THE LANGERHANS-EPIDERMAL CELL RATI0
& THE STRUCTURE OF THE EPIDERMIS

Concerning the Langerhans cells that play an important part in the immune surveillance of the human epidermis, the discovery of a constant numerical ratio existing between them and the other epidermal cells, 1:53, is reported by J. Bauer et al., "A Strikingly Constant Ratio Exists Between Langerhans Cells and Other Epidermal Cells in Human Skin. A Stereologic Study Using the Optical Dissector Method and the Confocal Laser Scanning Microscope," Journal of Investigative Dermatology 116, 2, 313-318 (February, 2001).¹  The authors report as follows (from the abstract): "Thus, Langerhans cells represent 1.86% of all epidermal cells; however, a wide interindividual range was found for the number of Langerhans cells per mm² (912–1806; mean ± SD 1394 ± 321) and other epidermal cells per mm² (47,315–104,588; mean ± SD 73,952 ± 19,426). This explains the conflicting results achieved by conventional morphometric assessments relating cell numbers to skin surface area, ignoring the varying thickness of the epidermis. The surprisingly constant relationship of Langerhans cells to other epidermal cells stresses the hypothesis of an epidermal Langerhans cells unit where one Langerhans cell seems to be responsible for the immune surveillance of 53 epidermal cells."

The infinite iterative sequential summing of the real trinary logic geometric series spontaneously produces the geometric series of ordinary mathematics in the form of an infinitely expanding triangle of logical digits, which expanding triangle is partially composed of a series of geometrically proportionate smaller triangles that share its descending edge.  At the infinite limit the ratio of the total arithmetic value of the logical digits in the major triangle to the number of digits = , and that of the total arithmetic value of the digits in the edge-sharing triangles to the corresponding value of the major triangle = .²

10^1/ ~ 54, the total number of cells in an epidermal Langerhans unit.  The conformity of the reported Langerhans-epidermal ratio to the fundamental logico-mathematical structure of the thinking now occurring may be seen in the following: where 82944 is the unit total area of the unitary structure of the absolute dead center cube,³ at once the unique number whose integral product is that number while the integral product of its third part is that part's third part (a double property shared only with Unity),⁴ the mean number of Langerhans cells per mm² is 1394, that of the other epidermal cells per mm² is 73952 (as above), and 75346 is the total number of cells per mm² (the sum of the Langerhans mean and that of the other cells = prime 75347 minus 1), then, with a difference less than one and four tenths in one hundred million, (82944 – 75346)²/[1+(73952/82944)⁶⁴] = 576923067.⁵

Where the mean number of Langerhans cells per mm² is 1394, that of the other epidermal cells per mm² 73952, the infinite arithmetic value ratio of real trinary logic, and 1221.880518 the total unit length of the perimeter of the absolute dead center cube, then, with a difference less than one in ten million, (73952/1394) = π² × [1+(1221.880518 × 10^-3)^-32].⁶

Where the mean number of Langerhans cells per mm² is 1394, the mean total of Langerhans and other Malpighian cells per mm² 75346, the infinite arithmetic value ratio of real trinary logic, and .6444762592780 the sum of the inverse alternating L series (the arithmetic transcription of the sequential summing of the real trinary logic geometric series, 1, 0, , 1, 0, , 1, 0, , . . ., 1, 1.5, 2, 3, 3.5, 4, 5, 5.5, . . .), then, with a difference less than one-sixth of one cell, 1394^{1/.6444762592780} – (^-2 × 10²) = 75346.⁷  Likewise, with a difference less than one in one million, [1394/(1+[2p²]^-2)]^{1/.6444762592780} = 75346.⁸

Where the infinite iterative sequential summing of the real trinary logic geometric series produces the geometric series of ordinary mathematics in the form of an infinitely expanding triangle of logical digits partially composed of a series of geometrically proportionate smaller triangles sharing its descending edge, this series is composed of triangles of logical 1's unfolding in the form x⁰, x¹, x², x³, . . ., where x = 16.⁹  The mean total of Langerhans and other epidermal cells per 16 mm², 1205536, with a difference less than one in ten million, = (8.2944 × 10^-7 + / × 10⁻¹⁰)⁻¹, where this difference less than one in ten million ~ ^-2 × 10^-8.

