1.7.1 Prime Numbers and the Geometry of the Minimal Element

Pre­vi­ous­ly in this book, we in­tro­duced Cayley's the­o­rem, which states that for a group $G$, there is a sym­met­ric group that acts on $G$, and $G$ is then iso­mor­phic to a sub­group of the sym­met­ric group. In the last sec­tion, the min­i­mal el­e­ment, $\mathcal{me}$, was used in the ex­pres­sion to count the number of Lie al­ge­bra gen­er­a­tors a par­tic­u­lar group had. This sec­tion will show geo­met­ri­cal­ly how Cayley's the­o­rem re­lates to the min­i­mal el­e­ment. First, we define the first four min­i­mal el­e­ments as points on the com­plex unit circle, seen in the figure below, and as fol­lows

On the unit circle, the min­i­mal el­e­ment, $\mathcal{m}\mathcal{e}_{\epsilon} = e^{2i\pi k}$, rep­re­sents the iden­ti­ty set $A\left\{ \mathcal{m}\mathcal{e}_{\epsilon} \right\}$

having one sole Lie al­ge­bra gen­er­a­tor. By ro­tat­ing $i\pi$ ra­di­ans on the unit circle in either the pos­i­tive or neg­a­tive di­rec­tion, we arrive at $\mathcal{m}\mathcal{e}_{j} = e^{i(\pi + 2\pi k)}$, which as the set of per­mu­ta­tions, $A\left( \mathcal{m}\mathcal{e}_{j} \right)$, now con­tains two Lie al­ge­bra gen­er­a­tors

which is in­clu­sive of the iden­ti­ty el­e­ment rep­re­sent­ing $A\left( \mathcal{m}\mathcal{e}_{\epsilon} \right)$. If we define the iden­ti­ty group $G(\epsilon)$, con­tain­ing the per­mu­ta­tion set $A\left\{ \mathcal{m}\mathcal{e}_{\epsilon} \right\}$, and the split com­plex group $G(j)$, with the per­mu­ta­tion set $A\left( \mathcal{m}\mathcal{e}_{j} \right)$, it holds that

Since the per­mu­ta­tion $A\left( \mathcal{m}\mathcal{e}_{\epsilon} \right)$ set for the group $G(\epsilon)$ is on the unit circle, we can define the group $G(\epsilon)$ as

and like­wise, the group $G(j)$, can be de­fined as

Given that $G(\epsilon)$ is de­fined by its el­e­ment in the set $A\left( \mathcal{m}\mathcal{e}_{\epsilon} \right)$, and $A\left( \mathcal{m}\mathcal{e}_{\epsilon} \right) \in A\left( \mathcal{m}\mathcal{e}_{j} \right)$, and both sets are in the set $U\left( 1,\mathbb{C} \right)$

it is also true that the $G(\epsilon)$ is a sub­group of $G(j)$

$G(j)$ can be seen as the next-or­dered sym­me­try of the unit circle, which is con­sis­tent with Cayley's the­o­rem, being that the group $G(\epsilon)$ is a sub­group of the sym­me­try group $G(j)$, based on the min­i­mal el­e­ment of choice $j.$

We can also con­sid­er the next-or­dered sym­me­try set, $A\left( \mathcal{m}\mathcal{e}_{j} \right)$, as the unions

which rep­re­sents the fact $A\left( \mathcal{m}\mathcal{e}_{j} \right)$ con­tains its min­i­mal el­e­ment, and its sub­group set $A\left( \mathcal{m}\mathcal{e}_{\epsilon} \right).$

We arrive at our next min­i­mal el­e­ment, $\mathcal{m}\mathcal{e}_{i} = e^{i\left( \frac{\pi}{2} + 2\pi k \right)} = i$, by ro­tat­ing $\frac{i\pi}{2}$ ra­di­ans in the pos­i­tive di­rec­tion from $\mathcal{m}\mathcal{e}_{\epsilon}$. The sym­me­try of the com­plex unit circle be­comes ap­par­ent, given that we could choose to rotate in­stead by $\frac{i\pi}{2}$ ra­di­ans in the neg­a­tive di­rec­tion to arrive at $- i$. We know the circle's sym­me­try, and this sym­me­try also ap­plies to the min­i­mal el­e­ment. This is an im­por­tant con­cept, and it is re­lat­ed to the on­to­log­i­cal con­cept of net-zero. Anal­o­gous to the cyclic group, which de­fines a group con­tain­ing at least one Lie al­ge­bra gen­er­a­tor ca­pa­ble of gen­er­at­ing all el­e­ments in the set, the pos­i­tive and neg­a­tive min­i­mal el­e­ments, $\mathcal{m}\mathcal{e}_{\pm i}$, are Lie al­ge­bra gen­er­a­tors, both of which can gen­er­ate all the el­e­ments in the $A\left( \mathcal{m}\mathcal{e}_{i} \right)$ set, shown for both cases as

Nested in these two ex­am­ples is the sym­me­try of the next lower order gen­er­a­tor set, $A\left( \mathcal{m}\mathcal{e}_{j} \right),$which is rep­re­sent­ed by the sym­me­tries

which are the Lie gen­er­a­tors on $U(1)$ co­in­cid­ing with the gen­er­a­tors of the $A\left( \mathcal{m}\mathcal{e}_{j} \right)$ set. By stat­ing it in these terms

sub­group nest­ing of the min­i­mal el­e­ment sets is ap­par­ent when con­sid­er­ing the ex­po­nents.

