1.2.9 Numeri Mirabiles

The last two sec­tions re­vealed how the tan­gent space and Lie al­ge­bras are con­nect­ed, that being, the group de­riv­a­tive is a phase shift that de­pends on the Lie al­ge­bra of the said spe­cif­ic group. Al­ge­bra­ic layers exist be­neath the tan­gent space, which will be con­sid­ered in this sec­tion as we push deeper into the func­tion­al spaces. In the last sec­tion, we con­sid­ered the fol­low­ing group de­riv­a­tive op­er­a­tion

which served to con­cep­tu­al­ize the tan­gent space and its geo­met­ric arrange­ment to that of the Lie al­ge­bra. The de­riv­a­tive op­er­a­tion $G^{'}\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$ rep­re­sents a ve­loc­i­ty in the group space, which we denote as $G\left( \widehat{v} \right)$, and set it equal to $G^{'}\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$ from the ex­am­ple above

Matrix mul­ti­pli­ca­tion of the group $G\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$ by the Lie al­ge­bra re­veals the phase-shift and dy­nam­i­cal trigono­met­ric de­pen­den­cy con­ferred from this sub­group onto $G\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$, as seen in the con­cise de­riv­a­tive form

which is equiv­a­lent to $G\left( \widehat{v} \right)$ in a phase-shift­ed form

As we are aware, the group $G\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$ is the ex­po­nen­tial map­ping of its Lie al­ge­bra $A_{1}^{\lbrack 3\rbrack}(it)$, and this re­la­tion­ship holds for $G\left( A_{1}^{\lbrack 3\rbrack}\left( it + \frac{i\pi}{2} \right) \right)$with its Lie al­ge­bra $A_{1}^{\lbrack 3\rbrack}\left( it + \frac{i\pi}{2} \right)$. The matrix log­a­rithm rep­re­sents the in­verse of the ex­po­nen­tial map­ping, and eval­u­at­ing the log­a­rithm of $G\left( A_{1}^{\lbrack 3\rbrack}\left( it + \frac{i\pi}{2} \right) \right)$ yields the Lie al­ge­bra we ar­rived at via the tan­gent space con­sid­er­a­tion

This should not be con­sid­ered solely as the phase-shift­ed $A_{1}^{\lbrack 3\rbrack}(it)$ al­ge­bra, but in­stead as a phase-shift­ed group rep­re­sent­ing the tan­gent space ve­loc­i­ty, $\mathfrak{T}\left( \widehat{v} \right)$

The tan­gent ve­loc­i­ty is equal to the sum of its Lie al­ge­bra $A_{1}^{\lbrack 3\rbrack}(it)$ and the phase-shift de­pen­den­cy $A_{1}^{\lbrack 3\rbrack}\left( \frac{i\pi}{2} \right)$, which is con­ferred on $A_{1}^{\lbrack 3\rbrack}(it)$ by the sub­al­ge­bra de­not­ed as $\Phi_{1}\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$

which after set­ting $t = 1$, $\Phi_{1}$ be­comes

The $\Phi_{1}\left( A_{1}^{\lbrack 3\rbrack}\left( \frac{i\pi}{2} \right) \right)$ al­ge­bra re­sults from the eval­u­a­tion of the matrix log­a­rithm of a matrix log­a­rithm func­tion

and if we were to carry out re­peat­ed it­er­a­tive matrix log­a­rithm func­tions in­def­i­nite­ly

the re­sult­ing al­ge­bra will con­verge to the ap­prox­i­mate form

Per­haps more strik­ing is that all num­bers, except for one, zero, $e$, and any num­bers re­lat­ed to $e$ in the form $e^{e^{\ddots^{e}}},$appear to con­verge to either

for $Im(z) \in \mathbb{R}^{+}$, or to

for $Im(z) \in \mathbb{R}^{-}$. The in­tu­ition behind these two num­bers is that they are the two so­lu­tions to the equa­tion

which means they also rep­re­sent the two so­lu­tions to

given that for the de­fined $z$

The num­bers $\phi_{\infty}(z)$, ${\overline{\phi}}_{\infty}(z)$ are re­lat­ed to the mul­ti­val­ued Lam­bert $W$ func­tion, also known as the prod­uct log func­tion, which, for $w,z\in \mathbb{C}$, have the prop­er­ty that the ex­pres­sion

holds if and only if for some in­te­ger $k$, the fol­low­ing ex­pres­sion holds

where $W_{k}(z)$ is con­verse­ly re­lat­ed to the $we^{w}$ func­tion. For real number values, the ex­pres­sion

has so­lu­tions for $y$ if $x \geq \frac{- 1}{e}$. This end­point at $- \frac{1}{e}$ occurs when $y = - 1$,

and the pos­si­ble so­lu­tions for $x$ are well-demon­strat­ed in the ex­pres­sion's plot in the Figure 1.2.7.

