the deep epidermal surface:
the Cornerstone construction order,
Minimum Order Tetrahedron hypercube,
& Absolute Dead Center hypercube
The real trinary logic Cornerstone construction order is as here illustrated (for the transcendental deduction of this order starting with the power body analogue of the power set, see, on this web, The Real Beyond the Void: the Beginning/the Power Body, Note 4):
On June 27, 2006, Dr. SB Hoath forwarded the results of his meditation on the above construction order.
He wrote, "Where the fourfold logical foundation is constructed by rotation of the elements of the Beginning/Power Body ordered cornerstone, the resulting construction has the following characteristics:
1) The sums of the elements of each of the two innermost columns and each of the two innermost rows each equal 36.
2) The sum of the squares at the edges = the sum of the squares along the diagonal = 26 such that the ratio of the side to the diagonal is unity = 1.
"The question arises: What does this nicely balanced unitary construction signify? Or, better stated, what does it measure?"
Dr. Hoath then went on to remark that in researching the experimental finding that the water content of vernix caseosa is very accurately the cube of the golden section ratio, one of his graduate students came upon a paper by N.A. Bulienkov, "The Role of Systemforming Modular Water Structures in Selforganization of Biological Systems," Journal of Molecular Liquids 106/23 [2003] 257275, in which the following statement appears:
"Two more conditions are important for the self organization of hierarchical systems: 1) the ratios of the dimensions of the parts to the dimension of the whole system must be powers of the golden ratio φ = 1.618034… (the golden ratio condition) and 2) the integration of parts in the whole is done solely by weak bonds, that control the system by a 'feedback' interaction with the environment."
He then continued, ". . . this paper contains the interesting premise that 'bound water structures' serve as φbased scaffolds for biological organization. Moreover, the above statement seems interpretable in terms of . . . a φ proportional 'power body'. Consider, for example, the following:
"Where the elements of the Beginning/Power Body ordered cornerstone are powers of φ, the resulting 'φ power body' foundational construction is:
"Given the above construction, where the sum of the squares on the side of the φ power body is φ^{2 }+ φ^{5 }+ φ^{1 }+ φ^{9 }+ φ^{7 }+ φ^{2} = 122.9919, the sum of the squares on the diagonal is φ^{2 }+ φ^{3 }+ φ^{8 }+ φ^{8 }+ φ^{3 }+ φ^{2} = 107.6656, 16 is the number of cell layers in the stratum corneum, and 14 is the number of layers in the Malpighian epidermis [SB Hoath and DG Leahy, "Formation and function of the stratum corneum," in R. Marks (ed.): Proceedings of the Stratum Corneum III Conference, Martin Dunitz Ltd, London, 2002] , then,
122.9919/107.6656 ~ 16/14.
"Where the above φ power body containing 36 elements is reduced to a 9fold square by addition,
then the resulting side of this new 9fold 'supersquare' = 46.97972 + 102.43 + 46.97972 = 196.387, and the diagonal = 46.97972 + 187.91 + 46.97972 = 281.8723.
"Given the above construction, where the LES+ coupling constant = φ^{1/2} [see, on this web, The v_{maxwell}^{1/16} × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe] is the face of a cube whose volume is 1.434633, then
281.8723/196.387 ~ 1.434633.
"Alternative constructions: where 1394 = average number of Langerhans cells per mm^{2}, 1999 = average number of melanocytes per mm^{2} [see SB Hoath and DG Leahy, "The Organization of Human Epidermis: Functional Epidermal Units and Phi Proportionality," The Journal of Investigative Dermatology 121, 14401446 (2003)],
281.8723/196.387 ~ 1999/1394 ~ φ^{3/4}.
"Where 1.131865032 is the 3rd axis segment in the golden rectangle construction, 0.014879624 is the 'tip' of the vector directly opposite to the .038955361 segment of the 1.170820394 vector in the logarithmic spiral construction, or alternatively the inverse square root of the total length of the perimeters of the 8 exterior triangles that lie on the triplecube's nonsquare faces and the 4 interior triangles that reflect these exterior surface triangles = 1/(12^{3} × φ^{2})^{1/2} [see, on this web, Sorted Diagonals, Addendum 12]. . . , then
1.131865032^{3 } 281.8723/196.387 ~ 0.014879624.