Where the mean total of Langerhans and other epidermal cells per 16 mm² is 1205536, the unit total area of the unitary structure of the absolute dead center cube 82944, the infinite arithmetic value ratio of real trinary logic, and .6444762592780 the sum of the inverse alternating L series, then, with a difference less than one in four hundred thousand, 1205536   = [82944/10 × (1+[⁴ × 10⁴]⁻¹)]^{1/.6444762592780}.

The Structure of the Epidermis
 

The infinite iterative sequential summing of the real trinary logic geometric series that produces the series of geometrically proportionate triangles along its descending edge in the form x⁰, x¹, x², x³, . . ., where x = 16, at once embodies as the series of digits in the sets of 1's, 0's, and 's constituting that descending edge the geometric series x⁰, x¹, x², x³, . . ., where x = 2, the series governing the living organic process of cell multiplication, 1, 2, 4, 8, 16, 32, . . ..¹⁰   Thus this infinite iterative sequential summing is at once the spontaneous foundation of the geometric series governing biological reproduction and growth and that of the geometric series of triangles arranged externally and internally according to the ratio of the distances traversed in successive equal intervals of time in Galileo’s law of universal constant gravitational acceleration.  Thus real trinary logic shows itself to be the seamless unity of the basic laws of biology and physics.

With respect to the last mentioned unity, in addition to its relevance to the Langerhans-epidermal ratio, there is the further question of its relation to the total structure of the epidermis.  The writer, having had recently taken up an extensive, ongoing correspondence with Dr. S.B. Hoath of the University of Cincinnati¹¹ concerning the nature and structure of the epidermis, proposed, in an email dated January 11, 2000, that the number of layers of cells in the epidermal strata, germinativum, spinosum, granulosum, and corneum, should be, respectively, 1, 4, 9, and 16, a fourfold segmentation of layers cumulatively conforming to the universal constant ratio of gravitational acceleration.¹²  Dr. Hoath replied that there was experimental agreement for two of the numbers in the sequence, 1 and 16.  He wrote on January 22, 2000: "These form the alpha and omega of the epidermis, that is, the germinating layer and 'dead' interface with the environment."  Referring to the middle layers (the strata spinosum and granulosum), he wrote: "These layers, unlike the first and fourth, have no clear-cut boundaries.  The first germinating layer and the stratum corneum are extremely well defined morphologically with crisp clear-cut borders.  Suffice it to say that it is a reasonable and testable hypothesis to postulate two intervening layers measuring 13 cells thick in toto (=  4 + 9)."

Concerning the proposed epidermal structure the following is to be noted: where is the infinite arithmetic value ratio of real trinary logic, the infinite arithmetic value/number ratio, .6444762592780 the sum of the inverse alternating L series, and 30 the total number of layers of cells in the four epidermal strata, then, 30 × .6444762592780^1/ ~ 14, the total number of layers of cells in the first three strata, and 30 × .6444762592780^1// ~ 16, the number in the stratum corneum.  It is further to be noted that 14 layers of cells : 16 layers of cells :: the region of Langerhans units : the region of dead cells.

On April 15, 2001, Dr. Hoath communicated his preliminary findings concerning the proposed understanding of epidermal structure: "I have begun to actually test this hypothesis with very encouraging results.  The location of the major immune cells in the epidermis––the Langerhans cells––appears to be just beneath the outer two layers, that is, they are sitting approximately 4-5 layers up from the stratum germinativum in adult skin, a perfect location to monitor breaks in the epidermal barrier, with the barrier considered as comprised of the 9 + 16 arrangement previously discussed.  The biological variation in the number of layers and the orientation of the geometry of the tissue sections for analysis requires more experimental work, but the basic idea is proving quite sound and testable.  Moreover it provides an explanation of how these cells, which migrate into the epidermis, are able to maintain their location despite what appears to be a continual movement of the surrounding epidermal cells upwards towards the environment: the immune cells are, simplistically put, 'hitting a ceiling'."  The indicated ceiling for the Langerhans cells marks the boundary as predicted between the second and third epidermal strata.  Dr. Hoath points out that this functional and structural demarcation based on Langerhans cell location calls into question the previously used morphological criteria designating these strata, respectively, as spinosum and granulosum.