Next, the min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{c} = e^{i\left( \frac{\pi}{4} + 2\pi k \right)}$ rep­re­sents a ro­ta­tion in the pos­i­tive di­rec­tion of $\frac{i\pi}{4}$ ra­di­ans from the iden­ti­ty

whose coun­ter­part ro­ta­tion in the neg­a­tive di­rec­tion, $\mathcal{m}\mathcal{e}_{- c}$, is

Like $i$ and $j$, we will define the pos­i­tive and neg­a­tive-valued $\mathcal{m}\mathcal{e}_{\pm c}$ as

the vari­able choice of $c$ de­fines these as cone num­bers but also light, which we will con­sid­er later in the chap­ter. As was the case for $\mathcal{m}\mathcal{e}_{\pm i}$, both Lie al­ge­bra gen­er­a­tors $\mathcal{m}\mathcal{e}_{\pm c}$, gen­er­ate all the set's el­e­ments

and

The nested min­i­mal el­e­ment sym­me­try equiv­a­lences for $\mathcal{m}\mathcal{e}_{\pm c}$ are

where the even ex­po­nent pat­tern that has emerged is due to our choice of sub­se­quent min­i­mal el­e­ments that are half the pre­vi­ous one. Our se­lec­tion of min­i­mal el­e­ments has led to the sym­met­ric di­vi­sion of the com­plex unit circle into small­er and small­er por­tions. The nested min­i­mal el­e­ment sym­me­try equiv­a­lences have ex­po­nents re­lat­ed by mul­ti­ples of the prime number $2$. This prime number re­la­tion­ship re­lates to our ini­tial manner of sec­tion­ing the com­plex unit circle, a sym­met­ric halv­ing of suc­ces­sive sec­tions, which rep­re­sents taking suc­ces­sive square roots of the $U(1)$ group

Ac­cord­ing­ly, the in­verse of the above process of these suc­ces­sive square roots is taking the square of a min­i­mal el­e­ment, which re­sults in the ad­ja­cent min­i­mal el­e­ment of a lower order, seen as fol­lows

The number $2$ is the only even prime number, and it is re­lat­ed to the ex­po­nent values for the $U(1)$ sec­tion­ing as an ever-in­creas­ing set of min­i­mal el­e­ments, $A\left( \mathcal{m}\mathcal{e}_{k\lbrack 1/2\rbrack} \right),$which is based on the sec­tion­ing ratio $\lbrack 1/2\rbrack$, and the $k^{th}$ it­er­a­tion, which runs from $0 \rightarrow k$, of the min­i­mal el­e­ment

and we note for any $\mathcal{m}\mathcal{e}_{k\lbrack 1/2\rbrack}$, we have

In the ex­pres­sion above, all the ex­po­nent values for the sec­tion­ing of $U(1)$ in the set $A\left( \mathcal{m}\mathcal{e}_{k\lbrack 1/2\rbrack} \right),$are built upon the rec­i­p­ro­cal com­po­si­tions of the prime number $2$. The min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{k\lbrack 1/2\rbrack}$ with $k = 3$, is equal to the com­po­si­tion

which has the in­te­ger prime number com­po­si­tion coun­ter­part

This leads us to see that all min­i­mal el­e­ments, $\mathcal{m}\mathcal{e}_{k\lbrack 1/2\rbrack},$of the $U(1)$ group are re­lat­ed to the rec­i­p­ro­cal prime number $\frac{1}{2}$, and its ex­po­nen­tial com­po­si­tion, except the sec­tion­ing iden­ti­ty $\left( U(1) \right)^{1}$, which is anal­o­gous to the in­te­ger number $1$. The rec­i­p­ro­cal prime number $\frac{1}{2}$, should also be con­sid­ered the second el­e­ment in the har­mon­ic series, and the iden­ti­ty as the first in the series, $\epsilon = 1$.

Con­sid­er­ing the first prime number, $2$, we move on to the next prime number, $3$, and sec­tion $U(1)$ into three parts, where the set of el­e­ments, $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack} \right)$, is based on the suc­ces­sive powers of min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack}$

around one ro­ta­tion of the com­plex unit circle. Each power of the min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{k\lbrack 1/3\rbrack}$, is ex­plic­it­ly

and like pre­vi­ous ex­am­ples, the di­rec­tion­al choice of ro­ta­tion away from the iden­ti­ty yields a min­i­mal el­e­ment, $\pm \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack},$which can gen­er­ate all Lie al­ge­bras in the set. We should also note that $\left( \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack} \right)^{2} = e^{\left( \frac{4i\pi}{3} \right)}$ also gen­er­ates all el­e­ments of the set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack} \right)$, given that

We con­tin­ue the process for the fol­low­ing three min­i­mal el­e­ment sets, $\left\{ A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/4\rbrack} \right),A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/5\rbrack} \right),A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/6\rbrack} \right) \right\}$ which rep­re­sent the rec­i­p­ro­cal num­bers $\frac{1}{4},\frac{1}{5},\frac{1}{6},$and along with the set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack} \right)$, we plot each below as a poly­gon in­scribed in the com­plex unit circle, where the poly­gon ver­tices rep­re­sent the pos­i­tive­ly ori­ent­ed path the min­i­mal el­e­ment takes around the unit circle. The number of ver­tices is equiv­a­lent to the re­spec­tive num­bers $4,5,6$.

$A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack} \right)$(green tri­an­gle), $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/4\rbrack} \right)$(purple square), $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/5\rbrack} \right)$(blue pen­ta­gon), $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/6\rbrack} \right)$(red hexa­gon).