"Where 1000 is the H_{2}Oresolution of the gravitational constant [see, on this web, Theorem for a Flat Universe: The ρ_{crit} = 3H_{0}^{2}/8πG Building Block 1000.^{125}], 785.547964 is the sum of all the elements in the φ power body, 1.27201965 is the square root of φ, or, alternatively, the LES+ coupling constant [see, on this web, The v_{maxwell}^{1/16} × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe], then,
1000/785.547964 ~ φ^{1/2}."
The writer's response to Dr. Hoath's observations follows.
The 26/26 side/diagonal ratio reflects the side/diagonal identity of the RTL cornerstone itself (D.G. Leahy, Foundation: Matter the Body Itself, pp. 534ff.):
"Insofar as the truth of THE PLACE identical with creation is the measure of an infinite order of squares (as constructed above), or cubes (in the third dimension), each one of which, in the absolute exteriority of the within now actually existing for the first time, here in the form of the foundational segments of the exploded central square (cube) of any square (cube), measures the diameter of a circle (sphere), there exists for the first time in thought, not 'a single sphere which expands in size without limit', but an infinite number of infinitely transparent absolute actualities: the sphere of absolute objectivity now existing: the sphere of spheres infinitely newly beginning, the sphere of infinitely new, infinitely separate spheres: the sphere the surface of which is the beginning of an absolutely transparent depth. This is the inception of the infinite proportionality of the body. This is the limit of the infinite expansion of the 'single sphere'. This is the beginning of the circumference of the infinite circle. This is the line for the first time. This is the time of beginning. This is creation displacing the abyss itself: the body of the Living God in the form of the beginning, depth absolutely surface, the infinite identical with the finite: the absolute incompatibility of the infinitely numbered points of the circumference of the circle: the absolutely transparent circle. The points of the infinite circle are squares between squares: the point, the square, is the side identical with the diagonal. The logical foundation of the last is most simply and directly evident in the structure of the cornerstone: where the extreme 1 of any side, 10or 01 or 10 or 01, never appears as the middle, but is both the extremes and the middle of the 111 diagonal, which has significance if this 111 diagonal can be identified with the sides, which in fact it can be, since the alternate 01 diagonal which unites the alternate extremes of the sides, so that 0 and are only in this diagonal at once extremes of one middle (the diagonal 1), has the discrete arithmetic value of any side, 10or 01 or 10 or 01, as 1 + 0 + = 2, while sharing with the 111 diagonal (arithmetically 1 + 1 + 1 = 3) the identical logical value, 01= 1 and 111 = 1. The logical identification of 2 and 3, 'appearance' and 'essence', is the logical identification of side and diagonal, itself at once the finite identified with the infinite."
Where the ratio 01/111
is the side/diagonal ratio of the cornerstone (at once the right/lefthanded
diagonals ratio), the SzentGyorgyi law of nonadditivity (N.A. Bulienkov, "The
Role of Systemforming Modular Water Structures in Selforganization of
Biological Systems," Journal of Molecular Liquids
106/23 [2003] 257275) –– 1+1 > 2
–– reflects the arithmetic
transcription of the underlying RTL structure: 1 + 1 + 1 (> 2) predicated of 1 +
0 +
(2). The fact that
the law of nonadditivity in relation to the discontinuity of water
structures leads at once a) to new structures and b) to fractal
replication (ibid., pp. 267ff.) is just so far consistent with the thinking
now occurring's understanding of the "absolute discontinuity of the
continuum" (Leahy,
Foundation, pp. 614, 627, and 628, et passim)—absolutely polyontological reality.
Where the arithmetic transcription of the real trinary logic diagonal/side ratio, 111/01, is 3/2 = 1.5, the equivalent ratio in ordinary mathematics 2^{1/2}, and the specific gravity of the human body is 1.06 (see, on this web, Theorem for a Flat Universe: The ρ_{crit} = 3H_{0}^{2}/8πG Building Block 1000.^{125}),
1.5/2^{1/2} ~ 1.06.
