Epidermal Unit and Organization of Human Interfollicular Epidermis," the authors examine the concept of functional epidermal units (FEU's) in human skin in which the maintenance of φ (= 1.618033989) proportionality provides a central organizing principle.1
Assuming the following empirical measurements taken from the scientific literature: 75346 nucleated epidermal cells/mm2, 1394 Langerhans cells/mm2, 1999 melanocytes/mm2, 16 stratum corneum layers, 900 μm2 corneocyte surface area, 17778 corneocytes/mm2, 14 day stratum corneum turnover time, and 93124/mm2 (75346 + 17778) total epidermal cells, in their paper the authors provide the following results (from the abstract): "1) the average number of corneocytes is a mean proportional between the sum of the [Langerhans cell + melanocyte cell] populations and the total number of epidermal cells: 3393/17778 ~ 17778/93124; 2) the ratio of the number of nucleated epidermal cells over the number of corneocytes is φ proportional: 75346/17778 ~ φ3; 3) assuming turnover times of 14 days each for the SC and Malpighian epidermis, the number of corneocytes results from subtraction of a cellular fraction equal to ~ 2/φ2 × the number of living cells: 75346 – (2/φ2 × 75346) ~ 17778; 4) if the total epidermal turnover time equals the SC turnover time × the ratio of living/dead cells, then compartmental turnover times are unequal (14 days for SC to 45.3 days for nucleated epidermis ~ 1/2φ) whereas the normalized cellular replacement rates are 52.9 corneocytes/69.3 keratinocytes per mm2/hour ~ 2/φ2."
It is noted here that "in plants with Fibonacci phyllotaxis the angle between successive leaves or other botanical units is generally close to the golden angle," as illustrated:2
The epidermal cell reduction ratio put forth in results 3 and 4 in the abstract above quoted has a profound and substantive analogue in plants with Fibonacci phyllotaxis.
The golden angle = 360º/φ2 (= 137.507764º). The 360º supplement to the golden angle = 360º/φ (the reflex angle found in the referenced plants = 222.492236º).3
following structural property pertains to the circle comprised of these two
angles: 180º − (180º
= 42.492236º and 180º + 42.492236º = 222.492236º.
Thus, where 2/φ2 is the epidermal cell reduction ratio, 75346 the number of FEU Malpighian cells, 17786.77785 the number of FEU corneocytes, and 93124 the total number of FEU cells,
75346 − (75346 × 2/φ2) : 180º − (180º × 2/φ2) :: 17786.77785 : 42.492236º :: 93124 : 222.492236º :: 418 : 1.
The number of reduced epidermal cells 75346 × 2/φ2 = 57559.22215 corresponds to the golden angle opening 180º × 2/φ2 = 137.507764º between successive leaves on the referenced plants.
When the total number of corneocytes in the average human epidermis is assumed to be 17778 × 1.73 × 106 mm2 = 3.075594 × 1010, the recapitulation of the epidermal-phyllotaxis structural ratio in the structure of the stratum corneum is evident:
3.075594 × 1010 ~ 4184.4
Note that when one golden rectangle is superimposed upon another in the manner
where AE = φ, EF = 1, 82944 is the total area of the unitary structure of the absolute dead center cube, 1055704.767 the latter's volume, 418 the epidermal-phyllotaxis structural ratio, and 2/φ2 the epidermal cell reduction ratio,6
AH/AI ~ 1.055704767,
AH × 829441/2 ~ 418,
AH/AF = 2/φ2.
Where 93124 is the total number of FEU cells, 75348 the number of FEU Malpighian cells, and 17778 the number of FEU SC cells, the thoroughgoing conformity of FEU epidermal structure to the logarithmic spiral and the golden rectangle is evident:
AC : AF :: 93124 : 75346 :: AB: AE :: BC : 1 :: (51/2 − 1) : 1,
AC : FC :: 93124 : 17778 :: (51/2 + 3) : 1,
AF : FC :: 75346 : 17778 :: φ3 : 1.
Where the number of degrees in the golden angle = 137.507764, 418 is the epidermal-phyllotaxis structural ratio, and 1/4, 5/9, 14/16, and 30/25 are the respective segmentation ratios associated with the four epidermal strata,7
137.507764/418 ~ 102 × AI/418 ~ (1/4)/(5/9) × (14/16)/(30/25).
Where AEFG and ABCD are the rectangular areas containing, respectively, the Malpighian AF, and the Malpighian + SC AF + FC, and 55/36 is the first environmental ratio in the series of epidermal segmentation ratios,8
ABCD : AEFG :: 55 : 36,
or, where (BC − MC)4 = (FC/AF)4 = (EK × BM × GD × LN),9 exactly
ABCD : AEFG :: [55 + (EK × BM × GD × LN)] : 36.
Thus the first environmental ratio precisely & explicitly accommodates the logarithmic measure of the epidermal surface. Here, neatly visible, the surface identity of environment and body.
golden angle θ = 137.507764º ~
OPC = 137.256610º (as illustrated),10
OT = OC/φ, 1/4, 5/9, 14/16, and 30/25 are the
segmentation ratios associated with the four epidermal strata (as above),
2/φ2 is the epidermal cell
(1/4)/(5/9) × (14/16)/(30/25) × 2/φ2 × [1 + (OT × 10-11)1/4] ~ (θ − OPC).
This paper was presented by Hoath at
"Bioengineering for the New Millennium," 2002 U.S. Symposium of the
International Society for Bioengineering and the Skin, Baltimore, MD,
October 2002. Cf.
SB Hoath and DG Leahy, "The Organization of Human Epidermis: Functional
Epidermal Units and Phi Proportionality,"
The Journal of Investigative
Dermatology 121, 1440-1446 (2003). See also, on this web,
The Langerhans-Epidermal Cell
Ratio & the Structure of the Epidermis and
Epidermal Immunity Structure and the Absolute Dead Center Hypercube
2 Text and image from "Phyllotaxis: An Interactive Site for the Mathematical Study of Plant Pattern Formation." Online. Smith College Department of Mathematics. Available: http://www.math.smith.edu/%7Ephyllo/Assets/Images/ExpoImages/ExpoTour/VirtualStations/4 /4.1.0.html. April 24, 2003.
3 Cf. ibid., http://www.math.smith.edu/~phyllo/CourseMaterial/Teachingmodules/phyltut6.html. April 24, 2003.
4 Where 1.73 × 106 mm2 is the average human body surface area, (1.73 × 106)1/2/π ~ 418. Where 360º is the circumference of the circle, α the fine structure constant, and 47.507764º the angle between 90º and 137.507764º, log 418 ~ 47.5077641/4 ~ φ2 ~ 360α.
5 Further for the logarithmic spiral and golden rectangle, see, on this web, The Golden Bowls & the Logarithmic Spiral, et passim.
6 For 82944 and the absolute dead center cube, see DG Leahy, Foundation: Matter the Body Itself (Albany, 1966), passim; see also material elsewhere on this web. Further for 418 and 2/φ2, cf. also, on this web, Pentagonal Light & Gravity & Epidermal Structure.
7 For the epidermal segmentation ratios, where the next in order following the SC 30/25 ratio would be the first environmental ratio 55/36, see SB Hoath and DG Leahy, “Formation and Function of the Stratum Corneum,” in The Essential Stratum Corneum, ed. R. Marks, J-C. Lévêque, R. Voegeli (London, 2002), pp. 31-40; cf. also, on this web, Pentagonal Light & Gravity & Epidermal Structure.
9 BC − MC = FC/AF = EK = BM = GD = LN = 1/φ3.
10 The writer thanks Mr. J. Cossentino for calculating OPC.