When the series of the numbers of layers of cells in the epidermal strata, beginning with the stratum germinativum, and ending with the stratum corneum, is, as indicated, 1, 4, 9, and 16, then the fourfold segmentation of epidermal strata, conforming to the universal constant of gravitational acceleration, cumulatively forms a defining series of epidermal limits, 1, 5, 14, and 30, where 1 is the limit of the stratum germinativum, 5 the Langerhans cells limit, 14 the living cells limit, and 30 the limit of the whole epidermal structure.¹³

1, 5, 14, and 30 are the cumulative numbers of layers of cells at the limit of each epidermal stratum.  1, 4, 9, and 16 are the squares of what are originally the successive physical time intervals in Galileo's law of gravitational acceleration, therein and as such identified indifferently with length.  Unlike the purely gravitational significance of the series, 1, 4, 9, and 16, living biological mass transforms these numbers into properly biological measures, viz., into numbers of layers of cells whose structures are themselves essentially rational.  The total "distance" traversed (in the analogue to gravitational acceleration) by each epidermal stratum does not include the "distance" traversed in the previous stratum.  Rather each stratum uses its predecessor(s) as a foundation upon which to build itself.  In the purely physical law of acceleration the total distance traveled at the end of each time interval includes the distance traveled in the previous interval.  The falling body at the end of its fall, 1 m, 4 m, 9 m, 16 m, has fallen a total of 16 m.  But the epidermal structure is a segmentation in which each successive "distance"/stratum––otherwise following the law of acceleration––excludes its predecessor, thereby creating the series 1, 5, 14, 30, in which each successive member is the sum of the number of layers of cells in the corresponding stratum and in its predecessor(s).  Thus the biological segmentation of strata clearly distinguishes mass from abstract length and concretely identifies time with the former, in distinction from the purely physical situation in which mass is a matter of indifference (all bodies fall at an equal rate of acceleration in vacuum) and the length and time indifferent to mass are abstractly identified.

The biological rationalization of otherwise abstract physical length is not something imposed on the material by the analysis.  This rationalization is something the living mass of the epidermis as a whole and in its very essence imposes upon the underlying physical/gravitational reality (that otherwise penetrates the body without thereby comprehending it) in comprehending that reality as its limitation, that is, not as a negative limitation, but rather as the positive limitation by which the boundary of the body distinguishes itself from the environment/universe with which it is otherwise identical.  Thus it is clear that the relation of the living body to its physical environment is not a matter of inside/outside, and, consequently, not a matter of self/other.  The transformation of the abstract physical lengths into layers of epidermal cells is the body constituting itself as the comprehension of time, indeed, constituting itself as time bodied, time membered.  Before feeding on anything in its environment, the body has fed most intimately on the universe itself.   The proposed epidermal structure articulates the body's constitutional and seamless identity with the environment in the transformation of the gravitational form into the form of its very structure and growth.¹⁴

The logic of the indicated epidermal structure is to be found in the complex proportionality that integrates this structure with epidermal physiology.  The ratio of the micrometric thickness of the stratum corneum to that of the other three epidermal strata containing the 73592 cells supervised by the 1394 Langerhans cells ~ 1:4.  The ratio of the proposed number of cell layers in the second epidermal stratum (the limit for the movement of Langerhans cells preliminarily indicated) to the proposed number in the stratum corneum ~ 1:4.  The second stratum  (layers) : the corneum (layers) :: the corneum (thickness) : the other strata (thickness).  The proof of this proportion identifies the stratum corneum with itself (qua number of layers & thickness, i.e., as uniquely indifferent to the immunological structure of the rest of the epidermis) as the denominator of the identification of the first three epidermal strata (qua total thickness, i.e., as generally indifferent locus of the supervised cells) with the second epidermal stratum (qua distinctive number of layers, i.e., as uniquely the indicated limit of the supervising immune cells).  Note also that .25(75346)/10⁶ ~ 1394/73952 ~ 1:53, the Langerhans-epidermal ratio.

