1.7.2 The Geometric Primes and the Higher-Ordered Forms

This book fo­cus­es on the equi­lat­er­al tri­an­gle's group rep­re­sen­ta­tion as our reg­u­lar poly­gon form $\in \mathbb{C}^{3}$ space. Higher-di­men­sion­al forms exist that are simple to define given the work we have al­ready done with the $S_{3\mathbb{C}}^{\infty}$ sym­me­try group. In the process of con­sid­er­ing these forms, we find the as­so­ci­a­tions that are in­her­ent to odd and even-num­bered vertex sets of the reg­u­lar poly­gon. This will have future im­pli­ca­tions re­lat­ed to the Poin­caré-Hopf the­o­rem, which states that there are no smooth vector fields on an even-di­men­sion­al $n$-sphere with­out sources or sinks.

Before pro­ceed­ing, we con­sid­er the eigen­val­ues for the $A\left( gp_{2} \right)$ Lie al­ge­bra gen­er­a­tors

which have in di­ag­o­nal­ized eigen­val­ue ma­tri­ces

We recall the pri­ma­ry gen­er­a­tors of the $A\left( p_{2} \right)$ set are

find­ing these Lie al­ge­bras have com­plex eigen­val­ues equal to the values of the ver­tices of the in­scribed equi­lat­er­al tri­an­gle in the com­plex unit circle.

The fun­da­men­tal theory of al­ge­bra states that at least one com­plex root exists for every single vari­able and non-con­stant poly­no­mi­al having com­plex co­ef­fi­cients. Given that real co­ef­fi­cients are com­plex num­bers with imag­i­nary por­tions equal to zero, the theory also in­cludes poly­no­mi­als with real co­ef­fi­cients. When the fun­da­men­tal theory of al­ge­bra is ap­plied to an $n \times n$ matrix and its char­ac­ter­is­tic poly­no­mi­al of degree $n$, it im­plies that this poly­no­mi­al can be fac­tored into a prod­uct of $n$ linear terms.

For clar­i­ty in regard to ap­ply­ing the fun­da­men­tal theory of al­ge­bra with matrix theory, we take a closer look at cal­cu­lat­ing the eigen­val­ues for the matrix $Y\left( gp_{2} \right)_{2}$. We start by set­ting $\left| Y\left( gp_{2} \right)_{2} - \lambda I \right|$to equal zero

and it is ap­par­ent that for the eigen­val­ues of the matrix $Y\left( gp_{2} \right)_{2}$ to exist, the de­ter­mi­nant of the matrix $\left| Y\left( gp_{2} \right)_{2} - \lambda I \right|$must be zero. The char­ac­ter­is­tic poly­no­mi­al of $Y\left( gp_{2} \right)_{2}$ is found by cal­cu­lat­ing the de­ter­mi­nant of $\left| Y\left( gp_{2} \right)_{2} - \lambda I \right|$

We sub­sti­tute this poly­no­mi­al result back into the ini­tial ex­pres­sion $\left| Y\left( gp_{2} \right)_{2} - \lambda I \right| = 0$

and after solv­ing for $\lambda^{3}$, we find

which has the three so­lu­tions in com­plex space $\lambda = \left\{ e^{\left( \frac{2i\pi}{3} \right)},e^{\left( \frac{4i\pi}{3} \right)},e^{(2i\pi)} \right\}$. Having the eigen­val­ue so­lu­tions of the char­ac­ter­is­tic poly­no­mi­al, and by the im­pli­ca­tions of the fun­da­men­tal theory of al­ge­bra, we can ex­press the poly­no­mi­al $\lambda^{3} - 1$ as the fol­low­ing prod­uct of linear terms

We plot this prod­uct of linear terms in the figure below, for which the func­tion cross­es the $y$-axis at $- 1$ when $\lambda = 0$, having roots at the point on the graph when the func­tion cross­es the $\lambda$-axis at $1$.

with roots equal to the eigen­val­ues of the matrix $Y\left( gp_{2} \right)_{2}$.

In Figure 1.7.11, the roots in the com­plex plane have the same arrange­ment as the in­scribed equi­lat­er­al tri­an­gle de­fined as the $gp_{2}$ geo­met­ric prime in the last sec­tion. In con­trast, the matrix gen­er­a­tors' eigen­val­ues, which com­prise the set $\epsilon^{*} \times A\left( gp_{2} \right) = \left\{ X_{1},X_{3},X_{2} \right\}$

are all real-valued, with char­ac­ter­is­tic poly­no­mi­als of the form

and the so­lu­tions to the equa­tion

rep­re­sent the eigen­val­ues of the $\epsilon^{*} \times A\left( gp_{2} \right)$ set of ma­tri­ces, $\lambda = \left\{ - 1,1,1 \right\}$.

which are equal to the eigen­val­ues of the matrix $Y\left( gp_{2} \right)_{2}$.

As pre­vi­ous­ly dis­cussed, these Lie al­ge­bra gen­er­a­tors will take on com­plex values, and for any matrix de­fined as the prod­uct $\kappa\left( \epsilon^{*} \times A\left( gp_{2} \right) \right)$, for $\in \mathbb{C}$, its eigen­val­ues will equal

We con­tin­ue to con­sid­er the higher-di­men­sion­al forms by first look­ing at the in­scribed square in Figure 1.7.13, which, like the equi­lat­er­al tri­an­gle, can be ro­tat­ed uni­form­ly in the com­plex plane, rep­re­sent­ing an or­der­ly cyclic shift­ing of the ver­tices.