With respect to the 9fold supersquare reduction of the φ version of the cornerstone construction order rotation, it is noted that the sum of the products of the left and right diagonals of that construction is (46.97972 × 187.91 × 46.97972) × 2 ~ 829440 + 16.
Where the volume of a regular tetrahedron (the minimum polyhedron and the shape
of the H_{2}O molecule) = 2,
the rational product of the minimum order 4 : 1 :: 2 : 3 (Leahy, Foundation
III.1), the total length of its edges = 82944^{.25}, the V/BSA
resistance ratio of the absolute dead center cube (see, on this web,
Physical Variables,
Placedness, & the Absolute Dead Center Cube
and
The v_{maxwell}^{1/16}
× 1 s/m Cube, Human Body Surface & BSA, &
the Infinitely Flat Structure of the Universe) = the edge of a 4dimensional
hypercube whose volume
is 82944, the area of the unitary structure of the absolute dead center cube
(Leahy, Foundation, pp. 433ff.). Where √162 is the ratio of the
volume of the 4dimensional absolute dead center hypercube to that of its
boundary (see below), 82944^{.25}/2
= √162/2.
Where the (n–1)volume of the boundary of ncube is 2 n s^{n–1} (http://gregegan.customer.netspace.net.au/APPLETS/29/HypercubeNotes.html), and 82944^{.25} is the V/BSA resistance ratio of the absolute dead center cube, the latter's (n–1)boundary volume is
62208 = 10368^{1.5} × 1/82944^{.25}.
Where 1000 is the H_{2}Oresolution of the gravitational constant (see, on this web, Theorem for a Flat Universe: The ρ_{crit} = 3H_{0}^{2}/8πG Building Block 1000.^{125}), 785.547964 the sum of all the elements in the φ version of the power body construction order rotation (see above), 1.27201965 the square root of φ ~ the LES+ coupling constant (see, on this web, The v_{maxwell}^{1/16} × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe), the absolute dead center 4dimensional hypercube volume 10368^{2} (n–1)boundary volume is
8445638.14 = 10368^{2} × 1/162^{1/2 }~ 10368^{2} × 785.547964/10/1000 ~ 10368^{2} × 1/(10φ^{1/2}).
Where the ratio of the absolute dead center cube (n–1)volume of its boundary to
its volume is 1/82944^{.25}, the ratio of the 4dimensional hypercube volume (= the
area of the unitary structure of the absolute dead center cube, Leahy,
Foundation, pp. 433ff.) whose edge is the consequent of the last ratio, to
the absolute dead center cube volume is
82944/10368^{1.5} = √162^{1} = (√2 × 9)^{1} = 8445638.14/10368^{2}.
Note that the property demonstrated in the last equation, viz., that the 4dimensional hypercube volume whose edge is the V/BSA ratio of a base cube is to the base cube volume exactly as the boundary of the 4dimensional hypercube of the base cube is to its volume––such a perfectly proportionate identity––is unique to the absolute dead center cube.
The aptness of the designation "the absolute dead center cube" is manifest. For all cubes (x/6)^{4}/x^{3} × (8x^{3}/x^{4}) = 162^{1}. Uniquely for the absolute dead center cube (x/6)^{4}/x^{3 } = 8x^{3}/x^{4} = 162^{1/2} (see, on this web, Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube).
Where X = (x/6)^{4}/x^{3} and Y = 8x^{3}/x^{4}, for all other cubes X ≠ Y and XY = Z. For the absolute dead center cube X = Y and XY = X^{2} = Y^{2} = Z. The real trinary logic analogue is 0 = 00 = = 1 (cf. Leahy, Foundation, III.1, et passim). See also, on this web, The Magnitude of Being.
For all cubes (XY)^{2} = Z^{2}. Uniquely for the absolute dead center cube (XY)^{2} = X^{4} = Y^{4} = Z^{2}. It is constitutionally uniquely fourthdimensional. It is constitutionally perfectly hypercubic. All other cubes are imperfectly hypercubic––constitutionally hyperrectangular, that is, for them X ≠ Y.