We can con­sid­er the range of the Lam­bert $W$ func­tion in the com­plex plane by set­ting $w = (a + ib)$, for $a,b\in \mathbb{R}$, to ex­press $we^{w} = z$, as

and re­arrange the ex­pres­sion

where one por­tion of the result is real, and the other part is imag­i­nary. The imag­i­nary por­tion will equal zero when

and the real branch­es of the Lam­bert $W$ func­tion's range are found by solv­ing for the $a$ vari­able

Sim­i­lar­ly, by set­ting the real por­tion equal to zero, we find the imag­i­nary branch­es in the range are

We plot the range of the Lam­bert $W$ func­tion in Figure 1.2.8, where imag­i­nary branch­es are shown in red, and the real branch­es are shown in blue. Re­turn­ing to the num­bers $\phi_{\infty}(z)$ and ${\overline{\phi}}_{\infty}(z)$, we find they are en­dowed with the ex­tra­or­di­nary prop­er­ty that they main­tain the same result for their proper log­a­rithm and ex­po­nen­tial map­pings

im­ply­ing these num­bers rep­re­sent a meet­ing point in com­plex space of the ex­po­nen­tial and log­a­rithm func­tions. These num­bers can be ex­pressed as equiv­a­lent to the fol­low­ing neg­a­tive-valued $W_{k}(z)$ func­tions

To fur­ther con­sid­er the unique prop­er­ties of these num­bers, we start by find­ing the polar form of $\phi_{\infty}(z)$

where the radial com­po­nent is $r \approx 1.374557010743707$, and since $Re\left( \phi_{\infty}(z) \right) > 0$, the angle ar­gu­ment is found by the in­verse tan­gent func­tion

What stands out at first glance in the ap­prox­i­mat­ed $\phi_{\infty}(z)$ polar form

is that $\theta$ is equal to the imag­i­nary part of the com­plex number, $\phi_{\infty}(z)$,

co­in­cid­ing with the polar angle ar­gu­ment result $\phi_{\infty}(\theta)$ de­not­ed.

We con­sid­er the $\tan^{- 1}\left( \frac{b}{a} \right)$ func­tion by look­ing at its plot Figure 1.2.7 for com­plex num­bers $a + ib$, and $a,b\in \mathbb{R}$. The point with co­or­di­nates $\left( \frac{b}{a},\theta \right)$, co­in­cid­ing with the angle vari­able $\phi_{\infty}(\theta),$is dis­cerned

Eval­u­a­tion of the radial com­po­nent for $\phi_{\infty}(z)$, yields

having the unique prop­er­ty that its log­a­rithm equals the real part of the $\phi_{\infty}(z)$ func­tion

When com­bined with the polar form's angle, this means that $\phi_{\infty}(z)$ can be ex­pressed as the ex­po­nen­tial prod­uct of its real and imag­i­nary parts

and given that

we have a better math­e­mat­i­cal un­der­stand­ing of how $\phi_{\infty}(z) = \ln\left( \phi_{\infty}(z) \right) = e^{\phi_{\infty}(z)}.$Con­cern­ing the ex­pres­sion $we^{w} = z$, and the Lam­bert $W$ func­tion, the num­bers $\phi_{\infty}(z)$ and ${\overline{\phi}}_{\infty}(z)$ rep­re­sent the two con­form­ing and con­ju­gate­ly-re­lat­ed so­lu­tions for $w$

mean­ing these are the two so­lu­tions of the ex­pres­sion $we^{w} = w^{2} = \ln\left( w^{w} \right).$The well-known omega con­stant de­fines the unique real number so­lu­tion to the ex­pres­sion

rep­re­sent­ing the $W_{0}(1)$ Lam­bert $W$ func­tion with an ap­prox­i­mat­ed value

Anal­o­gous to $\phi_{\infty}(z)$ and ${\overline{\phi}}_{\infty}(z),$the fol­low­ing $\Omega$-re­lat­ed iden­ti­ties are a direct con­se­quence of $\Omega e^{\Omega} = 1$