It is im­por­tant to note that this cyclic shift of the ver­tices occurs in a single com­plex plane. In a sim­i­lar manner to pre­vi­ous cyclic per­mu­ta­tion ma­tri­ces, the set of $Y\left( \mathbb{Z}_{4} \right)_{j}$ ma­tri­ces are

but what dif­fers with the $gp_{2}$ ma­tri­ces, $Y\left( p_{2} \right)_{j}$, is that the third matrix in the set, $Y\left( \mathbb{Z}_{4} \right)_{3}$, like the iden­ti­ty, is not ca­pa­ble of gen­er­at­ing all the per­mu­ta­tion el­e­ments of the set. This can be seen by taking odd and even in­te­ger powers of the $Y\left( \mathbb{Z}_{4} \right)_{3}$ matrix

which re­sults in either the iden­ti­ty matrix, $Y\left( \mathbb{Z}_{4} \right)_{1}$, or itself. The reason behind this is that the matrix $Y\left( \mathbb{Z}_{4} \right)_{3}$ shifts all ver­tices by two lo­ca­tions and given that the four ver­tices of the square are an even number, only half of the cyclic ro­ta­tions are pos­si­ble. What stands out about the $Y\left( \mathbb{Z}_{4} \right)_{3}$ matrix is that, like the iden­ti­ty matrix, it is sym­met­ric about both the main di­ag­o­nal and the anti-di­ag­o­nal, and this dif­fers from the lone sym­me­try $Y\left( \mathbb{Z}_{4} \right)_{2},Y\left( \mathbb{Z}_{4} \right)_{4}$ have about the anti-di­ag­o­nal.

The matrix-vector prod­uct rep­re­sen­ta­tion of the cyclic ro­ta­tion of ver­tices by the $Y\left( \mathbb{Z}_{4} \right)_{3}$ matrix is

for which, the vector en­tries rep­re­sent the ver­tices of the reg­u­lar square poly­gon. Graph­i­cal­ly, we rep­re­sent this cyclic ro­ta­tion in the com­plex plane below, where the left square has its ver­tices po­si­tioned as $(a,b,c,d)^{T}$, and the matrix-vector prod­uct $Y\left( \mathbb{Z}_{4} \right)_{3} \times {\widehat{v}}_{4}$ trans­forms the po­si­tions of the ver­tices to that found in the right square. The pri­ma­ry gen­er­a­tors of the set $A\left( Y\left( \mathbb{Z}_{4} \right)_{j} \right)$ are the subset

These have the fol­low­ing char­ac­ter­is­tic poly­no­mi­als

which is like the char­ac­ter­is­tic poly­no­mi­al, $\lambda^{3} - 1$, of the pri­ma­ry gen­er­a­tors $Y_{2},Y_{3} \in A\left( Y\left( gp_{2} \right)_{j} \right)$. Solv­ing the char­ac­ter­is­tic poly­no­mi­al ex­pres­sion

we also find eigen­val­ue so­lu­tions, $\lambda = \left\{ 1,i, - 1, - i \right\}$, aligned with the dis­tinct vertex values of the in­scribed sphere on the com­plex unit circle. In con­trast to the pri­ma­ry gen­er­a­tors, the char­ac­ter­is­tic poly­no­mi­als of the other Lie al­ge­bras in the set $A\left( Y\left( \mathbb{Z}_{4} \right)_{j} \right)$ are found to be

whose eigen­val­ues are so­lu­tions to the equa­tions

which are real-valued and not en­tire­ly dis­tinct.

To find the Lie al­ge­bra gen­er­a­tors that are re­spon­si­ble for the higher-di­men­sion­al ro­ta­tions around the axes of the square par­ti­tion­ing it into equal sec­tions, we define the anti-iden­ti­ty $\epsilon\left( \mathbb{Z}_{4} \right)^{*}$ for the group as

and cal­cu­late the matrix-matrix prod­uct set $A\left( X\left( \mathbb{Z}_{4} \right)_{j} \right) = \epsilon\left( \mathbb{Z}_{4} \right)^{*} \times A\left( Y\left( \mathbb{Z}_{4} \right)_{j} \right)$

The prod­ucts $\epsilon\left( \mathbb{Z}_{4} \right)^{*} \times \left\{ Y\left( \mathbb{Z}_{4} \right)_{1},Y\left( \mathbb{Z}_{4} \right)_{3} \right\}$ in­volve the Lie al­ge­bra gen­er­a­tors that had dual sym­me­try along the matrix di­ag­o­nal and anti-di­ag­o­nal, and these also yield Lie al­ge­bra gen­er­a­tors in the set $A\left( X\left( \mathbb{Z}_{4} \right)_{j} \right)$, char­ac­ter­ized by this dual sym­me­try, $\left\{ X\left( \mathbb{Z}_{4} \right)_{1},X\left( \mathbb{Z}_{4} \right)_{3} \right\}$. The com­mu­ta­tor of the ma­tri­ces $X\left( \mathbb{Z}_{4} \right)_{1}$ and $X\left( \mathbb{Z}_{4} \right)_{3}$ is equal to zero, given the dual sym­me­try they have in common