This relation of the absolute dead center cube to the infinite array
of all cubes––at once like (XY = Z)
and unlike (XY = X^{2} = Y^{2}
= Z) all others––conforms precisely
to the positive and negative metaidentity that constitutes the
unity of the fourfold array of the minimum of real trinary logic (Foundation,
pp. 276f.), to wit: “The fourfold of relations at the heart of
the foundationstone as it turns into the foundation is visualized
thusly:
“where if i = the string , 1, 0, and j = the string 0, 1, , the foundation proportion, (4) : (1) :: (2) : (3), is:
i/j : j/j :: j/i : i/i,
“the proof of which is:
i^{2}/ij · ij/j^{2} = i^{2}/j^{2} · ij/ij = i^{2}/j^{2} · 1 = 1.
“This proof is the inductive demonstration of the logically deduced definition of positive metaidentity, i.e., i^{2}/j^{2} = 1 = (i : j) + (i : j), the positive absolute aliorelative 1_{ij} = 1, which metaidentified in the proof as negative metaidentity (where ij/ij = 1̃_{ii} = 1 = [i : i] + [j : j]) is identically the proven unity of the minimum order of the elements of the foundation.”
The hypercubic rational constitution of the absolute dead center cube interrupts, once and for all, at a particular place, the infinite continuum of cubes hither and thither that place, for each of which it is true that its rational constitution is hyperrectangular.
This discontinuity of the continuum of imperfectly hypercubic cubes—the unique rational constitution of the absolute dead center cube in their midst—is the mathematical manifestation of the real trinary logic ‘the not yet’ as ‘beyond beyond x’ as ‘the absolute now’. Stated as a proportion, the rationally hypercubic constitution of the absolute dead center cube : the rationally hyperrectangular constitution of all other cubes :: fourth dimensional infinite flatness : second dimensional finite flatness :: the squaring of the squared rectangle : the squaring of the notsquared rectangle :: beyond beyond x : beyond and not beyond x :: the absolute Da : hither and thither :: the absolute now : past and future :: Schranke : Grenze :: the infinite difference of x : the finite difference of x. Further for the logicalontological reality of which the absolute dead center cube is the mathematical instantiation par excellence, see, on this web, Note to Faith and Philosophy Further to the Ontology of Real Trinary Logic, Beyond the Good: Not Hither the Good (& Not Hither Beyond the Good), The Real Beyond the Void: the Beginning/the Power Body, and Power Body Cornerstone Construction Order: Absolute Dead Center Hypercube: The Three Unique Transdecimal Number Identities.
The infinite approach of hyperrectangular rational cubic
constitutions to the hypercubic rational constitution of the
absolute dead center cube is to their infinite recession thereafter
as breadth is to depth, forming a hyperlinear angle whose
vertex/diagonal is the hypercubic rational constitution of the
absolute dead center cube. Let the relation of the squaring of
the squared rectangle to the squaring of the notsquared rectangle
be reduced to the relation of square to rectangle, then there is a
finitely flat, abstract image of the relation of the constitutional twodimensional to
fourdimensional flatness, the hypercubic rational constitution of
the absolute dead center cube at once the perfect discontinuity of
the continuum of hyperrectangular rational cubic constitutions, here
illustrated: The Mathematics of the Absolute Dead Center Cube : The Mathematics of All Other Cubes :: Real Trinary Logic : Dyadic Trinary Logic(s) :: Proportionate Division in Extreme & Mean Ratio : Proportionate Division in All Other Ratios. Note that, where φ is the division in extreme and mean ratio, the definition of the absolute dead center cube, (x/6)^{4}/x^{3 } = 8x^{3}/x^{4 }(= √162^{1}), = [(φ×10^{162})^{1/162}]^{1/[162}^{×(√162/2)}^{2]}/10 × π/4 accurate to 1 part in 100 million. Cf., on this web, The Beautiful Rationality of Π, √2, & Φ and Quantum Gravitational vs. Quantum Logic: Virtually Lefthanded Real Trinary Logic.