With the $\Omega$ iden­ti­ties in mind, we define the $\phi_{\infty}^{- 1}\left( iA_{1}^{\lbrack 3\rbrack} \right)$ al­ge­bra as the in­def­i­nite­ly re­peat­ed neg­a­tive matrix log­a­rithm func­tion

which in­ter­est­ing­ly, "con­verges" to two al­ter­nat­ing con­ju­gate­ly-re­lat­ed points com­posed by the al­ge­bras

and each result $\phi_{\infty}^{- 1}\left( iA_{1}^{\lbrack 3\rbrack} \right)$ switch­es be­tween the two forms each it­er­a­tion. Specif­i­cal­ly, the number $\phi_{\infty}^{- 1}(z)$, within six­teen digits, is ap­prox­i­mate­ly equal to

for which, $\phi_{\infty}^{- 1}\left( \overline{z} \right)$ is the com­plex con­ju­gate of $\phi_{\infty}^{- 1}(z)$. The num­bers $\phi_{\infty}^{- 1}(z)$ and $\phi_{\infty}^{- 1}\left( \overline{z} \right)$ rep­re­sent so­lu­tions of the Lam­bert $W$ func­tion for $we^{w} = z$, and sat­is­fy the uni­tary con­di­tion that $|z| = 1$. The two points on the unit circle char­ac­ter­iz­ing their re­sult­ing $z$ values are

The $\phi_{\infty}^{- 1}(z),\phi_{\infty}^{- 1}\left( \overline{z} \right)$ num­bers have ex­po­nen­tial map­pings which equal the in­verse of their re­spec­tive com­plex con­ju­gates

lead­ing to the fol­low­ing iden­ti­ties

­

For com­plete­ness, there are also two ad­di­tion­al nu­mer­ic coun­ter­parts

which are al­ter­nat­ing "con­ver­gence" points of the in­fi­nite­ly it­er­at­ed log­a­rithm func­tion

Next, we would like to con­sid­er where these num­bers sit in the range of the Lam­bert $W$ func­tion. Since $\Omega$ is a real con­stant, it is straight­for­ward to see that it sits on the pos­i­tive real-axis of the range. It turns out the nu­mer­ic values of $\phi_{\infty}(z)$ and ${\overline{\phi}}_{\infty}(z)$ sit on the $({- W}_{- 1}, - W_{0})$ branch­es of Lam­bert $W$ func­tion's range, which we recall were the so­lu­tions de­rived from the ex­pres­sion $(a + ib)e^{(a + ib)} = z$

and neg­a­tive branch­es in blue, where the fol­low­ing con­stants lie on the branch cuts:

Figure 1.2.8 rep­re­sents a plot of the range of the Lam­bert $W$ func­tion cen­tered around the origin, where $\phi_{\infty}(z)$ and ${\overline{\phi}}_{\infty}(z)$ are lo­cat­ed on the neg­a­tive real-valued branch­es, cor­re­spond­ing to the num­bers $W_{- 1}( - 1),$and $W_{0}( - 1)$ on pos­i­tive real-valued branch­es. These num­bers, when taken to­geth­er, have the fol­low­ing in­verse re­la­tion­ships

In Figure 1.2.9, we plot the points rep­re­sent­ing the in­verse num­bers $\phi_{\infty}^{- 1}(z),\phi_{\infty}^{- 1}\left( \overline{z} \right),$ $\phi_{\infty}^{- 1}( - z),\phi_{\infty}^{- 1}\left( - \overline{z} \right)$, which also sit on the $\pm \left( W_{0},W_{- 1} \right)$ real branch cuts in the Lam­bert $W$ func­tion's range. The take­away from the pre­vi­ous two plots is that the $\pm \left( W_{0},W_{- 1} \right)$ branch cuts rep­re­sent the con­ver­gence points of the in­fi­nite­ly it­er­at­ed log­a­rithm of num­bers and their in­vers­es, and these branch cuts also serve as the con­ver­gence point of the log­a­rithm and ex­po­nen­tial func­tions. The ex­pres­sion, $we^{w} = z$ is rooted in the log­a­rithm func­tion, and it is not dif­fi­cult to see that it is a par­tic­u­lar form of the log­a­rith­mic spiral. For com­plex num­bers, the log­a­rith­mic spiral is de­fined as

where $z$ rep­re­sents the dis­tance from the origin in the com­plex plane, $\theta$ is the angle from the real-axis, and $a > 0,k\in \mathbb{R}$ are con­stants.