Each poly­gon in the figure above shares the same iden­ti­ty min­i­mal el­e­ment, $\mathcal{m}\mathcal{e}_{\epsilon}$, which is anal­o­gous to how every in­te­ger has the number $1$ as a common factor. The first six in­te­gers con­tain the iden­ti­ty in­te­ger, $1$, three prime num­bers, $2,3,5$, and two com­pos­ite num­bers, $4,6$. The num­bers $4$ and $6$ result from the fol­low­ing prime number prod­ucts

which can be fur­ther clas­si­fied as even com­pos­ite num­bers. All even com­pos­ite num­bers will be com­posed of at least one factor of the number $2$, the only even prime number. The poly­gons that rep­re­sent the even com­pos­ite num­bers $4$ and $6$ have ver­tices at points on the unit circle, which their com­pos­ing prime num­bers have al­ready passed through. For in­stance, the set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/4\rbrack} \right)$, rep­re­sents the com­pos­ite number 4, and shares a vertex with the prime number $2$-re­lat­ed set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/2\rbrack} \right)$, at the point $\left( \mathcal{m}\mathcal{e}_{\lbrack 1/2\rbrack} \right)^{2}= \mathcal{m}\mathcal{e}_{\lbrack 1/2\rbrack}$. The set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/6\rbrack} \right)$, rep­re­sent­ing the com­pos­ite number $6$, shares ver­tices with the prime number $3$-re­lat­ed set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack} \right)$ at the points $\left( \mathcal{m}\mathcal{e}_{\lbrack 1/6\rbrack} \right)^{2}= \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack},\left( \mathcal{m}\mathcal{e}_{\lbrack 1/6\rbrack} \right)^{4} = \left( \mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack} \right)^{2},$and a vertex with its other prime number re­lat­ed set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/2\rbrack} \right)$ at $\left( \mathcal{m}\mathcal{e}_{1/6} \right)^{3}= \mathcal{m}\mathcal{e}_{\lbrack 1/2\rbrack}.$These shared ver­tices rep­re­sent com­pos­ite num­bers' beau­ti­ful geo­met­ric re­la­tion­ship with their prime number gen­er­a­tors in the com­plex space of the $U(1)$ group.

The dis­tri­b­u­tion of prime num­bers is high­est in the range of the lower-valued nat­ur­al num­bers. In the set of num­bers from $1$ to $100$, there are $25$ prime num­bers, $49$ even com­pos­ite num­bers, and $25$ odd com­pos­ite num­bers, the first of which is the number 9.

The radian mea­sures for the ver­tices of the first thir­teen poly­gons$\ (n)$ are shown in the table below, rep­re­sent­ing both the $n^{th}$ el­e­ment of the har­mon­ic series and the $n^{th}$ rec­i­p­ro­cal in­te­ger, where each $n$-gon has a total of $n$ ver­tices, num­bered $0$ to $n - 1$.

$\displaystyle n$	$\displaystyle V_{0}$	$\displaystyle V_{1}$	$\displaystyle V_{2}$	$\displaystyle V_{3}$	$\displaystyle V_{4}$	$\displaystyle V_{5}$	$\displaystyle V_{6}$	$\displaystyle V_{7}$	$\displaystyle V_{8}$	$\displaystyle V_{9}$	$\displaystyle V_{10}$	$\displaystyle V_{11}$	$\displaystyle V_{12}$
$\displaystyle 1$	$\displaystyle \epsilon$
$\displaystyle 2$	$\displaystyle \epsilon$	$\textcolor[HTML]{FF0000}{\displaystyle i\pi}$
$\displaystyle 3$	$\displaystyle \epsilon$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{3}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{3}}$
$\displaystyle 4$	$\displaystyle \epsilon$	$\displaystyle \frac{i\pi}{2}$	$\textcolor[HTML]{FF0000}{\displaystyle i\pi}$	$\displaystyle \frac{3i\pi}{2}$
$\displaystyle 5$	$\displaystyle \epsilon$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{5}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{5}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{6i\pi}{5}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{8i\pi}{5}}$
$\displaystyle 6$	$\displaystyle \epsilon$	$\displaystyle \frac{i\pi}{3}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{3}}$	$\textcolor[HTML]{FF0000}{\displaystyle i\pi}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{3}}$	$\displaystyle \frac{5i\pi}{3}$
$\displaystyle 7$	$\displaystyle \epsilon$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{7}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{7}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{6i\pi}{7}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{8i\pi}{7}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{10i\pi}{7}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{12i\pi}{7}}$
$\displaystyle 8$	$\displaystyle \epsilon$	$\displaystyle \frac{i\pi}{4}$	$\textcolor[HTML]{1E48FF}{\displaystyle \frac{i\pi}{2}}$	$\displaystyle \frac{3i\pi}{4}$	$\textcolor[HTML]{FF0000}{\displaystyle i\pi}$	$\displaystyle \frac{5i\pi}{4}$	$\textcolor[HTML]{1E48FF}{\displaystyle \frac{3i\pi}{2}}$	$\displaystyle \frac{7i\pi}{4}$
$\displaystyle 9$	$\displaystyle \epsilon$	$\displaystyle \frac{2i\pi}{9}$	$\displaystyle \frac{4i\pi}{9}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{3}}$	$\displaystyle \frac{8i\pi}{9}$	$\displaystyle \frac{10i\pi}{9}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{3}}$	$\displaystyle \frac{14i\pi}{9}$	$\displaystyle \frac{16i\pi}{9}$
$\displaystyle 10$	$\displaystyle \epsilon$	$\displaystyle \frac{i\pi}{5}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{5}}$	$\displaystyle \frac{3i\pi}{5}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{5}}$	$\textcolor[HTML]{FF0000}{\displaystyle i\pi}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{6i\pi}{5}}$	$\displaystyle \frac{7i\pi}{5}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{8i\pi}{5}}$	$\displaystyle \frac{9i\pi}{5}$
$\displaystyle 11$	$\displaystyle \epsilon$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{6i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{8i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{10i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{12i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{14i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{16i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{18i\pi}{11}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{20i\pi}{11}}$
$\displaystyle 12$	$\displaystyle \epsilon$	$\displaystyle \frac{i\pi}{6}$	$\textcolor[HTML]{1E48FF}{\displaystyle \frac{i\pi}{3}}$	$\textcolor[HTML]{1E48FF}{\displaystyle \frac{i\pi}{2}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{3}}$	$\displaystyle \frac{5i\pi}{6}$	$\textcolor[HTML]{FF0000}{\displaystyle i\pi}$	$\displaystyle \frac{7i\pi}{6}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{3}}$	$\textcolor[HTML]{1E48FF}{\displaystyle \frac{3i\pi}{2}}$	$\textcolor[HTML]{1E48FF}{\displaystyle \frac{5i\pi}{3}}$	$\displaystyle \frac{11i\pi}{6}$
$\displaystyle 13$	$\displaystyle \epsilon$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{2i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{4i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{6i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{8i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{10i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{12i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{14i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{16i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{18i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{20i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{22i\pi}{13}}$	$\textcolor[HTML]{FF0000}{\displaystyle \frac{24i\pi}{13}}$