and we can fur­ther clas­si­fy both ma­tri­ces as sym­met­ric, mean­ing the trans­pose of each matrix is equal to itself. We also find that the com­mu­ta­tor of the ma­tri­ces $X\left( \mathbb{Z}_{4} \right)_{2}$ and $X\left( \mathbb{Z}_{4} \right)_{4}$ is equal to zero

since the trans­pose matrix of each is equal to the matrix itself

Given that both are or­thog­o­nal ma­tri­ces, the fol­low­ing holds

Con­tin­u­ing, we find the fol­low­ing re­la­tion­ship with the com­mu­ta­tors

The set of ro­ta­tion ma­tri­ces $A\left( X\left( \mathbb{Z}_{4} \right)_{j} \right)$ flips the square by two dis­tinct axes, the first along axes that are vertex-to-vertex ori­ent­ed, and the second axis is formed by the equal­ly par­ti­tion­ing of two op­po­site poly­gon edges. The $X\left( \mathbb{Z}_{4} \right)_{2}$ and $X\left( \mathbb{Z}_{4} \right)_{4}$ ro­ta­tion ma­tri­ces flip ver­tices around a vertex-to-vertex ori­ent­ed axis, where the matrix-vector prod­ucts are

So, the en­tries along the main di­ag­o­nals of $X\left( \mathbb{Z}_{4} \right)_{2}$ and $X\left( \mathbb{Z}_{4} \right)_{4}$ act to leave the co­in­cid­ing vertex po­si­tion un­changed, while non-di­ag­o­nal en­tries work to flip the vertex po­si­tions. Graph­i­cal­ly we depict the $X\left( \mathbb{Z}_{4} \right)_{4} \times {\widehat{v}}_{4}$ matrix-vector prod­uct in the fol­low­ing figure, where the axis of ro­ta­tion is de­scribed as the $bd$ vector, and we see that the ver­tices $b$ and $d$ remain un­changed, while the $a$ and $c$ ver­tices have ex­changed po­si­tions.

In con­trast, the $X\left( \mathbb{Z}_{4} \right)_{1}$ and $X\left( \mathbb{Z}_{4} \right)_{3}$ ro­ta­tion ma­tri­ces flip ver­tices around an edge-to-edge ori­ent­ed axis, where the matrix-vector prod­ucts are

which leads to all ver­tices moving po­si­tion, given that nei­ther matrix has en­tries on the di­ag­o­nal. The next two fig­ures graph­i­cal­ly depict both ro­ta­tions, with the axis of ro­ta­tion for the $X\left( \mathbb{Z}_{4} \right)_{1} \times {\widehat{v}}_{4}$ ro­ta­tion being $bc|da$, and that of $X\left( \mathbb{Z}_{4} \right)_{3} \times {\widehat{v}}_{4}$ ro­ta­tion being $ab|cd.$

At this junc­ture, it is es­sen­tial to point out that the set of gen­er­a­tors for the in­scribed $n$-gons $A\left( Y\left( \mathbb{Z}_{n} \right)_{j} \right)$ rep­re­sent com­plex num­bers, and the eigen­val­ues of each gen­er­a­tor rep­re­sent the ver­tices that each number will in­scribe on the bound­ary of the unit circle. For the set $A\left( Y\left( \mathbb{Z}_{n} \right)_{j} \right)$, when $n$ is non-prime, the set of gen­er­a­tors will be com­pos­ite in terms of the num­bers they rep­re­sent. In the case for the set $A\left( Y\left( \mathbb{Z}_{4} \right)_{j} \right)$, the $Y\left( \mathbb{Z}_{4} \right)_{2}$ Lie al­ge­bra gen­er­a­tor is

whose set of eigen­val­ues is $\lambda\left( Y\left( \mathbb{Z}_{4} \right)_{2} \right) = \left\{ 1,i, - 1, - i \right\}$, and the $Y\left( \mathbb{Z}_{4} \right)_{2}$ gen­er­a­tor rep­re­sents the min­i­mal el­e­ment in either the pos­i­tive or neg­a­tive ro­ta­tion­al ori­en­ta­tion, $\mathcal{m}\mathcal{e}_{\left\lbrack \pm 1/\mathbb{Z}_{4} \right\rbrack}$. There are two pos­si­ble num­bers that $Y\left( \mathbb{Z}_{4} \right)_{2}$ may rep­re­sent; $i$ or $- i$, since both num­bers are pri­ma­ry gen­er­a­tors since they can gen­er­ate all the eigen­val­ues in the set $\lambda\left( Y\left( \mathbb{Z}_{4} \right)_{2} \right)$, given that

For clar­i­ty, we will define $\mathcal{m}\mathcal{e}_{\left\lbrack + 1/\mathbb{Z}_{4} \right\rbrack}$ as the first cyclic shift of the iden­ti­ty right­ward

and we define $\mathcal{m}\mathcal{e}_{\left\lbrack - 1/\mathbb{Z}_{4} \right\rbrack}$ as the first cyclic shift of the iden­ti­ty left­ward

