RUBIDIUM (37), LANTHANUM (57) &
 THE LOGICALLY OUTFITTED CUBE(S)

The crystal structure of Rubidium, atomic number 37,¹ is body-centered cubic, as here illustrated:²

This structure formally coincides that of the logically outfitted dead center cube with the addition of places for 8 atoms, each at the center of one of 8 cubic subdivisions, as here illustrated:

Let the formal structure of the logically outfitted triple dead center cube be superimposed on a triple-Rubidium-3-cube, as here illustrated:

Then Rubidium, atomic number 37, formally coincides the trinary logical structure whose cubic diagonals are 37 in number.   The body-diagonal of this crystal structure = 3704.66989082698 pm (the number of units 3700 + the Feigenbaum constant, d).  The triple-Rubidium-3-cube has in addition to the 63 occupied cube vertices of the logically outfitted triple dead center cube 24 occupied cubic centers.  The ratio of the 63 atoms occupying the vertices of this triple-cube to the 24 atoms occupying its cubic centers is approximately the square of the ratio of the atomic weight of Lanthanum, atomic number 57, to that of Rubidium, atomic number 37, (La_w/Rb_w)², or, where j is the golden section ratio, approximately φ².³

Further, when the logically outfitted triple dead center cube is formally superimposed on the triple-Rubidium-3-cube as indicated, the places occupied by the 24 cubic centers assimilated to the logical structure are centers for 24 3-place diagonals, as here illustrated:

These diagonals are formally shorter than the 3-place diagonals of the logically outfitted triple dead center cube.  Also, unlike the latter, the 3-place diagonals in this set are of identical length.  The total number of diagonals in the two sets (noting that the 6 diagonals supported by logically blue atomic positions follow the path of 3 of the blue diagonals in the original logical arrangement) is 61 (the original 37 + these 24).   Neatly, the 24 diagonals in the triple-Rubidium-3-cube occupy 61 atomic places, as illustrated, exactly 37 of which belong to the original structure of the logically outfitted triple dead center cube.  As in the purely logical analysis the 37 diagonals naturally sort themselves into a set of 20 exterior-interior surface and 17 interior (non-surface) diagonals, so, analogously, the 37 atomic places here coinciding the logical positions sort themselves into 20 occupying the outer lengthwise sides of the triple-Rubidium-3-cube crystal structure and 17 occupying the inner lengthwise plane.  Also, 48 of the 61 atoms supporting the 24 additional diagonals occupy logically yellow or green places while 13 occupy logically blue places.   Just as the blue diagonals in the triple dead center cube form a unit uniting the poles of the triple cube, so the atoms occupying logically blue places in the Rubidium crystal structure do likewise.⁴  The ratio 48/13 = (37/10)-(13×10)⁻¹.

If the total number of logical diagonals in the triple-Rubidium-3-cube crystal structure is 61 = 37+24, then, where 82944 is the common coefficient of the four fundamental force constants,⁵ and 216 pm is the value of the covalent radius of the Rubidium atom, and 169 pm is the value of the covalent radius of the Lanthanum atom, and 37 is the original number of diagonals in the logically outfitted dead center cube, and 24 is the number of additional diagonals in the logically outfitted Rubidium crystal structure, the deep structure of the ratio of the covalent atomic radius of Rubidium (37) to that of Lanthanum (57) is the following:

10^{82944/(37×242)}/10²(37+24) = 216/169.⁶

Notes

¹  The resonance frequency of the rubidium atom (⁸⁷Rb) = 6834682608 Hz ~ 82944² Hz.  GPS satellites carry both cesium and rubidium clocks.  The writer is indebted to Mr. J. Iuliano for calling attention to the numerical relation of the elements Rubidium and Lanthanum.  On July 11, 2000, he communicated to the writer his observation concerning the numbers 37 and 57 as they appear in the periodic table.  He discovered (the latest values for the physical constants have been substituted by the writer) that, where Rb_n = 37 (the atomic number of Rubidium), and La_n = 57 (the atomic number of Lanthanum), and Rb_w = 85.4678 (the atomic weight of Rubidium), and La_w = 138.9055 (the atomic weight of Lanthanum), and 2807/1579 is the ratio of certain eigenvalues in Mr. Iuliano's research on Fermat's last theorem, e^(16/π)(Rbw^/Law^)(2807/1579) × (1/Rb_n+1/La_n) α^−1/2, the inverse square root of the fine structure constant.   For the equivalent formulas, e^{2π−1(e2807.000184[2/1579])} × (1/57+1/37) = 11.7062372 . . . , and e^{2π−1(e[2808-(1-[(37-37−1)10πδ]−1)][784+5.5]−1)} × (1/57+1/37) = 11.7062372007985 . . . , see, on this web, 82944 & the Four Fundamental Forces & the God Particle.   Mr. Iuliano also noted (with similar substitutions by the writer) that, where the critical exponent β = .37, and the Feigenbaum bifurcation constant δ = 4.669201609102990. . ., and the gravitational constant G = 6.67390 × 10⁻¹¹, (β × 10^-5)/(δRb_wLa_w) G.  For the critical exponent β = .37, see, below, note 3.
²  Illustration from an interactive viewing at M. Winter, "WebElements^TM Periodic Table."  WebElements Ltd, UK.  Online.  Available: http://www.webelements.com.  July 14, 2000.
³  While other elements have Rubidium's cubic body-centered form, only in the case of Rubidium (37) and Lanthanum (57) does the weight ratio correspond to the structural ratio as indicated.  These elements also have the neat weight/number interrelational property such that (Rb_w+La_w)(1/Rb_n+1/La_n) 10.  Similarly, on the analogy to the composition of the value (above, in the text) of the triple cubic crystal body-diagonal, 57^.37(Rb_w+La_w) = 1000 + δ/π.  The inverse of δ/π appears in the fine structure formula, [7800-(π/δ)⁴]^1/2/.6444762592780 = α⁻¹ = 137.035999594927.  For the formulas equivalent to this last, see, on this web, 82944 & the Four Fundamental Forces & the God Particle, note 14, and also, on this web, Report on a Dream & Related Matters & the God Particle, passim.  Also, as pointed out by Mr. Iuliano, where α = 2.502907875095892, the Feigenbaum distance constant, (La_n/Rb_n)(La_w/Rb_w) α.   For the fundamentality of the occurrence of 37, both alone & in its ubiquitous conjunction with 57, see, on this web, 82944 & the Four Fundamental Forces & the God Particle, including additional references passim.  Further to the point, on July 16, 2000, Mr. Iuliano called the writer's attention to the fact that (La_w - Rb_w)16/15 = 57.  The writer noticed the relationship between 16/15, the critical exponent .37, the fine structure constant, α, and the edge of the dead center cube, 10368^.5, as follows: [([16/15]^.37)² × 1E33]^−1/256/10368^.5 = α.  For the equivalent of this formula in terms of the product of the ratios of the nuclear and universal force exponents of 82944, cf. ibid., note 9.  In this last connection, note here also that, where α is the fine structure constant, and β is the critical exponent .37, and φ is the golden section ratio, 82944^1/p + φ(.82944-[2807/1579]⁻¹⁶) = (α⁻¹)^2b.  (For .82944, cf., below, note 4.)  Likewise, 82944^1/π + φ(.82944-[(Rb_nLa_n)/(Rb_n+La_n)]^-8β) = 82944^1/π + φ(.82944-[(Rb_w+La_w)/10]^-8β) = (α⁻¹)^2β.  Where 3704.66989082698 is the value (above, in the text) of the triple cubic crystal body-diagonal of Rubidium, 82944^1/π + φ(.82944-[82944/3704.66989082698]^-8β) = (α⁻¹)^2β.  Coincidentally, on July 27, 2000, Mr. Iuliano called the writer's attention to the values of the six quarks as presented in B. Greene, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory (New York, 1999).  The writer noted that, where (following Greene) u is the atomic mass unit value of the up quark, .0047, and d is the value of the down quark, .0074, and c is the value of the charm quark, 1.6, and s is the value of the strange quark, .16, and t is the value of the top quark, 189, and b is the value of the bottom quark, 5.2, it is also the case that 82944^1/π + φ(.82944-[([udcstb/(u+d+c+s+t+b)]10³)⁻¹]^-8β) = (α⁻¹)^2β.  For the length of the edge of the dead center cube, cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), p. 434, et passim.  For the fundamental role of the ratio 22.4 (Rb_nLa_n)/(Rb_n+La_n) (Rb_w+La_w)/10 82944/3704.66989082698 [(udcstb/[u+d+c+s+t+b])10³]⁻¹, uncovered in the logico-mathematical analysis of Foundation, as well as for that of 16/15 (= ), and of its square (= ), see, ibid., Index of Numbers and Symbols, pp. 687ff.  Note that the body-diagonal of the triple-Rubidium-3-cube crystal structure, 3704.66989082698 pm, ~ one10-billionth of the Egyptian remen (cf., on this web, Sorted Diagonals, Addendum 5).
⁴  Cf., on this web, Sorted Diagonals.  Where (y+g)/b is the ratio of the total length of the yellow & green diagonals to the total length of the blue diagonals in the logically outfitted triple dead center cube, and (Y+G)/B is the analogous ratio in the triple-Rubidium-3-cube, and a_e is the electron magnetic moment anomaly, then [b(Y+G)/B(y+g)]^-.1(37/20) × (1 + α_e) .82944 × (1 + α_e)^.25.  (For .82944, cf., above, note 3.)  For more on the lengths of the diagonal sets in the logically outfitted triple dead center cube, see, on this web, Sorted Diagonals, Addendum 10.
⁵  Cf., on this web, 82944 & the Four Fundamental Forces & the God Particle and The v_maxwell^1/16 × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe.
⁶  This formula is a transcription into logically significant terms, as demonstrated, of a formula discovered by Mr. Iuliano and forwarded to the writer on August 29, 2000.   Mr. Iuliano's formula relating the covalent radii of Rubidium and Lanthanum in nanometers is as follows:

10^{82943.99305/(666×32)}/10²(37+24) = .215997356/.169.

This formula should be compared to that first forwarded by Mr. Iuliano on February 23, 1999 (cf. on this web, 82944 & the Four Fundamental Forces & the God Particle).  Mr. Iuliano also reminds the writer that the covalent atomic radius of Hydrogen (1) is 37 pm.  Note also that 10^{82944/(37×242)}/10²(37+24) 8^.25/3^.25, the square root of the ratio of the relative length of the shortest diagonals in the logically outfitted triple cube to that of the diagonals in the triple-Rubidium-3-cube.  The latter ratio is at once the ratio of the diagonal of the base to the height of the smallest square-based pyramid the area of whose face is a whole number not 1 and one half the area of the base of the pyramid, and the sum of whose ascending edge and one half the base of its face is to the height of its face as φ : 1 (cf., on this web, Thinking Creation Ex Nihilo; cf., also, Leahy, Foundation, Section IV.2). 

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