In the table, re­peat­ing prime number ver­tices are sym­met­ri­cal­ly dis­trib­uted about the real-axis for each poly­gon, where, for ex­am­ple the set for the $9$-gon is

where the set starts at the iden­ti­ty, then passes through two novel ver­tices, the first vertex for the geo­met­ric prime number $3$, two more novel ver­tices, the second vertex for the prime number $3$, and fi­nal­ly, the last two novel ver­tices. These novel ver­tices will repeat start­ing with the $18$-gon, and this can be seen best by fol­low­ing the in­te­ger pro­gres­sion for $2n$, where start­ing at the prime number $2$, all its el­e­ments repeat in $4$-gon, and all its el­e­ments will then repeat in the $8,12$-gons. The reason for this is straight­for­ward since it is re­lat­ed to re­peat­ing prime num­bers of the same value in com­pos­ite num­bers, seen by con­sid­er­ing the prime number gen­er­a­tors for each of the poly­gons

where $p_{n}$ rep­re­sents the $n^{th}$ prime number. We see above that the num­bers $4$ and $8$ are com­pound num­bers com­posed solely of the prime number $p_{1} = 2$, while the number $12$ is a com­pound number com­posed of two prime num­bers $p_{1} = 2,p_{2} = 3$. All prime number ver­tices can be ex­pressed as

where $p_{n}$ is the $n^{th}$ prime number, with an in­te­ger value $Z_{k}$, and except for the prime number $2$, is a ra­tio­nal that is always $even \times i\pi k$ over odd prime

making them ir­re­ducible. This dif­fers from the in­te­ger num­bers, $Z_{k}$, which will have ver­tices in two forms

based on whether $Z_{k}$ is an odd or even in­te­ger. Except for the number $2$, when $Z_{k}$ is even, we im­me­di­ate­ly know it is a com­pound number, given it will con­tain re­ducible ra­tio­nal num­bers in its set of ver­tices. The $Z_{k}(odd)$ in­te­gers will be prime num­bers if all the ver­tices of its in­scribed $n$-gon rep­re­sen­ta­tion in the unit circle are ir­re­ducible. Im­por­tant­ly, this en­ables us to geo­met­ri­cal­ly see how prime num­bers differ from com­pound num­bers since a prime number $p_{n} = Z_{k}$, will not pos­sess in­ter­nal sym­me­tries al­low­ing any lower in­te­ger-valued prime poly­gon less than$\ {p_{n} = Z}_{k}$, to rotate in a vertex-to-vertex fash­ion in the com­plex unit circle.

To show this geo­met­ric char­ac­ter­is­tic of the prime num­bers, in the fol­low­ing figure we graph the in­scribed $9$-gon rep­re­sent­ing the set of ver­tices with a min­i­mal el­e­ment basis of $\mathcal{m}\mathcal{e}_{\lbrack 1/9\rbrack}$

and its geo­met­ric prime number gen­er­a­tor, the equi­lat­er­al tri­an­gle, $3$-gon, with a min­i­mal el­e­ment basis of $\mathcal{m}\mathcal{e}_{\lbrack 1/3\rbrack}$

three dis­tinct ori­en­ta­tions of its prime number gen­er­a­tor $P_{2} = 3$.

We see the equi­lat­er­al tri­an­gle has three unique ori­en­ta­tions in the $9$-gon, which is the geo­met­ric result of the number $9$'s com­po­si­tion

We now have a better geo­met­ric sense of how prime num­bers can be ex­pressed in com­plex space, lead­ing us to for­mal­ly define the geo­met­ric primes.

De­f­i­n­i­tion (1.7.38) - The Geo­met­ric Primes, $gp_{n}$, are the geo­met­ric rep­re­sen­ta­tion of the prime num­bers $p_{n}$ as in­scribed reg­u­lar poly­gons in the com­plex unit circle, with $p_{n}$ ver­tices, $V$, sym­met­ri­cal­ly arranged around the com­plex unit circle, $U(1)$, by dis­tance in­ter­vals equal to the min­i­mal el­e­ment basis, form­ing the set of ver­tices $A\left( \mathcal{m}\mathcal{e}_{\left\lbrack 1/p_{n} \right\rbrack} \right)$ de­fined as

and these ver­tices have unique lo­ca­tion values for all geo­met­ric in­te­gers less than $gp_{n}$, and are unique to any geo­met­ric prime in the set of all geo­met­ric primes.

in­scribed in the com­plex unit circle, rep­re­sent­ing reg­u­lar poly­gons.

