According to given results of the set-theoretical approach of E-infinity describing the five-dimensional Kaluza-Klein spacetime [4] [11] , the amount of baryonic matter of the universe emerges as

Ω b = 1 2 φ 5 = 0.04508 … (about 4.51%). (4)

The dark matter amount can be recast in the very simple form of

Ω d = 1 100 2 φ − 5 = 0.22180 … (about 22.18%). (5)

It may not be pure fortuity but nevertheless surprising that both amounts show a reciprocity relation. This can be seen, if we write down the remaining dark energy amount as the difference to the entire mass in a more persuasive form

Ω Λ = 1 − 1 10 ( 5 φ 5 + ( 5 φ 5 ) − 1 ) (about 73.31%). (6)

Such coincidence means that both mass constituents should not be considered independent of each other. Mathematically, reciprocity is found, for instance, if one considers volume in comparison to surface, or particles in comparison to waves. Relevant topological arguments from the set theory are summarized by El Naschie [11] . However, the denominator factor of 100 in Equation (5) may be interpreted as a coupling term equal to the normed dimensions of the universe [12] .

The entire dark constituents yields

Ω d + Λ = 5 2 φ 2 = 0.954915 … (7)

By the way, the simple result of 5φ² represents the five-dimensional surface of the pre-quantum wave being the cobordism of the topological volume of φ⁵ of the Kaluza-Klein five-dimensional manifold [4] [13] [14] .

The estimated constituents are fairly well consistent with measurements of the Wilkinson Microwave Anisotropy Probe mission (WMAP) [15] .

El Naschie recently pointed out that D = φ⁻⁵ = 11 + 0.0901699… obviously represents the fractal dimension D_M of Witten’s M-theory [16] [17] .

Because φ⁻⁵ = 11 + φ⁵, its beautiful hierarchical form (or continued fraction representation) can be expressed as

φ − 5 = 11 + 1 11 + 1 11 + 1 11 + … . = 11 + 0.0 9 0 1699 … ,

in contrast to the most unique representation for the Golden mean

φ = 5 − 1 2 = 1 1 + 1 1 + 1 1 + … . = 0. 618 0 33989 …

An alternative approach for the mass constituents of the universe by applying the Archimedes’ constant π instead of φ yields almost equal amounts in comparison to results according to Equations (4) to (7) [18] [19]

Ω b = π − 3 π = 0.0450703 … , (8)

Ω d = π ( π − 3 ) 100 = 0.221875 … , (9)

Ω Λ = 1 − 1 10 ( 10 ( π − 3 ) π + π 10 ( π − 3 ) ) = 0.733054 … , (10)

and finally

Ω d + Λ = 3 π = 0.9549296 … (11)

for the sum of the dark constituents (matter and energy). It may turn out in the future whether this replacement may be of any physical importance. However, some numerical approximations connect Hardy’s entanglement probability φ 5 with Archimedes’ constant π and the inverse of Sommerfeld’s fine structure constant α ¯ 0 , respectively [20]

π − 3 ≈ 16 137 − 24 = 0.14159292 (12)

α ¯ 0 ≈ 137 + 2 5 φ 5 = 137.03606 … (13)

If one deals with particle entanglement, superconductivity represents a physical phenomenon suspecting such property. Some years ago, the present author suggested linking the optimum hole doping σ_o of high-T_c superconductors with Hardy’s φ⁵ entanglement probability [18] . The fractal-hierarchical structure of electrons entangled in pairs obviously determines this optimum near a quantum critical point that can be linked on the one hand with the universal fractal constant δ₁ = 8.7210972 … of the renormalized quadratic Hénon map (remember the quadrilateral layer structure of the cuprates)

σ o ≈ 2 δ 1 = 0.2293 , (14)

on the other hand with Hardy’s entanglement probability

σ o ≈ 8 π φ 5 = 0.2296 , (15)

this time connecting φ and π, the most important universal numbers of the cosmos.

The quotient of the Fermi speed to the Klitzing speed would yield

ν F / ν K ≈ 2 π φ 5 = 0.0571 , (16)

which is again proportional to φ 5 [20] .

The fractal nature of electronic response in superconductors was documented some years ago by scanning tunneling microscopy [21] [22] and is consistent with generated d-wave fractal patterns in superconductors as a consequence of antiferromagnetism [23] .