1.6.1 The Singularity

At the end of Chap­ter 5 we touched upon the sphere of tan­gent spaces as the sin­gu­lar­i­ty sit­u­at­ed at the origin $\in \mathbb{C}^{3}$ space. In math­e­mat­ics, the term sin­gu­lar­i­ty de­notes a point where a given math­e­mat­i­cal object, such as an equa­tion or a sur­face, is not de­fined or ceases to be well-be­haved in the an­a­lyt­ic sense.¹ As an in­di­vid­ual math­e­mat­i­cal object, the in­fi­nite sphere of tan­gent spaces $\mathfrak{T}\left( S_{3\mathbb{C}}^{\infty} \right)$ is con­tin­u­ous and well-be­haved, as shown in the last chap­ter, and would not fit the math­e­mat­i­cal de­f­i­n­i­tion of a sin­gu­lar­i­ty in this regard. We use the term sin­gu­lar­i­ty mind­ful of the per­spec­tive from which it is being an­a­lyzed, and when using real analy­sis, this would meet the cri­te­ria for a math­e­mat­i­cal sin­gu­lar­i­ty. To demon­strate this point, we can take the func­tion $f(x) = \frac{1}{x^{2} - 1},\ x\in \mathbb{R}$, and plot this in the Carte­sian plane, where it is read­i­ly seen, for $x = \pm 1$, the func­tion goes to in­fin­i­ty (Figure 1.6.1.a), for $x$ on the in­ter­val $\lbrack - 1.5,1.5\rbrack$.

By taking the same func­tion for $x$ as imag­i­nary-valued, we find the com­plete­ly dif­fer­ent result of a con­tin­u­ous curve, with­out the sin­gu­lar poles for the real-valued func­tion, as seen above in Figure 1.6.1.b. A sur­face plot of the func­tion $f(x) = \left| \frac{1}{X^{2} - 1} \right|$, for $X = (x + ix),\ x\in \mathbb{R}$, as a com­bi­na­tion of the real and imag­i­nary func­tion com­po­nents, re­veals the con­tin­u­ous func­tion plot­ted in Figure 1.6.2.

By the de­f­i­n­i­tion, for $x\in \mathbb{R}$ , the func­tion $f(x) = \left| \frac{1}{X^{2} - 1} \right|$admits no real so­lu­tions for $x$, in its de­nom­i­na­tor, when solved for zero

which re­strains the func­tion from going off to in­fin­i­ty. For $x\in \mathbb{C}$, in the above, we could choose $x = \left( - \frac{1}{2} + \frac{i}{2} \right)$, or $\left( \frac{1}{2} - \frac{i}{2} \right)$ to find, once again, sin­gu­lar­i­ties lurk­ing in the func­tion. The case in point, the func­tion $f(z) = \frac{1}{z^{2} - 1},$for $z = (x + iy),$ $x,y\in \mathbb{R}$, has sin­gu­lar­i­ties which form on its sur­face for $y = (ix - i),(ix + i)$, and $x = ( - 1 - iy),(1 - iy)$, seen in Figure 1.6.3, below.

We con­sid­er a sim­i­lar con­cept for the $D_{3}/S_{3}$ group, a sub­group $\subset S_{3\mathbb{C}}^{\infty}$, and if our per­spec­tive was con­fined to the real num­bers, rep­re­sen­ta­tive of the dis­joint hy­per­bo­lae in the $X_{1},X_{2},X_{3}$-planes, it would appear as dis­con­nect­ed for ab­solute number values $< 1$ (Figure 1.6.4).

From the per­spec­tive of $\mathbb{C}^{3}$ and the $S_{3\mathbb{C}}^{\infty}$ group, we know that the hy­per­bo­lae are not dis­con­nect­ed, when taken in the con­text of their Lie al­ge­bra sub­group rep­re­sen­ta­tion, and on the sur­face of the con­tin­u­ous sphere of tan­gent spaces group $\in \mathbb{C}^{3}$.

1. "The De­fin­i­tive Glos­sary of Higher Math­e­mat­i­cal Jargon — Sin­gu­lar­i­ty". Math Vault. 2019-08-01.