Plot­ted above are the polyg­o­nal rep­re­sen­ta­tions of the first four geo­met­ric primes, g$p_{1},{gp}_{2},{gp}_{3},{gp}_{4} = 2,3,5,7$, where the number of ver­tices co­in­cides with the prime number's in­te­ger value. All ver­tices have unique lo­ca­tion values com­pared to all in­te­gers of lower value, in­clud­ing geo­met­ric primes and geo­met­ric com­pos­ites. The vertex lo­ca­tion values of all geo­met­ric primes con­tin­ue to be unique com­pared to other geo­met­ric primes in the set of all geo­met­ric primes. Com­pos­ite num­bers will share vertex values with geo­met­ric primes, given that these num­bers are com­posed of prime number prod­ucts.

The­o­rem (1.7.39) – Except for the iden­ti­ty vertex $V_{0}= \mathcal{m}\mathcal{e}_{\epsilon}$, all ver­tices of the geo­met­ric prime, $gp_{n}$, for $n > 1$, can gen­er­ate all ver­tices of its in­scribed reg­u­lar $n$-gon.

Proof – By de­f­i­n­i­tion, the first vertex is lo­cat­ed a dis­tance in ra­di­ans of $\ln\left( \mathcal{m}\mathcal{e}_{\left\lbrack \pm \frac{1}{p_{n}} \right\rbrack} \right)$ from the iden­ti­ty vertex $V_{0}= \mathcal{m}\mathcal{e}_{\epsilon}$

The set of pos­i­tive­ly and neg­a­tive­ly ori­ent­ed ver­tices on the bound­ary of the com­plex unit circle, with the min­i­mal el­e­ment basis $\mathcal{m}\mathcal{e}_{\pm \left\lbrack \frac{1}{p_{n}} \right\rbrack}$, is the set

for which each vertex is sym­met­ri­cal­ly spaced around the bound­ary of the unit circle $\in \mathbb{C}^{1}$ by an ar­clength dis­tance, $\Delta V$

Given that $\mathcal{m}\mathcal{e}_{\left\lbrack \frac{1}{p + n\ } \right\rbrack} = e^{\left( \frac{2i\pi}{p_{n}} \right)},$ each vertex will be lo­cat­ed at an ar­clength dis­tance from the iden­ti­ty at $\mathcal{m}\mathcal{e}_{\lbrack\epsilon\rbrack} = e^{2i\pi k}$ of

for in­te­ger values of $k = 0$ through $p_{n} - 1$. There­fore, the nu­mer­a­tor re­mains $even \times (i\pi)$, the de­nom­i­na­tor is $p_{n}(odd)$ for $n > 1$, and ir­re­ducible given it is prime. The set gen­er­at­ed by a vertex lo­cat­ed at $\left\lbrack \left( \mathcal{m}\mathcal{e}_{\left\lbrack \pm \frac{1}{p_{n}} \right\rbrack} \right)^{k} \right\rbrack$ is

which is equiv­a­lent to the set

As the set of ver­tices $A\left( \mathcal{m}\mathcal{e}_{\left\lbrack \pm \frac{1}{p_{n}} \right\rbrack} \right)$ is on the com­plex unit circle and de­fined as modulo $2i\pi$, the set is equiv­a­lent t

which can be stated as

given that

$\square$

This proof shows that the geo­met­ric primes rep­re­sent cyclic groups by the cri­te­ria that all vertex el­e­ments, except for the iden­ti­ty vertex, can gen­er­ate all vertex el­e­ments.

The prime num­bers are in­ti­mate­ly con­nect­ed to har­mon­ic series and the Rie­mann zeta func­tion, $\zeta(s),$for $s\in \mathbb{C}$, ev­i­denced by the Euler prod­uct for­mu­la

which was proved by Euler and pub­lished in $1737.$To show how Euler ar­rived at his result, we take the fol­low­ing two re­lat­ed zeta func­tion in­fi­nite series rep­re­sen­ta­tions

and sub­tract the second series from the first

Next, we take the series $\frac{1}{3^{s}}\left( 1 - \frac{1}{2^{s}} \right)\zeta(s)$

and sub­tract this from $\left( 1 - \frac{1}{2^{s}} \right)\zeta(s)$ to find

and this is a series with all rec­i­p­ro­cal prime number fac­tors of $1/2$ and $1/3$ re­moved. The process re­peats in­fi­nite­ly, re­sult­ing in

and di­vid­ing both sides by the in­fi­nite prod­uct of with terms of the form $\left( 1 - \frac{1}{p_{n}^{s}} \right)$, we find that

For $s = 1$, the zeta func­tion, $\zeta(1),$simply re­duces to the har­mon­ic series

and the Euler prod­uct is

To show how the Euler prod­uct re­lates to geo­met­ric primes, we first look at the fol­low­ing con­tour in­te­gral

We con­sid­er the unit circle by set­ting $z(t) = e^{it},$to find $\frac{dz}{dt} = ie^{it}$, and the in­te­gral above be­comes

Sim­pli­fy­ing and eval­u­at­ing the in­te­gral, we find the result

which is con­sis­tent with the perime­ter dis­tance of the com­plex unit circle in ra­di­ans. The min­i­mal el­e­ment rep­re­sents the first vertex of each $n$-gon lo­cat­ed in the coun­ter­clock­wise di­rec­tion from the iden­ti­ty vertex. The perime­ter dis­tance of this vertex is

where $Z_{n}$ is the rec­i­p­ro­cal in­te­ger value and $\frac{1}{Z_{n}}$ is the $n^{th}$ el­e­ment of the har­mon­ic series. We can ex­press the zeta func­tion as a prod­uct with the pre­vi­ous con­tour in­te­gral as

which is equiv­a­lent to

where the Euler prod­uct holds for the har­mon­ic series terms in paren­the­sis, and we can ex­press the zeta func­tion as a prod­uct with the con­tour in­te­gral of the com­plex unit circle

