The Spiral of Theodore and Cosmic Numbers

This interruption becomes evident if we observe that, having arrived at $\sqrt{17}$, the Spiral of Theodore completes a complete revolution (Figure 4). Here classical Mathematics stops and no Pythagorean has ever contemplated what we are now about to show you. We have said that the spiral of Theodore completes the first revolution upon itself having arrived at the construction of $\sqrt{17}$, the first revolution therefore passes from a length equal to 1 to a length equal to $\sqrt{17}$. We note that $\sqrt{17} - 1 = 3.1231...$. (3) We proceed with a second revolution which will therefore start from $\sqrt{18}$ and arrive at $\sqrt{55}$. We note that $\sqrt{55} - \sqrt{18} = 3.1735...$. (4) The third revolution will start from $\sqrt{56}$ and arrive at $\sqrt{113}$ thus obtaining $\sqrt{113} - \sqrt{56} = 3.1468...$; (5) then we will have at the fourth revolution $\sqrt{191} - \sqrt{114} = 3.1431...$. (6) The most astute will have already understood the mathematical marvel, only recently demonstrated², for which the difference between one revolution and another of the Spiral of Theodore tends precisely to $\pi$. So then our friend Tabit will have thus found a classical mirror of his contemporary result. If with the regularization he indicated we had the notable algebraic relation $1^2 + 2^2 + 3^2 + 4^2 + 5^2 + ... = \frac{\pi^2}{2}$; (7) now we have a classical geometric construction that links together not the squares, but the square roots $\sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}...$; (8) but which, again, leads to the result $\pi$. ² Hahn K. Harry, Schoenberger K. The Ordered Distribution of Natural Numbers on the Square Root Spiral. Arxiv. 2007. The cosmic numbers 5, 6 and 8 (Niko) In the first Number of Mathesis we highlighted how, despite the cardinality of Natural Numbers being infinite, or rather indefinite, the quality of all Numbers can be synthetically collected by the first 12 Numbers, exactly as the indefinite chromatic spectrum can be symbolically classified through the use of the number 7 defining the 7 colors of the rainbow. The understanding of the symbolism of the first 12 Numbers is therefore fundamental to have an intuition of the meaning of all other Numbers as well. In our article we had suspended at Number 4 our exposition due to the fact that $\sigma(4) = 1 + 2 + 3 + 4 = 10$, that is, the triangular of Number 4 produces Number 10 thus referring to a first conclusion of a synthetic cycle that has 4 terms and that unfolds in 10 phases. In this article we want to continue our exposition, but in a slightly different way. Instead of proceeding progressively, we will proceed thematically speaking of those Numbers that can rightfully be called Cosmic Numbers, namely 5, 6 and 8 and which are respectively associated with the Microcosm, the Macrocosm and the Metacosm. These three Numbers progressively explain the interaction of the binary 2 with the ternary 3 being $2 + 3 = 5$; $2 \times 3 = 6$; $2^3 = 8$: They therefore refer to the relationship between Man-Woman (expressed by Number 2) and the Divine Trinity (expressed by Number 3). This interaction is the origin of the Cosmos in the sense of created order that unfolds first in the individual with Number 5, then in a universal sense with Number 6 and finally in an uncreated or divine sense with Number 8. Number 5 or Microcosm. Number 5 is the number of the Microcosm, that is, of Man awakened in the Spirit. It is the symbol of Light, particularly of the light that in Man is the intellect. In symbolic mathematics addition is not always commutative, so $5 = 1 + 4$ is not equivalent to $5 = 4 + 1$. In the first case number 1 is the will of Man that dominates the 4, which in this case represents the 4 interior and exterior elements, and arranges them by illuminating them. It is the dominion of spirit over matter, of will over fatality. Man here is a "son of Light". He is human word initiated into the Divine Word. In this case meditation on 5 is operated with the representation of a pentagram with the point facing upward and a central point. Eventually a crown on the upper point of the star indicates the royalty achieved by the adamantine will. Conversely, $5 = 4 + 1$ means matter, fatality that dominates over the will of man here impotent and miserable, a slave. In this case the point of the pentagram is directed downward and from symbol of Light becomes symb