1.4.6 The Conic Group

At this point, we de­scribe the sym­me­try group rep­re­sen­ta­tive of all pos­si­ble ways to per­mute the equi­lat­er­al tri­an­gle in the $\mathbb{C}^{3}$ space, in­clud­ing the eight points on the unit circle point set $\left\{ 1,( - 1)^{\left( \frac{1}{4} \right)},i,( - 1)^{\left( \frac{3}{4} \right)}, - 1, - ( - 1)^{\left( \frac{1}{4} \right)}, - i, - ( - 1)^{\left( \frac{3}{4} \right)} \right\}$, for three or­thog­o­nal­ly arranged unit cir­cles. To do this, we use gen­er­a­tors with cone number de­ter­mi­nants. To start, we choose two el­e­ments, the first $\in \mathcal{X}$, and the second $\in \mathcal{Y}$

for which, $\det\left( X_{2,1} \right) = - ( - 1)^{\left( \frac{3}{4} \right)},$ $\det\left( Y_{2,1} \right) = ( - 1)^{\left( \frac{3}{4} \right)}$. The ra­tio­nale for choos­ing el­e­ments having cone number de­ter­mi­nants is that the prod­uct re­la­tion­ships for $\mathcal{Y},\mathcal{X}$, result in el­e­ments having de­ter­mi­nants which are either a cone number or an axis number. After con­sid­er­ing how we went about find­ing all el­e­ment per­mu­ta­tions in the cone group, it might seem like an even more daunt­ing task to do so for the sym­me­try group the cone group be­longs to. How­ev­er, it turns out that it can be much more straight­for­ward, given that el­e­ments $\in \mathcal{Y}$ can be cone num­bers. The el­e­ments that we will be study­ing are in the $Y_{1,k}$ group. Defin­ing a set $\mathcal{C}$ with el­e­ments having the fol­low­ing arrange­ment of num­bers:

Since we are deal­ing with finite el­e­ments, using this set amounts to turn the dial on a com­bi­na­tion lock to arrive at any pos­si­ble per­mu­ta­tion in the sym­me­try group. Now, the task is to show how to do just this.

De­f­i­n­i­tion (1.4.105) – We define the Sym­met­ric Conic Group $\mathbf{S}_{\mathbf{3}\mathbb{C}}^{\boldsymbol{\Delta}}$, as group under com­po­si­tion that sits in the com­plex geo­met­ric space $\mathbb{C}^{3 \times 3}$, with a set of per­mu­ta­tion el­e­ments $A^{\Delta}$ with su­per­script no­ta­tion $\Delta$, rep­re­sent­ing the cone.

The­o­rem (1.4.106) – For $A^{\Delta}$as a set, the set of per­mu­ta­tions ${\in S}_{3\mathbb{C}}^{\Delta}$, forms a group under com­po­si­tion

Proof – Ver­i­fi­ca­tion of group axioms

• Clo­sure: All el­e­ments $\in A^{\Delta}$are bi­jec­tive, and the com­po­si­tion of any el­e­ments $\phi \circ \psi\ \in A^{\Delta}$ is also bi­jec­tive.

• As­so­cia­tiv­i­ty: Per­mu­ta­tions are func­tions, and their com­po­si­tions are as­so­cia­tive.

• Iden­ti­ty: The iden­ti­ty func­tion $\epsilon_{A^{\Delta}}$ bi­jec­tive­ly maps all el­e­ments of the set $A^{\Delta}$ to them­selves $\left( \epsilon_{A^{C\Delta}}:A^{\Delta} \rightarrow A^{\Delta} \right)$.

• In­verse: $\phi$ is a per­mu­ta­tion and is thus bi­jec­tive, and there exists a func­tion $\phi^{- 1}$ that is also bi­jec­tive.

$\square$

The­o­rem (1.4.107) – The Di­he­dral Conic Group $\mathbf{S}_{\mathbf{3}\mathbb{C}}^{\boldsymbol{\Delta}}$ con­tains 3,072 unique per­mu­ta­tion el­e­ments $\in \mathbb{C}^{3}$, where el­e­ments of $X_{j}\in \mathcal{X}$ have 512 per­mu­ta­tions, el­e­ments of $Y_{j}\in \mathcal{Y}$ have 512 per­mu­ta­tions, and all de­ter­mi­nants are cone number/axis number-valued.

$\mathbf{Proof}\mathbf{-}$ Star­ing with the el­e­ments

we find all three el­e­ments in the $\mathcal{C}$-subset, where mul­ti­ple com­bi­na­tions of per­mu­ta­tions can be con­sid­ered, of which, we present the fol­low­ing prod­ucts that result in the el­e­ments $\in \mathcal{C}$:

Since each of the three matrix en­tries in the per­mu­ta­tion may take any value in the set

$\left\{ 1,( - 1)^{\frac{1}{4}},i,( - 1)^{\frac{3}{4}}, - 1, - ( - 1)^{\frac{1}{4}}, - i, - ( - 1)^{\frac{3}{4}} \right\}$,

and given there are six per­mu­ta­tion forms, a simple cal­cu­la­tion shows the total number of per­mu­ta­tions is

To show the power of these el­e­ments using the el­e­ments of $\mathcal{C}$, it be­comes pos­si­ble to arrive at all our orig­i­nal el­e­ments from the cone group, by the fol­low­ing prod­ucts

for $X_{2,1}$, a prod­uct el­e­ment.