Note that the basic structure of the hyperlinear angle's vertex/diagonal––the hypercubic rational constitution of the absolute dead center cube qua perfect interruption of the continuum of hyperrectangular rational cubic constitutions––is that of the 4123 minimum order (see above, and below) of the fundamental form of all logical propositions: pq̅ = + pq = + p̅q = + p̅q̅ = . Thus the hyperlinear displacement of the 1234 order of the Cartesian coordinate system––the displacement of the notion of the origin (= zero = nothing) with the notion of the law of absolute unity (= zero = not nothing)––indeed, the displacement of space itself. Cf. D.G. Leahy, Novitas Mundi: Perception of the History of Being (Albany, 1994), pp. 326f. and 390f.; also Leahy, Foundation, III.1, et passim, and IV.2, Excursus Circularis. See also, on this web, Logical Unity & the Four Physical Forces, Physical Variables, Placedness, & the Absolute Dead Center Cube, and The Simplicity & Syntax of the Concepts, Immediacy, Mediation, Omnipotence, & Beginning.
For another unique logically related mathematical property of the absolute center cube, viz., that the circumference in which the face of this cube is inscribed = the area of the circumference circumscribing each of the 36 equal subsquares of that face, cf. Leahy, Foundation, IV.2, p. 524, n. 98; also, on this web, Physical Variables, Placedness, & the Absolute Dead Center Cube.
Where φ is the golden section, and (8 × 3^{1/2})/4 the area of the base of the regular tetrahedron whose volume = 2, the rational product of the minimum order proportion 4 : 1 :: 2 : 3 (see above, and Leahy, Foundation, III.1),
100φ × [(8 × 3^{1/2})/4]^{1/1024} = 161.9998395 ~ 162 accurate to 1 part in 1 million.
For the purely numerical uniqueness of 82944 (= [x/6]^{4} in the case of the geometrically unique absolute dead center cube), cf. ibid., III.6, III.7, and IV.2, p. 524, n. 98. See also, on this web, Measure Beyond Beyond Reach.

In addition to the appearance of 10 times the LES+ boundary number φ^{1/2} as the ratio of the dead center hypercube 10368^{2} to its (n–1)boundary 8445638.14, note that where the number of cells in the Langerhans Epidermal Unit (LEU) immunological barrier is 54 (see, on this web, The LangerhansEpidermal Cell Ratio & the Structure of the Epidermis, and SB Hoath and DG Leahy, "Formation and function of the stratum corneum," in Proceedings of the Stratum Corneum III Conference, ed. R. Marks [London, 2002]),
8445638.14 × (10368^{1.5}/10^{6})^{.125} = 8503060.715 ~ 54^{4} accurate to 1 part in 1 million.
On September 12, 2007, Mr. J. DePompeo called the writer's attention to the fact that where 1, 4, 9, and 16 are the respective numbers of layers of cells in the four epidermal strata (ibid.), the number of cells in the LEU barrier 54 (ibid.), and the integral product of το σωμα του Χριστου 4.29981696 × 10^{40 }(see, on this web, Dali Crucifixion [Corpus Hypercubus]),
(1/4 × 9/16)^{2} × 54^{4} × 10^{32 }= 4.29981696 × 10^{40}.
On September 14, 2007, Mr. DePompeo noted that where 321.9937888. . . is the diagonal of the rectangular face of the logically outfitted absolute dead center triplecube, and 17 the number of the triplecube's interior diagonals (see, on this web, Sorted Diagonals),
429981696^{1/17} × 10^{2} = 321.99564518. . . = (1.618033466. . .)^{12} ~ φ^{12} ~ 321.9937888. . .,
and, where 1.170820393. . . is the locus of the logarithmic spiral (see, on this web, The Golden Bowls & the Logarithmic Spiral), and × 10^{7} the rational product of Genesis 1:1, בראשית ברא אלהים את השמים ואת הארץ, “In the beginning God created the heaven and the earth," (see, on this web, passim; also Leahy, Foundation, p. 505),
(429981696 × × 10^{7})^{1/12}/2^{1/2 }= 1.170818511. . . ~ 1.170820393. . ..
Deep is the superficiality of the boundary.