When $k = 0$, the log­a­rith­mic spiral ex­pres­sion de­fines the com­plex circle with the radial com­po­nent $a.$We start by set­ting $z = \left( \phi_{\infty}(z) \right)^{2}$ in the log­a­rith­mic spiral ex­pres­sion

and given that $\phi_{\infty}(z) = e^{\phi_{\infty}(z)}$

Taking the ex­po­nent por­tion from the ex­pres­sion above, we can solve to de­ter­mine the $\theta$ and$\ k$ values

and then define the log­a­rith­mic spiral ex­pres­sion

which has the vari­able values

After the ad­di­tion of an ex­po­nent vari­able $\tau$, the log­a­rith­mic spiral ex­pres­sion be­comes

which is plot­ted in Figure 1.2.10, where char­ac­ter­is­tic log­a­rith­mic spi­rals can be ap­pre­ci­at­ed.

The Swiss math­e­mati­cian Jacob Bernoul­li (1655-1705) re­ferred to the log­a­rith­mic spiral in Latin as the spira mirabilis, which trans­lates to mirac­u­lous spiral. Mirac­u­lous to him was the self-sim­i­lar­i­ty prop­er­ty that char­ac­ter­izes the log­a­rith­mic spiral, where the over­all shape of the spiral re­mains intact with each curve. Based on their math­e­mat­i­cal prop­er­ties, the num­bers we have con­sid­ered in this sec­tion are as mirac­u­lous as the log­a­rith­mic spi­rals were to Bernoul­li.

Ini­tial­ly, we start­ed this sec­tion con­sid­er­ing the phase shifts of the $G\left( \widehat{v} \right)$ group ve­loc­i­ty, which were de­pen­dent on both the tan­gent space and the sub­space which de­fined the phase space al­ge­bra,

The tan­gent space is re­vealed for $G\left( \widehat{v} \right)$, by set­ting $t = 0$

and the phase space al­ge­bra by taking the matrix log­a­rithm of $\left. \ G\left( \widehat{v} \right) \right|_{t = 0}$

The group de­riv­a­tive and Lie al­ge­bras con­sid­ered in matrix rep­re­sen­ta­tion theory are as­so­ci­at­ed with the Lam­bert $W$ func­tion. This is seen by set­ting $t = 1$ in the group ve­loc­i­ty ex­pres­sion $G\left( \widehat{v} \right) = iA_{j}^{\lbrack 3\rbrack}G\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$, and ob­serv­ing that this is a matrix form of the ex­pres­sion $we^{w} = z$

noting the tan­gent space al­ge­bra $iA_{j}^{\lbrack 3\rbrack}$ rep­re­sents a matrix ver­sion of $w$ from the ex­pres­sion $we^{w} = z$. The tan­gent space also has a unique and spe­cif­ic sub­tan­gent space, con­sid­ered by eval­u­at­ing the de­riv­a­tive of $\mathfrak{T}\left( A_{1}^{\lbrack 3\rbrack}(it) \right)$

for which we set $t = 0$, find­ing the smooth paths through the iden­ti­ty char­ac­ter­iz­ing the sub­tan­gent space vec­tors

We assign a vari­able $t$ to the $A_{1}^{\lbrack 3\rbrack}\left( \frac{i\pi}{2} \right)$ sub­tan­gent al­ge­bra, and define the group $G_{\mathfrak{T}}\left( A_{1}^{\lbrack 3\rbrack}\left( \frac{i\pi}{2}t \right) \right)$ as the ex­po­nen­tial map­ping

where $G_{\mathfrak{T}}\left( A_{1}^{\lbrack 3\rbrack}\left( \frac{i\pi}{2}t \right) \right)$ is con­tin­u­ous, having a group ve­loc­i­ty $G_{\mathfrak{T}}\left( \widehat{v} \right)$, found by eval­u­at­ing its de­riv­a­tive

Like the $G\left( \widehat{v} \right)$ group ve­loc­i­ty, $G_{\mathfrak{T}}\left( \widehat{v} \right)$ can be ex­pressed as the prod­uct

which, after set­ting $t = 1$, takes on the Lam­bert $W$ func­tion's as­so­ci­at­ed ex­pres­sion