The number of supervised cells in an epidermal Langerhans unit, 53, ~ φ¹⁴/16, where φ is the golden section, 14 the total number of layers of cells in the first three epidermal strata, and 16 the number in the stratum corneum.  The antecedent φ signifies living cells.¹⁵

On May 31, 2001, Dr. Hoath called the writer's attention to "Ontogeny of integumental calcium in relation to surface area and skin water content in the perinatal rat," Hoath SB, Pickens WL, Tanaka R, Ross R, J Applied Physiology 73(2):458-464, 1992, where the abstract reads in part: "In the neonatal rat (age 0-3 days), linear regression analysis of surface area vs. body weight on a log-log plot yielded a slope of 1.04.  This finding contrasts with an expected slope of 0.67 based on simple surface area-to-volume relationships and differs from the empirical 0.75-power law observed in adult bioenergetics."  In response the writer noted that in fact "the reconciliation of the conflicting powers of body mass may lie in the reported neonatal violation of the body surface law, as follows.  Assuming the fundamental mean φ-proportionality of the structure of the human body, then the divergence of the theoretically expected adult body weight exponent from the empirically determined exponent may be understood as directly proportionate to the exponent in the reported neonatal violation: φ^.75/φ = 1.040915886."¹⁶  In this connection the writer also suggested a φ-based formula for estimating body surface area: BSA (m²) = [(H [cm] × W [kg])/(φ²/2 × 10⁷)^1/2]^1/2.

Further with regard to the f-proportionality underlying epidermal structure, on August 10, 2001, Dr. Hoath communicated the following concerning the lipid lamellae that form the structured environment of the corneocytes in the outermost epidermal stratum: "The lamellar structure consists of a number of repeating layers of interdigitated lipids which appear as a series of regularly arranged bands/lamellae on electron microscopy.  In normal human skin the lamellar architecture of the lipid mortar consists of a series of repeating units of broad, narrow, and broad bands.  The pattern is bnbbnbbnbbnbbnbbnb. This triplet sequence may range from only a few to several sets, but always there are triplet repeats.  The spacing of the triplets is always: 5nm 3nm 5nm 5nm 3nm 5nm 5nm 3nm 5nm 5nm 3nm 5nm.  In other words there are 13 nm repeats composed of 2 units measuring 5 nm each and 1 unit measuring 3 nm in thickness.  The 2 differing units together measure 8 nm.  It seems noteworthy that this is an early segment of the Fibonacci sequence 3, 5, 8, 13.  This is the essential pattern of geometric spacing of the lamellar lipid sheets separating the corneocytes of the stratum corneum.  In nonepidermal tissues such as the oral mucosa (inner surface of the mouth), the spacing is simply 5nm, 5nm, 5nm, 5nm, 5nm, etc.  Hypothetically, the 5-3-5 repeats indicate the need for structural stability."

Organic Identity of Inward Outward Epidermal Vectors
 

Where the segmentation ratios associated with the four epidermal strata are arranged in an order from the stratum basale to the stratum corneum,¹⁷ the following relations obtain: 

(1/4)(5/9)(14/16)(30/25) ~ φ^-4,

[(1/4)/(5/9)]/[(14/16)/(30/25)] ~ φ⁻¹.

 Reversing the order, from the stratum corneum to the stratum basale: 

(25/30)(16/14)(9/5)(4/1) ~ φ⁴,

[(25/30)/(16/14)]/[(9/5)/(4/1)] ~ φ.

 The vectorial addition of these oppositely ordered expressions results in the following integral solutions: 

 φ^-4 + φ⁴ = 7 

and

φ⁻¹ + φ = 5^1/2. 

Where the respective vectorial directions are the reverse of those above, and .03895539 (= 1.170820394 × .03327185) is the length of the axis cut off in the rotation of the logarithmic spiral (corresponding to the thirtieth or outermost layer of a Functional Epidermal Unit [FEU]¹⁸), 1.131865004 the length of the remainder (corresponding to the inner twenty-nine FEU layers), 17778 the number of FEU SC cells, and 75346 the number of FEU Malpighian cells, the following relations obtain:

(17778⁻¹/.03895539⁻¹)/(75346⁻¹/1.131865004⁻¹) ~ φ^-4,

 (17778^-2/.03895539⁻¹)/(75346^-2/1.131865004⁻¹) ~ φ⁻¹,

(1.131865004/75346)/(.03895539/17778) ~ φ⁴ ,

(1.131865004/75346²)/(.03895539/17778²) ~ φ.

