Epidermis Golden Angle

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

The following structural property pertains to the circle comprised of these two angles: 180º − (180º × 2/φ²) = 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:

Note that when one golden rectangle is superimposed upon another in the manner here illustrated:⁵

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/φ²the epidermal cell reduction ratio,⁶

AH/AI ~ 1.055704767,

and

AH × 82944^1/2 ~ 418,

and

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 :: (5^1/2 − 1) : 1,

and

AC : FC :: 93124 : 17778 :: (5^1/2 + 3) : 1,

and

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.

____________________________________________________

Where the 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), and 2/φ²is the epidermal cell reduction ratio:

(1/4)/(5/9) × (14/16)/(30/25) × 2/φ² × [1 + (OT × 10⁻¹¹)^1/4] ~ (θ − OPC).