In the last chap­ter, we con­sid­ered the series of the nat­ur­al num­bers as the el­e­ment-to-el­e­ment in­verse of the har­mon­ic series. The tri­an­gu­lar num­bers, $T_{n}$, are de­fined as the sums of suc­ces­sive se­quences of the nat­ur­al num­bers to $n$, as fol­lows

These num­bers are tri­an­gu­lar since they count ob­jects arranged in the equi­lat­er­al tri­an­gle, with one el­e­ment in each row/column po­si­tion. For ex­am­ple, the fifth tri­an­gu­lar number $T_{5}$ is seen in the figure below, where $T_{5} = 15$ re­sults from count­ing all the white cir­cles from each row.

Eval­u­at­ing the first five tri­an­gu­lar num­bers, each rep­re­sent­ing a suc­ces­sive sum of rows from the equi­lat­er­al tri­an­gle from the figure, we find

Next, we con­sid­er how the number $n$ re­lates to its re­spec­tive $T_{n}$ sum, by eval­u­at­ing the ratio $\frac{T_{n}}{n}$ for each row, as fol­lows

find­ing the dis­tinct pat­tern that each $\frac{T_{n}}{n}$ ratio in­creas­es by $\frac{1}{2}$ com­pared to $\frac{T_{n - 1}}{(n - 1)}.$This ratio gen­er­al­izes as the ex­pres­sion

where­by solv­ing for $T_{n}$, we find the tri­an­gu­lar num­bers have the closed form

Given that a bi­no­mi­al co­ef­fi­cient is de­fined as a ratio of fac­to­ri­als

the nu­mer­a­tor of $T_{n}$, can also be ex­pressed as a ratio of fac­to­ri­als

which leads to the bi­no­mi­al co­ef­fi­cient ex­pres­sion

and the tri­an­gu­lar num­bers can be de­fined as

As we found pre­vi­ous­ly, the radian dis­tance for each vertex of the $Z_{n}$ in­scribed poly­gon rep­re­sen­ta­tion in the unit circle can be ex­pressed as the min­i­mal el­e­ment re­la­tion­ship

and by taking the prin­ci­pal valued log­a­rithm for each el­e­ment in the set above, we can define the set $\Delta V\left( \mathcal{m}\mathcal{e}_{\lbrack 1/n\rbrack} \right)$, to rep­re­sent the set of radian dis­tance values from $\epsilon$, for each vertex of the $Z_{n}$ poly­gon, as

Next, we choose to find the total vertex dis­tance, but sum­ming each dis­tance el­e­ment in the set $\Delta V\left( \mathcal{m}\mathcal{e}_{\lbrack 1/n\rbrack} \right)$, to find the series

for which we factor out $\frac{2i\pi}{n}$

and find the pres­ence of the tri­an­gu­lar num­bers in the series. We can now sub­sti­tute our closed form for the tri­an­gu­lar num­bers into the series

find­ing that our total vertex dis­tance also has a closed form as fol­lows

Next, we solve the series by equat­ing the tri­an­gu­lar num­bers to their fac­to­r­i­al ratio equal­i­ty, as fol­lows

and like­wise, we find the total vertex dis­tance sum­ma­tion re­lates to the bi­no­mi­al co­ef­fi­cient

We return to the set $A\left( \mathcal{m}\mathcal{e}_{\lbrack 1/n\rbrack} \right)$, which rep­re­sents the ex­po­nen­tial form for each vertex lo­ca­tion on the com­plex unit circle

which is equiv­a­lent to the form

The fol­low­ing ex­po­nen­tial prod­uct sum

is anal­o­gous to the total vertex dis­tance sum­ma­tion, and specif­i­cal­ly, they are the fol­low­ing log­a­rithm and ex­po­nen­tial re­la­tion­ships

and

which shows the con­nec­tion these prod­ucts and sums have with the min­i­mal el­e­ment Lie al­ge­bra. As ex­pect­ed, the result of the prod­uct sum has the ex­po­nen­tial form

which for in­te­ger values of $n$, the value will os­cil­late be­tween $- 1$ for $n$ even and $1$ for $n$ odd. Sup­pose we extend the vari­able from the nat­ur­al num­bers to the reals, $n\in \mathbb{R}$. For this case, we plot the func­tion in the figure below, find­ing the fa­mil­iar wave­forms of the ex­po­nen­tial func­tion, which are phase-shift­ed by $i\pi$ ra­di­ans.

As we con­tin­ue study­ing the sin­gu­lar­i­ty by prob­ing deeper into it math­e­mat­i­cal­ly, the im­por­tance of tri­an­gu­lar num­bers be­comes more ap­par­ent. The exact way they are re­lat­ed to the el­e­ments of the har­mon­ic series is easy to see by start­ing with the pre­vi­ous­ly de­rived result

which we solve for $\frac{1}{n}$, to find the in­verse re­la­tion­ship it has with the tri­an­gu­lar num­bers

which leads to the sum­ma­tion ex­pres­sion for the $n^{th}$ har­mon­ic number as

To show the beauty these num­bers have, we con­sid­er the sum­ma­tion of the in­verse tri­an­gu­lar num­bers, $\left( \mathcal{T}_{n} \right)^{- 1},$

which we see is a series com­posed of in­creas­ing sub­series of ra­tio­nal num­bers. To see if this series con­verges, we take each el­e­ment in the series

then take the par­tial frac­tion de­com­po­si­tion of this result to find

which we can now sum in­di­vid­u­al­ly to find $\left( \mathcal{T}_{n} \right)^{- 1}$