With $\mathcal{m}\mathcal{e}_{\left\lbrack + 1/\mathbb{Z}_{4} \right\rbrack}$ de­fined as the Lie al­ge­bra gen­er­a­tor $Y\left( \mathbb{Z}_{4} \right)_{2}$, this means they are rep­re­sen­ta­tions of the number $i$, and that $\mathcal{m}\mathcal{e}_{\left\lbrack - 1/\mathbb{Z}_{4} \right\rbrack}$ de­fines the $Y\left( \mathbb{Z}_{4} \right)_{4}$ gen­er­a­tor

rep­re­sent­ing the number $- i$. The other two gen­er­a­tors of the $A\left( Y\left( \mathbb{Z}_{n} \right)_{j} \right)$ set are $Y\left( \mathbb{Z}_{n} \right)_{1}$ and $Y\left( \mathbb{Z}_{n} \right)_{3}$, which rep­re­sent the gen­er­a­tors

The $Y\left( \mathbb{Z}_{n} \right)_{1}$ gen­er­a­tor has the eigen­val­ue set $\lambda\left( Y\left( \mathbb{Z}_{4} \right)_{1} \right) = \left\{ 1,1,1,1 \right\}$, and it is straight­for­ward to see it rep­re­sents the number $e^{2i\pi} = 1$ in the com­plex plane. The $Y\left( \mathbb{Z}_{n} \right)_{3}$ gen­er­a­tor has the eigen­val­ue set $\lambda\left( Y\left( \mathbb{Z}_{4} \right)_{3} \right) = \left\{ - 1, - 1,1,1 \right\}$ and rep­re­sents the split com­plex number $j = e^{\pm i\pi}$, given that

For the square of the matrix rep­re­sen­ta­tion $Y\left( \mathbb{Z}_{4} \right)_{3}$, we find

which is also con­sis­tent the split com­plex number de­f­i­n­i­tion $j^{2} = 1.$

We move on to the next geo­met­ric in­te­ger, $\mathbb{Z}_{5}$, which is also the next geo­met­ric prime, $gp_{3}$, which per our de­f­i­n­i­tion, is the pos­i­tive min­i­mal el­e­ment, $\mathcal{m}\mathcal{e}_{\left\lbrack + 1/gp_{3} \right\rbrack},$and rep­re­sents the $Y\left( gp_{3} \right)_{2}$ Lie al­ge­bra gen­er­a­tor

$Y\left( gp_{3} \right)_{2}$ has the dis­tinct set of eigen­val­ues $\lambda\left( Y\left( gp_{3} \right)_{2} \right) = \left\{ 1,e^{\left( \frac{2i\pi}{5} \right)},e^{\left( \frac{4i\pi}{5} \right)},e^{\left( \frac{6i\pi}{5} \right)},e^{\left( \frac{8i\pi}{5} \right)} \right\}$, rep­re­sent­ing the com­plex number $e^{\left( \frac{2i\pi}{5} \right)}$. As ex­pect­ed, since this is a geo­met­ric prime, all gen­er­a­tors have the same set of eigen­val­ues, where the set of $gp_{3}$ Lie al­ge­bra gen­er­a­tors is

The in­scribed reg­u­lar pen­ta­gon rep­re­sent­ing the $gp_{3}$ geo­met­ric prime is pre­sent­ed in Figure 1.7.18, with each vertex valued at an eigen­val­ue from the set

There are five ways to split the reg­u­lar pen­ta­gon into two equal-sized por­tions, all of which in­volve ex­tend­ing a seg­ment from each vertex to the mid­point of the op­pos­ing edge, known as the apothem. If this seg­ment ex­tends through the edge and to the bound­ary of the com­plex unit circle, it will in­ter­sect the bound­ary at the vertex's an­tipo­dal point, as seen in Figure 1.7.19.

We extend the $gp_{3}$ group to its higher-di­men­sion­al gen­er­a­tors by cal­cu­lat­ing the prod­ucts of the anti-iden­ti­ty with the set of Lie al­ge­bra gen­er­a­tors $\in$ $A\left( Y\left( gp_{3} \right)_{j} \right)$, which yields the $A\left( X\left( gp_{3} \right)_{j} \right)$ set

Each gen­er­a­tor in the set acts to rotate four of the five ver­tices about an axis of sym­me­try of the reg­u­lar pen­ta­gon seen in the pre­vi­ous figure. We take the second Lie al­ge­bra in the $A\left( X\left( gp_{3} \right)_{j} \right)$ set above to con­sid­er the matrix-vector prod­uct

The result leaves the $d$-vertex as un­changed, while the ver­tices $a$ and $b$ ex­change, as do $e$ and $c$. To best vi­su­al­ize the $X\left( gp_{3} \right)_{2} \times (a,b,c,d,e)^{T}$ro­ta­tion, we rep­re­sent the trans­form in the figure below, where the ver­tices re­flect across the axis of ro­ta­tion ex­tend­ing from $d$.

The set of $A\left( X\left( gp_{3} \right)_{j} \right)$ Lie al­ge­bra gen­er­a­tors act in a manner sim­i­lar to the $X$-type gen­er­a­tors, where they have an axis of ro­ta­tion and sym­met­ri­cal­ly rotate in the non-axis planes.

This unique sym­me­try of the reg­u­lar pen­ta­gon con­trasts with the two sym­me­tries of the reg­u­lar square, which were the even halv­ing of the square from vertex-to-vertex split­ting and mid-edge to op­pos­ing mid-edge split­ting. This process can be con­tin­u­al­ly re­peat­ed, by first cre­at­ing the set of cyclic Lie al­ge­bra gen­er­a­tors $A\left( Y\left( \mathbb{Z}_{n} \right)_{j} \right)$, then ro­tat­ing each el­e­ment in the set by the anti-iden­ti­ty $\epsilon\left( \mathbb{Z}_{n} \right)^{*},$to build all the higher-order groups.