Fit­ting­ly, the eval­u­a­tion of the $G_{\mathfrak{T}}\left( \widehat{v} \right)$ group ve­loc­i­ty for $t = 0$, yields the sub­tan­gent space al­ge­bra

If our start­ing point is from the per­spec­tive of the $G\left( \widehat{v} \right)$ group ve­loc­i­ty, its Lie al­ge­bra, $A_{1}^{\lbrack 3\rbrack}\left( i + \frac{i\pi}{2} \right),$is rooted in a de­pen­den­cy on the tan­gent space group. To find the dy­nam­i­cal state of $G_{\mathfrak{T}}\left( \widehat{v} \right)$ that yields this Lie al­ge­bra, we solve the fol­low­ing com­po­nent ex­pres­sion for the $t$ vari­able

and using this so­lu­tion, the $G_{\mathfrak{T}}\left( \widehat{v} \right)$ group ve­loc­i­ty is found to equal $\left. \ G\left( \widehat{v} \right) \right|_{t = 1}$

By chang­ing the $t$-vari­able to an it­er­a­tive form $t_{k}$, then defin­ing $t_{0}$ to be the in­de­pen­dent vari­able of the $G\left( \widehat{v} \right)$ group, the de­pen­den­cy value for $t_{k}$ can be gen­er­al­ized for any de­riv­a­tive sub­tan­gent group as

where $a_{0}^{'}$ is the non-vari­able al­ge­bra from the axis entry of the group de­riv­a­tive Lie al­ge­bra $A_{1}^{\lbrack 3\rbrack}\left( a_{0}^{'} \right)$, and $a_{k}$ is the non-vari­able axis entry al­ge­bra from the sub­al­ge­bra group. We apply this gen­er­al­iza­tion to the last ex­am­ple, which start­ed with the ex­pres­sion

where, $t_{1}$ is the vari­able that co­in­cides with the tan­gent space. We mul­ti­ply both sides by $e^{- \ln\left( \frac{i\pi}{2} \right)}$

then take the log­a­rithm of both sides

noting in this case, the de­riv­a­tive axis al­ge­bra is equal to

and the axis sub­al­ge­bra is equal to

for the vari­able set as $t_{1} = 1.$We sub­sti­tute the $a_{0}^{'}$ and $a_{1}$ sym­bol­ic forms into the ex­pres­sion

and solve for $t_{1}$

which match­es our gen­er­al­ized $t_{k}$ ex­pres­sion. The gen­er­al­ized $t_{k}$ de­pen­den­cy so­lu­tions for non-de­riv­a­tive sub­al­ge­bras are

which can be demon­strat­ed for the tan­gent space group

and the $A^{\lbrack 3\rbrack}\left( it_{0} \right)$ group's de­pen­den­cy for $t_{0} = 1$, is

For this case, the $t_{1}$ so­lu­tion is

which leads to the equal­i­ty

We come full circle and back to these in­trigu­ing and mar­velous num­bers to frame them in the con­text of group theory by defin­ing the Mirabilis Group.

De­f­i­n­i­tion (1.2.261) – The Mirabilis Group $\mathbf{G}_{\mathbf{\phi}}\left( \mathbf{A}_{\mathbf{j}}^{\left\lbrack \mathbf{3} \right\rbrack}\mathbf{t}_{\mathbf{\phi}}\mathbf{,}{\overline{\mathbf{A}}}_{\mathbf{j}}^{\left\lbrack \mathbf{3} \right\rbrack}\mathbf{t}_{\mathbf{\phi}} \right)$ is de­fined by the al­ge­bra sets

for $t_{\phi}\in \mathbb{C}$, for which, the num­bers are based on the prod­uct log func­tions

and char­ac­ter­is­tic to these num­bers, the fol­low­ing iden­ti­ties hold for the $A_{j}^{\lbrack 3\rbrack}\left( \phi_{\infty}(z)t_{\phi} \right)$, ${\overline{A}}_{j}^{\lbrack 3\rbrack}\left( {\overline{\phi}}_{\infty}(z)t_{\phi} \right)$ al­ge­bras