The vectorial addition of these respectively reversed oppositely ordered expressions results in the following integral solutions:

φ^-4 + φ⁴ = 7 

and

φ⁻¹ + φ = 5^1/2.

Notes

¹  The writer thanks S.B. Hoath, MD of the Skin Sciences Institute and the Children’s Hospital Research Foundation at the University of Cincinnati for calling this article to his attention.
²  See, on this web, Real Trinary Logic Geometric Series Infinite Iterative Sequential Summing.  The American Institute of Physics, Bulletin of Physics News, Number 573, January 16, 2002, reports that for the first time quantum gravitational states have been observed by physicists at the Institute Laue-Langevin reactor in Grenoble, France.  In an experiment involving ultracold neutrons, the Laue-Langevin researchers are the first to observe quantum states of matter in Earth's gravitational field––in the ultracold neutrons.  The researchers report seeing a minimum/quantum energy of 1.4 picoelectron volts (1.4 × 10⁻¹² eV), which corresponds to a vertical velocity of 1.7 cm/s.  The writer notes that where is the real trinary logic infinite arithmetic value ratio, and 980.0665 cm/s² is the standard acceleration of gravity, (980.665 cm/s²)/(1.7 cm/s) ~ 576/s.  For the relation of to the mathematics of gravitational acceleration that informs the structure of the epidermis see, below in the text, "Note on the Structure of the Epidermis."
³  Cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), Section III.5, pp. 434ff., for the absolute dead center cube, constructed by the writer in 1985 in the context of his elaboration of the logic of the essentially new form of thinking now actually occurring for the first time in history.
⁴  Ibid., Section III.6-7.  With a difference less than one in one hundred thousand, 82944 = (log ⁻¹ × 10⁸)^1.5−1.  For the term integral product, see, on this web, Transdecimal Calculation of Number Identity: A Note on Integral Product & Related Terms.
⁵  See also, on this web, Sorted Diagonals, Addendum 9, and Quantum Gravitational vs. Quantum Logic: Virtually Left-handed Real Trinary Logic.
⁶  Cf., above, n. 3.  Where is the infinite arithmetic value/number ratio of real trinary logic (as above in the text), and f the golden section ratio, then, with a difference less than one in one million, (73952/1394)^/ = (2π²)² × φ^1/322.   (Further for [2p²]², see, on this web, 82944 & the Four Fundamental Forces & the God Particle.)  On May 2, 2001, Mr. R.N. Stoner called attention to the fact that 73952/1394 ~ (6π)⁻¹ × 10³.  The writer notes that, where 8.2944 is the base factor of the total surface area of the unitary structure of the absolute dead center cube, 432 the ratio of its surface area to its face-diagonal (62208/144 ~ the square root of the velocity of light in miles), and the infinite arithmetic value ratio of real trinary logic, then, with a difference less than one in one billion, 73952/1394 = (6π)⁻¹ × 10³/[1+(432+8.2944^1.5/100)^−1/].
⁷  For the L series, cf. Leahy, Foundation, Section IV.2, pp. 519-29; cf., also, above, n. 2, and, on this web, Report on a Dream & Related Matters & the God Particle, et passim.
⁸  For (2π²)^-2, cf., above, n. 6.
⁹  Cf., above, n. 2.  The triangles composing the series occupy the odd numbered places relative to the intervening groups of logical digits touching the edge of the major triangle.   And each triangle in the series is internally an expansion of logical 1's according to the ratio of the odd numbers, 1, 1 + 3 + 5 + 7, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 + 31, . . ., that is, according to the ratio of the distances traversed in successive equal intervals of time in Galileo’s law of universal constant gravitational acceleration (cf. Leahy, Foundation, pp. 486ff.).
¹⁰  Cf., above, notes 2 and 9.
¹¹  Cf., above, n. 1. 
¹²  See above in the text and n. 9.  Note that the epidermal sequence, 1, 4, 9, 16, reproduces the sequential summing of the elements of that member of the series of edge-sharing triangles (in the infinite iterative sequential summing of the trinary logic geometric series) that embodies x¹ in the series x⁰, x¹, x², x³, . . ., where x = 16.