We return to con­sid­er the char­ac­ter­is­tic poly­no­mi­als of the pri­ma­ry Lie al­ge­bra gen­er­a­tors of the set $A\left( Y\left( gp_{3} \right)_{j} \right),$by find­ing the de­ter­mi­nant of the matrix $\left| Y\left( gp_{3} \right)_{2} - \lambda I \right|$

for the result $\lambda^{5} - 1$, rep­re­sent­ing the $Y\left( gp_{3} \right)_{2}$ char­ac­ter­is­tic poly­no­mi­al. Given that the set $A\left( Y\left( gp_{3} \right)_{j} \right)$ rep­re­sents a geo­met­ric prime, we know that all char­ac­ter­is­tic poly­no­mi­als will be the same except for the iden­ti­ty el­e­ment, whose char­ac­ter­is­tic poly­no­mi­al is

with a set of eigen­val­ues that are all equal to one, $\lambda\left( Y\left( gp_{3} \right)_{1} \right) = \left\{ 1,1,1,1,1 \right\}.$To make it clear to the reader, so­lu­tions to the char­ac­ter­is­tic poly­no­mi­al set equal to zero are the eigen­val­ues of the matrix. This also rep­re­sents a poly­no­mi­al ex­pres­sion with vari­ables that can take on matrix values. For in­stance, the $\lambda^{5} - 1$ char­ac­ter­is­tic poly­no­mi­al can be ex­pressed as

where $\Lambda$ rep­re­sents a matrix vari­able, and $\epsilon_{5}$ is the square $5 \times 5$ iden­ti­ty matrix. All the Lie al­ge­bra gen­er­a­tors from the set $A\left( Y\left( gp_{3} \right)_{j} \right)$ are so­lu­tions to this matrix poly­no­mi­al above, which we can demon­strate with the $Y\left( gp_{3} \right)_{2}$ gen­er­a­tor by set­ting $\Lambda = Y\left( gp_{3} \right)_{2}$, and cal­cu­lat­ing as fol­lows

For the above, $\left( Y\left( gp_{3} \right)_{2} \right)^{5}$ is equal to $\epsilon_{5}$, given that each suc­ces­sive power shifts the en­tries to the right by one off-di­ag­o­nal, which after five shifts re­turns to the iden­ti­ty gen­er­a­tor. In Figure 1.7.21, we plot the char­ac­ter­is­tic poly­no­mi­al in the real domain, where the func­tion has a root at $\lambda = 1$. We recall that the other so­lu­tions in the com­plex plane are the re­main­ing eigen­val­ues from the set

The plot of the char­ac­ter­is­tic poly­no­mi­al func­tion $\lambda^{5} - 1$, like the func­tion $\lambda^{3} - 1$, is anti-sym­met­ric about the ver­ti­cal axis, given that it is an odd-pow­ered poly­no­mi­al. We com­pare this to the even-pow­ered char­ac­ter­is­tic poly­no­mi­al of the $Y\left( \mathbb{Z}_{4} \right)_{2}$ Lie al­ge­bra, whose result was found to be

The char­ac­ter­is­tic poly­no­mi­al func­tion $\lambda^{4} - 1$ rep­re­sents an even degree poly­no­mi­al, and in Figure 1.7.22, we can see the func­tion's plot has sym­me­try about the ver­ti­cal axis, and in the real domain, the func­tion has two roots at $\lambda = 1, - 1$.

Given that $\mathbb{Z}_{4}$ is a com­pos­ite number, we find that there are, in ad­di­tion to the iden­ti­ty's char­ac­ter­is­tic poly­no­mi­al $(\lambda - 1)^{4},$two others: the first gen­er­at­ed by the pos­i­tive and neg­a­tive min­i­mal el­e­ments,$\ \mathcal{m}\mathcal{e}_{\left\lbrack \pm \frac{1}{\mathbb{Z}_{n}} \right\rbrack},$equal­ing the result $\lambda^{4} - 1$, and the second is based on prime com­po­si­tion $p_{1} \times p_{1} = \mathbb{Z}_{4}$, where $p_{1} = 2$. We find by cal­cu­lat­ing the fol­low­ing de­ter­mi­nant

a char­ac­ter­is­tic poly­no­mi­al equal to $\left( \lambda^{2} - 1 \right)^{2}$, where the por­tion in paren­the­sis is equal to the char­ac­ter­is­tic poly­no­mi­al of the pri­ma­ry gen­er­a­tor, $Y\left( gp_{1} \right)_{2},$be­long­ing to the $gp_{1}$ geo­met­ric prime set $A\left( Y\left( gp_{1} \right)_{j} \right)$.