Both sums are re­lat­ed to the har­mon­ic num­bers, and given that

we can see that this series con­verges to

which can be sim­pli­fied as fol­lows

and there­fore, we have that

A beau­ti­ful thing about these num­bers is that when the sum of the $\left( \mathcal{T}_{n} \right)^{- 1}$ num­bers is eval­u­at­ed as $n \rightarrow \infty$, we find it con­verges to 2

In­tu­itive­ly, we can see this con­ver­gence result by first look­ing at the set of suc­ces­sive values of $\left( \mathcal{T}_{n} \right)^{- 1}$, $A\left( \left( \mathcal{T}_{n} \right)^{- 1} \right),$

and after fac­tor­ing out the number $2,$we find

where the values in the set ap­proach the value of $1$, as the values of $n$ get larger. An ev­i­dent re­la­tion­ship exists be­tween this and the har­mon­ic series, seen when ex­press­ing the set as

where each el­e­ment is now rep­re­sent­ed as a dif­fer­ence of $1$ and the or­dered el­e­ments of the har­mon­ic series. This re­la­tion­ship yields the fol­low­ing par­tial sums of $\left( \mathcal{T}_{n} \right)^{- 1}$ num­bers, and how they relate to the har­mon­ic num­bers as

where we ar­rived at this ex­pres­sion by first con­sid­er­ing $\left( \mathcal{T}_{n} \right)^{- 1}$ as the series

which, when ex­pand­ed, is found to be a tele­scop­ing series

There­fore, the par­tial sum of $\left( \mathcal{T}_{k} \right)^{- 1}$ is

and after re­ar­rang­ing the ex­pres­sion by solv­ing for $H_{n}$, we find

which sim­pli­fies to

and by de­creas­ing the it­er­a­tion for the har­mon­ic num­bers by one, this yields the fol­low­ing ex­pres­sion for $H_{n}$

As we de­scribed pre­vi­ous­ly in the sec­tion, the har­mon­ic series is re­lat­ed to the prime num­bers by way of the Euler prod­uct

which in the case when $s = 1$, the Rie­mann zeta func­tion $\zeta(1)$ is re­lat­ed to the primes as the fol­low­ing equiv­a­lence

Our de­fined num­bers $\left( \mathcal{T}_{n} \right)^{- 1}$ are re­lat­ed to the tetra­he­dral num­bers, which are de­fined as the sum­ma­tion of the tri­an­gu­lar num­bers to $n$

with the dif­fer­ence being in the case of $\left( \mathcal{T}_{n} \right)^{- 1}$, we are sum­ming the in­verse values of the tri­an­gu­lar num­bers, $\frac{1\ }{T_{k}}.$

Our work show­ing the re­la­tion­ship be­tween the in­verse tri­an­gu­lar number sums $\left( \mathcal{T}_{n} \right)^{- 1}$, the har­mon­ic series, and the har­mon­ic num­bers, $H_{n}$, lets us learn pre­cise­ly how the in­verse tri­an­gu­lar number sums $\left( \mathcal{T}_{n} \right)^{- 1}$ are re­lat­ed to both the Rie­mann zeta func­tion and the prime num­bers. To start, since the har­mon­ic number $H_{n}$ is equal to

as $n \rightarrow \infty$, the fol­low­ing limit is equal to the zeta func­tion, $\zeta(s),$for $s = 1$

and based on our equal­i­ty, both of the fol­low­ing limits are equal to $\zeta(1)$

Taking the ex­pres­sion found pre­vi­ous­ly for $\left( \mathcal{T}_{n} \right)^{- 1}$,

and with simple al­ge­bra­ic ma­nip­u­la­tion, we find

and given that

it is also true that

This leads to the iden­ti­ty

where we note that

Using this iden­ti­ty, we expand the limit in­volv­ing the in­verse tri­an­gu­lar number sum­ma­tion

and equate it to the func­tion $\zeta(1)$, to find

In the ex­pres­sion above, the$\lim_{n \rightarrow \infty}(n)$ is offset term-by-term by the ones in the brack­et­ed terms, while the other limit is equal to one

given that

which once again ver­i­fies that

There­fore, $\zeta(1)$ is equal to the fol­low­ing series of $\left( \mathcal{T}_{n} \right)^{- 1}$ terms

With the ex­pres­sion above, we see the $\left( \mathcal{T}_{n} \right)^{- 1}$ num­bers are re­lat­ed to the $\zeta(1)$ func­tion by way of well-be­haved in­fin­i­ty minus the in­fi­nite series of the $\frac{\left( \left( \mathcal{T}_{n} \right)^{- 1} \right)}{2}$ num­bers. To fur­ther con­sid­er a more exact re­la­tion­ship be­tween the zeta func­tion and the $\frac{\left( \mathcal{T}_{n} \right)^{- 1}}{2}$ num­bers, we must first define the Tau func­tion $\mathrm{T}(s)$.

De­f­i­n­i­tion (1.7.120) – The Tau func­tion $\mathrm{T}(s)$, for $s\in \mathbb{C}$, rep­re­sents the in­fi­nite series of the fol­low­ing in­verse tri­an­gu­lar num­bers

for $\mathrm{T}(s)$, de­fined as

and as a series, is the el­e­ment-by-el­e­ment ad­di­tive in­verse of the re­spec­tive zeta func­tion $\zeta(s).$

It is easy to see the ad­di­tive in­verse prop­er­ty the $\mathrm{T}(s)$ func­tion has with the zeta func­tion, given that

and we will denote the in­fin­i­ty that re­sults from the sum as $\overline{n}$

Geo­met­ri­cal­ly, the num­bers of the $\mathrm{T}(s)$ func­tion travel in the op­pos­i­tive di­rec­tion in the com­plex plane com­pared to the series of num­bers from the $\zeta(s)$ func­tion. Specif­i­cal­ly, the num­bers are re­lat­ed as op­po­site-valued Euler prod­ucts

which we prove for $\mathrm{T}(s)$ in the the­o­rem that fol­lows.