The Mirabilis Group rep­re­sents the Lie al­ge­bra anchor point lo­cat­ed at the cross­roads of the log­a­rithm and ex­po­nen­tial func­tions for the $X$-type groups. Our gen­er­al­ized $t_{k}$ de­pen­den­cy so­lu­tion $t_{k} = \frac{a_{0}}{a_{k}}$ also holds for these al­ge­bras and groups

where $a_{0}$ can now rep­re­sent any in­fi­nite number of axis entry sub­al­ge­bras that exist. For our fa­mil­iar group case

its sub­group de­pen­den­cy is rooted in the $A_{1}^{\lbrack 3\rbrack}\left( \phi_{\infty}(z)t_{\phi} \right)$ Lie al­ge­bra of the Mirabilis Group, and has a value for $t\phi$, equal to

lead­ing to the fol­low­ing iden­ti­ties

A log­a­rith­mic path of de­scent exists for all num­bers, and for those matrix rep­re­sen­ta­tions not taking on sin­gu­lar values when ap­ply­ing the in­fi­nite it­er­a­tive log­a­rithm func­tion, the path can be de­fined as fol­lows:

De­f­i­n­i­tion (1.2.267) – The Log­a­rith­mic Path of De­scent $\mathbf{\mathrm{T}}\mathbf{}$ is the unique set of values that com­pose the set

where each $\tau_{n}$, is de­fined re­cur­sive­ly by the ex­pres­sion

the func­tion, $\ln_{\lbrack n\rbrack}\left( A_{0} \right)$, de­fines the $n^{th}$ order log­a­rithm of the Lie al­ge­bra, $A_{0}$,

for $A_{0}$, with non-sin­gu­lar values $\forall\ln_{\lbrack n\rbrack}\left( A_{0} \right)$, and it is de­fined that, $\ln_{\lbrack 0\rbrack}{\left( A_{0} \right) = A_{0}},\ln_{\lbrack - 1\rbrack}{\left( A_{0} \right) = e^{A_{0}}.}$

To better un­der­stand what $\mathrm{T}$ is, we choose the com­plex number ex­am­ple $A_{0} = (2 + 1.2i),$and re­cur­sive­ly cal­cu­late the first ten values of $\tau_{n}$ from its log­a­rith­mic func­tion de­scent

where the suc­ces­sive $t_{n}$ values appear to be con­verg­ing to the value of $1$. Indeed, by $\tau_{113}$, the de­scent has ap­proached the con­ver­gence value of $1$ to within six­teen-digit nu­mer­i­cal pre­ci­sion, seen by the values $\tau_{109:113}$

Each $\tau_{n}$ value holds the key to what the value for $\ln_{\lbrack n - 1\rbrack}\left( A_{0} \right)$ is, seen using the ex­am­ple

for $A_{0} = (2 + 1.2i)$, where we find that $\tau_{1}\left( \ln\left( A_{0} \right) \right) = A_{0}$ holds

The re­la­tion­ship be­tween $t_{\phi}$ and $\tau_{n}$ is char­ac­ter­ized by the fol­low­ing equiv­a­lence

which is simple to show, given that the prod­uct sum is a tele­scop­ing one

im­ply­ing the fol­low­ing iden­ti­ty holds

In Figure 1.2.11, we plot the log­a­rith­mic paths of de­scent for the in­te­gers two through ten, where a log­a­rith­mic-like spiral forms on the de­scent into the it­er­a­tions of the log­a­rithm func­tion. Unlike the ex­am­ple of the log­a­rith­mic spiral, this is an inward-ro­tat­ing spiral, where the values of $\tau_{n}$ con­verge to the value of one. The paths of these unique in­te­gers are dis­tinct from one an­oth­er, defin­ing a sig­na­ture for each number and its in­for­ma­tion­al con­tent. This sig­na­ture is em­bod­ied in the short-hand no­ta­tion of the $t_{\phi}$ vari­able, where the ex­pres­sion

yields the rep­re­sent­ed number by the prod­uct with $\phi_{\infty}(z)$.

This sec­tion's pur­pose is to high­light the in­ter­de­pen­dence that exists with all the sub­spaces com­posed of unique sub­al­ge­bras. Each group is con­nect­ed to its ad­ja­cent sub­group through its $\tau_{n}$ vari­able, and that of the sub­group's $\tau_{n + 1}$ vari­able. If an action occurs in one space due to an ex­ter­nal in­flu­ence, the effect is a math­e­mat­i­cal prop­a­ga­tion of value changes for the $\tau_{n}$ vari­ables through the in­fi­nite set of al­ge­bras that com­pose its su­per­group.