Like the prime number com­po­si­tion $p_{1} \times p_{1} = \mathbb{Z}_{4}$, $\left( \lambda^{2} - 1 \right)^{2}$ rep­re­sents a prime char­ac­ter­is­tic poly­no­mi­al com­po­si­tion, which we can ex­press as fol­lows

where $\mathcal{cp}\left( gp_{1} \right)$ de­notes the non-iden­ti­ty unique char­ac­ter­is­tic poly­no­mi­al of the $gp_{1}$ geo­met­ric prime. We also use the fol­low­ing no­ta­tion to ex­press powers of $\mathcal{cp}\left( gp_{1} \right)$

which will prove help­ful as we look at more non-prime com­pos­ite geo­met­ric num­bers. A look at the set of gen­er­a­tors rep­re­sent­ing the hexa­gon, $A\left( Y\left( \mathbb{Z}_{6} \right)_{j} \right)$, is seen below

and there are a total of four dif­fer­ent char­ac­ter­is­tic poly­no­mi­als in this set, which are the fol­low­ing

Of the four char­ac­ter­is­tic poly­no­mi­als, two rep­re­sent prime char­ac­ter­is­tic poly­no­mi­al com­po­si­tions as fol­lows

where we can gen­er­al­ize the $\mathcal{c}\mathcal{p}^{j}\left( gp_{n} \right)$ no­ta­tion to

which holds for all prime char­ac­ter­is­tic poly­no­mi­al com­po­si­tions.

To rep­re­sent the sum of all char­ac­ter­is­tic poly­no­mi­als in the group, we create a group char­ac­ter­is­tic poly­no­mi­al for the set $A\left( Y\left( \mathbb{Z}_{6} \right)_{j} \right)$, de­fined as

where the co­ef­fi­cients are de­ter­mined by the number of times each char­ac­ter­is­tic poly­no­mi­al ap­pears in the set. In ex­pand­ed form, the group char­ac­ter­is­tic poly­no­mi­al $\mathcal{CP}\left( Y\left( \mathbb{Z}_{6} \right) \right)$ is

and plot­ted in the figure below, where the blue-shaded region rep­re­sents where the func­tion dips below the hor­i­zon­tal axis for $\lambda$ values be­tween $0$ and $1$.

The area of this blue-shaded region is found by cal­cu­lat­ing the in­te­gral $\int_{0}^{1}{\mathcal{CP}\left( Y\left( \mathbb{Z}_{6} \right) \right)}d\lambda$,

whose result we see is greater than $- 1.$Below we list sev­er­al in­te­grals of the group char­ac­ter­is­tic poly­no­mi­als with bound­aries be­tween $0$ and $1$, up to the group $\mathcal{CP}\left( Y\left( \mathbb{Z}_{21} \right) \right)$, where prime nu­mer­a­tor re­sults are in de­pict­ed in bold:

At the onset, we find a high fre­quen­cy of prime num­bers for the nu­mer­a­tors in the re­sults, but after the $Y\left( gp_{7} \right)$ in­te­gral $,$this ten­den­cy ap­pears to de­crease. The non-prime in­te­gral re­sults are a mix of ra­tio­nals with even/odd and odd/odd values, while all the in­te­gral re­sults of geo­met­ric primes have re­sults that are odd/even ra­tio­nal number values.

We plot in the fol­low­ing figure, the group char­ac­ter­is­tic poly­no­mi­als for the first six even-num­bered in­te­gers, $\mathcal{CP}\left( Y\left( \mathbb{Z}_{\alpha} \right) \right),\ \alpha = \left\{ 2,4,6,8,10,12 \right\},$seeing that each $\mathcal{CP}\left( Y\left( \mathbb{Z}_{\alpha} \right) \right)$ func­tion has a global min­i­mum on the in­ter­val $\lambda = \left\lbrack \left. \ \frac{1}{2},1 \right) \right.\ ,$and the global min­i­mum of the geo­met­ric prime func­tion $\mathcal{CP}\left( Y\left( pg_{1} \right) \right)$ is lo­cat­ed pre­cise­ly at $\lambda = \frac{1}{2}.$

The group char­ac­ter­is­tic poly­no­mi­als of the geo­met­ric primes have the fol­low­ing gen­er­al­ized form

which is dis­tinct as a poly­no­mi­al and po­ten­tial­ly a means to dis­tin­guish a prime number from a com­pos­ite number by the graph of its $\mathcal{CP}\left( Y\left( gp_{n} \right) \right)$ func­tion. Plot­ting the first seven geo­met­ric prime num­bers below, the rapid flat­ten­ing of the curves for each func­tion cen­ters around values of $\lambda = \frac{1}{2},$co­in­cid­ing with ver­ti­cal axis values at ap­prox­i­mate­ly the even in­te­ger pre­ced­ing each $p_{n}$, or around $\left( p_{n} - 1 \right).$

We eval­u­ate how close­ly the value at $\lambda = \frac{1}{2}$ comes to each even number $- \left( p_{n} - 1 \right)$ by solv­ing the func­tion $\mathcal{CP}\left( Y\left( gp_{n} \right) \right)$, for $\lambda = \frac{1}{2}$

which for all primes except for $p_{1} = 2$, be­comes

where the neg­a­tive-valued in­te­ger term, $- \left( p_{n} - 1 \right),$ap­pears in the result, and the $\frac{\left( p_{n} - 2 \right)}{2^{\left( p_{n} \right)}}$ term re­duces in size ex­po­nen­tial­ly by $p_{n}$. Using the ex­am­ple for the fifth geo­met­ric prime $gp_{5}$, where $p_{5} = 11$, we find the result

which we can see is very close to the value $- \left( p_{n} - 1 \right)$, for $\lambda = \frac{1}{2}.$In the figure, it is also ap­par­ent that the odd-valued func­tions $\mathcal{CP}\left( Y\left( gp_{\alpha} \right) \right)$ cross the ver­ti­cal axis at their neg­a­tive in­te­ger values, and math­e­mat­i­cal­ly, we can un­der­stand why this is the case by solv­ing the fol­low­ing $\mathcal{CP}\left( Y\left( gp_{n} \right) \right)$ ex­pres­sion

for the triv­ial value $\lambda = 0$

and given $p_{n}$ is odd for all $n \geq 2$, we have that $( - 1)^{p_{n}} = - 1$, and we find