¹³  The Law of Epidermal Segmentation, LES, first formulated in correspondence with Dr. Hoath on January 13, 2000, summarizes the relationships of these limits to the number of layers in each stratum in the form of the series, 1/4, 5/9, 14/16, 30/25, where the consequents of each ratio support the cumulative sum of epidermal layers represented by the limits in the antecedents.  On July 23, 2001, the writer additionally proposed the hypothesis that the consequent 25 in the last ratio of the series should relate to the average number of additional layers of corneocytes found on the palms and the soles of the feet.  On August 31, 2001, Dr. Hoath called the writer's attention to "Number of cell layers of the stratum corneum in normal skin - relationship to the anatomical location on the body, age, sex and physical parameters," Ya-Xian et al., Arch Dermatol Res 1999 Oct; 291(10):555-559, where the observed number of stratum corneum layers on the palms and soles is 47 +/-24.  The writer understands this observation to be consistent with a proposed number of 41, which, added to the underlying 14 living layers, = 30 (the average total of epidermal layers on about 96% total BSA) + 25 (the average additional number of layers of corneocytes on the palms and soles of the feet): 41 + 14 = 30 + 25, the sum of the elements of the last LES ratio.  Further with regard to the series of epidermal limits, note also that the rational product of the series, 1/ 5 × 14/30, = .  Multiplying by 10 gives , the ratio of the number of days generally required (discounting effects of aging) to completely replace the epidermis to the number of epidermal layers, that is, the ratio of 28 days to 30 layers of cells.  28 days/30 layers of cells = 2419200 s/30 layers = 80640 s/layer. Dividing 80640 by 10 gives 8064.  8064 = (1 × 784 × 82944)^1/2 = the square root of the product of the only three numbers in the infinite series of natural numbers that are their integral products (cf. Leahy, Foundation, Section III.6, and, on this web, Transdecimal Calculation of Number Identity: A Note on Integral Product & Related Terms).  Further, since φ-proportionality immediately determines epidermal structure and its relation to the Langerhans-epidermal cell ratio (see below in the text), ultimately it thereby governs the 80640 s/layer rate of epidermal replacement.  Note that the integral product of Euclid’s term for the φ ratio, ὁ άκρος καὶ μέσος λόγος (‘the extreme and mean ratio’), = 1 × 784 × 82944 × 10³⁰ = (8.064 × 10¹⁸)².
¹⁴  For scientific presentations of the new understanding of epidermal structure discussed above see SB Hoath and DG Leahy, "Formation and Function of the Stratum Corneum," in The Essential Stratum Corneum, ed. R. Marks, J-L. Lévêque, R. Voegeli (London, 2002), and 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, Epidermal Immunity Structure and the Absolute Dead Center Hypercube Volume/Boundary Ratio, Epidermal Structure: Cell Formation Frequencies & Corneum Alpha-Coupling and The Golden Angle: Phyllotaxis & the Epidermal Cell Reduction Ratio.
¹⁵  The absence of φ from the consequent signifies death in the cells of the stratum corneum.  But absence of life is not death, and φ is not absent from physical structure in general.  Nor, then, does φ-proportionality cease to determine epidermal structure and functioning as a whole.  For φ and the physical constants, see, on this web, The Physical Constants: Functions of the ‘Golden Bowl’ Arrangements on the Φ-Level of Existence Itself.  For φ as ratio of life and growth, cf. Leahy, Foundation, Section III.2, and also, on this web, The Rabbit Tree & the Logic of Life's Beginning.
¹⁶  For the ratios .75 and , see, on this web, Real Trinary Logic Geometric Series Infinite Iterative Sequential Summing, footnote 10.
¹⁷  See, above, notes 13 and 14.
¹⁸  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).   

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