To show how the $\mathcal{CP}\left( Y\left( gp_{n} \right) \right)$ func­tions differ from the odd non-prime com­pos­ite func­tions $\mathcal{CP}\left( Y\left( \mathbb{Z}_{\alpha} \right) \right)$ in the in­ter­val $\left\lbrack 0,1\rbrack \right.\ \in \mathbb{R}$, we plot in Figure 1.7.26 the set of func­tions $\mathcal{CP}\left( Y\left( \mathbb{Z}_{\alpha} \right) \right),$for$\ \ \alpha = \left\{ 9,15,21 \right\}$, and com­pare this to the set $\mathcal{CP}\left( Y\left( gp_{\alpha} \right) \right)$, for $\alpha = \left\{ 4:8 \right\}$. We mark the point where the geo­met­ric prime char­ac­ter­is­tic poly­no­mi­al func­tions cross the strip at $\lambda = \frac{1}{2}$. All of the func­tions in­ter­sect the ver­ti­cal axis at the neg­a­tive value of their nat­ur­al number $- n$ at $\lambda = 0$. As men­tioned pre­vi­ous­ly, at the point of the plot when $\lambda = \frac{1}{2}$, the $\mathcal{CP}\left( Y\left( gp_{n} \right) \right)$ func­tions are all close to the in­ter­sec­tion point with the in­te­ger value $- \left( p_{n} - 1 \right)$. The non-prime group poly­no­mi­al func­tions cross the ad­ja­cent in­te­ger thresh­old for $- \left( \mathbb{Z}_{n} - 1 \right)$, on the in­ter­val $\lambda = \left\lbrack 0,\frac{1}{2} \right).$

There is a geo­met­ric basis to our de­f­i­n­i­tion of the group char­ac­ter­is­tic poly­no­mi­als re­lat­ed to the pri­ma­ry gen­er­a­tors. For com­pos­ite num­bered groups, lower-or­dered poly­gons will exist that evenly divide the $n$-gon de­fined by its $\mathcal{CP}\left( Y\left( \mathbb{Z}_{n} \right) \right)$ group, while the prime number groups $\mathcal{CP}\left( Y\left( gp_{n} \right) \right)$ con­sist solely of their unique $(n - 1)$ pri­ma­ry gen­er­a­tors, $\left( \lambda^{n} - 1 \right)$, along with the iden­ti­ty gen­er­a­tor $(\lambda - 1)^{n}$. The number of pri­ma­ry gen­er­a­tors for lower-or­dered poly­gons in­scrib­ing the $\mathcal{CP}\left( Y\left( \mathbb{Z}_{n} \right) \right)$ group's $n$-gon rep­re­sent the con­stants in that group's char­ac­ter­is­tic poly­no­mi­al. This is in ad­di­tion to the unique pri­ma­ry gen­er­a­tors of the $n$-gon itself, cal­cu­lat­ed as

where $\mathcal{p}\mathcal{g}_{n}$ de­notes the number of pri­ma­ry gen­er­a­tors of the $n$-gon and $\mathcal{p}\mathcal{g}_{k}$ are lower-or­dered pri­ma­ry gen­er­a­tors that in­scribe the $n$-gon geom­e­try. An ex­am­ple helps to vi­su­al­ize what this means we look at the $\mathcal{CP}\left( Y\left( \mathbb{Z}_{12} \right) \right)$ group, for which, the do­decagon is its de­fined $n$-gon, and its group char­ac­ter­is­tic poly­no­mi­al is

where we have arranged the poly­no­mi­al by order of the $\lambda$'s. Each term in the poly­no­mi­al rep­re­sents the pri­ma­ry gen­er­a­tors of the sub-or­dered poly­gons that in­scribe the do­decagon. The in­te­ger $12$ can be di­vid­ed with­out re­main­ders as fol­lows:

Each di­vi­sion rep­re­sents the poly­gon de­fined by this number of ver­tices and the number di­vid­ing $\mathbb{Z}_{12}$ will rep­re­sent the ex­po­nent of each pri­ma­ry gen­er­a­tor equa­tion. Each pri­ma­ry gen­er­a­tor equa­tion will have the form

where the ex­po­nent $k$ rep­re­sents the number of pri­ma­ry gen­er­a­tor ver­tices for the in­scribed poly­gon rep­re­sen­ta­tion, $\mathcal{k}$ is the number of pri­ma­ry gen­er­a­tors in the poly­gon, and the $\frac{n}{k}$ ex­po­nent rep­re­sents the number of times this lower-or­dered poly­gon di­vides the higher-order $n$-gon. For $\mathcal{p}\mathcal{g}_{k}\left( \mathbb{Z}_{12} \right)$, all the pri­ma­ry gen­er­a­tor poly­gon terms are pre­sent­ed below that evenly divide the geom­e­try of $Y\left( \mathbb{Z}_{12} \right)$,

where the lead­ing con­stants $a \rightarrow f$ for each term define the pri­ma­ry gen­er­a­tors that the poly­gon has. Each pri­ma­ry gen­er­a­tor, by way of suc­ces­sive powers, can gen­er­ate all the el­e­ments in its subset, and to ensure clar­i­ty for the reader, we present the pri­ma­ry gen­er­a­tors for each lower-order poly­gon above:

To de­ter­mine the value of the $f$ con­stant, we solve our sum

and geo­met­ri­cal­ly, the result rep­re­sents the four ver­tices of the do­decagon that are not vertex points of iden­ti­ty-aligned sub-poly­gons, which as a set are the point, the line, the equi­lat­er­al tri­an­gle, the square, and the reg­u­lar hexa­gon. With the con­stants de­ter­mined, we can now ex­press the $\mathcal{CP}\left( Y\left( \mathbb{Z}_{12} \right) \right)$ char­ac­ter­is­tic poly­no­mi­al as

which is pre­sent­ed in the figure below in the in­scribed geom­e­try rep­re­sen­ta­tion.

Group char­ac­ter­is­tic poly­no­mi­als are com­prised of terms stem­ming from the sub-geome­tries that in­scribe their geom­e­try, weight­ed based on the number of pri­ma­ry gen­er­a­tors in the set of el­e­ments from those sub-geome­tries. As we are aware, a geo­met­ric prime $gp_{n}$ will have a total of $(n - 1)$ pri­ma­ry gen­er­a­tors, and in the pre­vi­ous ex­am­ple, the $gp_{2},gp_{3}$ terms were $\left( \lambda^{2} - 1 \right)^{6}$ and ${2\left( \lambda^{3} - 1 \right)}^{4}$ re­spec­tive­ly. When higher-or­dered geo­met­ric primes $Y\left( gp_{n}\ \right)$ are present in a group's char­ac­ter­is­tic poly­no­mi­al, these tend to be weight­ed more heav­i­ly than a com­pos­ite Geo­met­ric Number $Y\left( \mathbb{Z}_{k} \right)$ for de­grees of $n$ and $k$ that are in prox­im­i­ty.

To finish the sec­tion, we look at the group char­ac­ter­is­tic poly­no­mi­als' terms, which rep­re­sent the em­bed­ded min­i­mal el­e­ments of the sub-geome­tries that com­prise that group's order. As stated before, the $\mathcal{p}\mathcal{g}_{k}$ terms in a group char­ac­ter­is­tic poly­no­mi­al will have the form

and the unique min­i­mal el­e­ment to the em­bed­ded degree $k$ geom­e­try is

rep­re­sent­ing either the first vertex of the $k$-gon that is pos­i­tive­ly ori­ent­ed from the iden­ti­ty, $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{1}{k} \right\rbrack} = e^{\left( \frac{2i\pi}{k} \right)}$, or the first vertex which is neg­a­tive­ly ori­ent­ed from the iden­ti­ty, $\mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{k} \right\rbrack} = e^{\left( - \frac{2i\pi}{k} \right)}.$The min­i­mal el­e­ments and the other pri­ma­ry gen­er­a­tors $\in Y\left( \mathbb{Z}_{k} \right)$ are so­lu­tions to the char­ac­ter­is­tic poly­no­mi­al equa­tion $\lambda^{k} - 1 = 0$. The min­i­mal el­e­ments yield a so­lu­tion to this equa­tion, having a wind­ing number of one around the com­plex unit circle for the eval­u­a­tion of the char­ac­ter­is­tic poly­no­mi­al equa­tion

For pri­ma­ry gen­er­a­tors, that lie on the upper half of the unit circle for pos­i­tive­ly ori­ent­ed so­lu­tions, or the lower half for neg­a­tive­ly ori­ent­ed so­lu­tions, the wind­ing num­bers will be equal to the value $j$

for which, the result rep­re­sents a dis­tance factor the pri­ma­ry gen­er­a­tor is from the min­i­mal el­e­ment

In *Fig­ures 1.7.28 & **1.7.***29, we con­sid­er the graphs of the char­ac­ter­is­tic poly­no­mi­al equa­tions $\lambda^{4} - 1$, $\lambda^{5} - 1$, where the blue areas rep­re­sent def­i­nite in­te­grals of these equa­tions from $0$ to $1$, with the fol­low­ing re­sults

The areas bound by the char­ac­ter­is­tic poly­no­mi­als be­tween $0$ and $1$ can be gen­er­al­ized to the form

and if we con­sid­er the neg­a­tive-valued char­ac­ter­is­tic poly­no­mi­al, $1 - \lambda^{n - 1},$

this equals one-half the value of the in­verse tri­an­gu­lar num­bers $\left( \mathcal{T}_{n}^{1} \right)^{- 1}$ found in the pre­vi­ous sec­tion

With the Euler prod­uct in mind, we can ex­press the har­mon­ic series $\zeta(1)$ in terms of the char­ac­ter­is­tic poly­no­mi­als of the geo­met­ric primes

or as the prod­uct

The $\left( p_{n} - 1 \right)$ term in the geo­met­ric prime group char­ac­ter­is­tic poly­no­mi­al

is the same as the $(p - 1)$ term in Wilson's the­o­rem, show­ing the con­nec­tion be­tween the geom­e­try of the min­i­mal el­e­ment and the more con­ven­tion­al ap­proach to in­tro­duc­to­ry number theory. De­tails about number theory and Wilson's the­o­rem are in the re­main­ing sec­tions that follow.