The­o­rem (1.7.124) – The fol­low­ing Euler Prod­uct de­fines the Tau func­tion

Proof – The func­tion $\mathrm{T}(s)$ is de­fined, for $s\in \mathbb{C}$, as the fol­low­ing sum­ma­tion

which, when ex­pand­ed, is equiv­a­lent to

this arranges to

where $\overline{n}$ rep­re­sents the in­fi­nite sum of iden­ti­ties

We pro­ceed to cal­cu­late $\left( 1 - \frac{1}{2^{s}} \right)\mathrm{T}(s)$ by al­ge­bra­ic ma­nip­u­la­tion of the series, where we cal­cu­late first

and sub­tract this from $\mathrm{T}(s),$

where all even-valued rec­i­p­ro­cals have can­celed, and the above sim­pli­fies to

We con­tin­ue this process for the sub­se­quent two it­er­a­tions as fol­lows

where all suc­ces­sive com­pound rec­i­p­ro­cals for the prime rec­i­p­ro­cals $3^{- s}$ and $5^{- s}$ have also been elim­i­nat­ed from the right-hand side. By ex­tend­ing this process in­fi­nite­ly, the result is

for which, fac­tor­ing and sim­pli­fi­ca­tion yields

and after di­vid­ing both sides by the in­fi­nite rec­i­p­ro­cal prime-re­lat­ed prod­uct series, we find

which equals the stated the­o­rem

Pre­vi­ous­ly, we ar­rived at the equal­i­ty for $\zeta(1)$

After sub­sti­tut­ing our de­fined $\overline{n}$ and the $\mathrm{T}(s)$ func­tion, the ex­pres­sion is equal to

and solv­ing for $\overline{n}$, we find the fol­low­ing ad­di­tive in­verse equal­i­ty

which is con­sis­tent with the ad­di­tive in­verse prop­er­ties in the de­f­i­n­i­tion for $\mathrm{T}(s).$This equal­i­ty, along with the spe­cif­ic so­lu­tion to our pre­vi­ous proof

present the means to un­der­stand the in­fin­i­ty of ones math­e­mat­i­cal­ly. Specif­i­cal­ly, how it splits into the el­e­ments of the har­mon­ic series with the el­e­ment-by-el­e­ment ad­di­tive in­verse series of in­verse tri­an­gu­lar num­bers. We can also define a very con­cise equal­i­ty for the iden­ti­ty by solv­ing for $1$ in the fol­low­ing ex­pres­sion from our proof

where the dif­fer­ence be­tween prod­ucts of two dis­tinct in­fi­nite series with the in­fi­nite prod­uct of rec­i­p­ro­cal primes yields the iden­ti­ty. Given that

this is also equiv­a­lent to our iden­ti­ty ex­pres­sion

and once again, math­e­mat­ics never fails to im­press with its beauty and co­he­sive­ness.

We turn to prime num­bers and their rep­re­sen­ta­tions in group theory. This book has fo­cused on the prime number $p_{2} = 3$, from the stand­point of groups, and the geo­met­ric prime is en­cap­su­lat­ed in matrix al­ge­bra as the set $A\left( gp_{2} \right)$, com­posed of the fol­low­ing two gen­er­a­tors

The ma­tri­ces are the same $Y_{2},Y_{3}$ gen­er­a­tors from our group theory study but differ re­gard­ing their di­rec­tion of ro­ta­tion when acting on a po­si­tion vector. To show this math­e­mat­i­cal­ly, we define a vector $v_{2} = \left( e^{2i\pi},e^{\frac{2i\pi}{3}},e^{\frac{4i\pi}{3}} \right)^{T}$, which rep­re­sents the three ver­tices of the in­scribed geo­met­ric prime $gp_{2}$ arranged in coun­ter­clock­wise order from left to right, and take the fol­low­ing matrix-vector prod­uct

which rep­re­sents a clock­wise ro­ta­tion of the ver­tices by $\frac{2i\pi}{3}$ ra­di­ans, shown in the figure below.

As stated, we find that the matrix-vector prod­uct $Y_{3} \times v_{2}$, ro­tates the plane in the op­po­site di­rec­tion, coun­ter­clock­wise, given that

whose trans­form we show in the figure below.

We direct the reader's at­ten­tion to the $3 \times 3$ matrix below

where $\alpha$ is just a dummy vari­able, and like the prime number $3$, the matrix cannot be split equal­ly into small­er sub­ma­tri­ces, except for $1 \times 1$ sizes and itself, $3 \times 3$. We com­pare this to the com­pos­ite geom­e­try of the in­te­ger nine$,\mathbb{Z}_{9}$, which has a square matrix rep­re­sen­ta­tion, but we can divide the matrix into nine even $3 \times 3$ sec­tions

which un­der­scores the fact it is com­posed of $gp_{2}$ ma­tri­ces, and this sec­tion­ing of the $\mathbb{Z}_{9}$ matrix is the only way to do so. Com­pare this to the com­pos­ite geom­e­try of the in­te­ger $6,\mathbb{Z}_{6}$, which has two unique ways to evenly sec­tion its matrix

where the first rep­re­sents a $2 \times 2$ sec­tion­ing of $3 \times 3$ ma­tri­ces, and the second a $3 \times 3$ sec­tion­ing of $2 \times 2$ ma­tri­ces, sig­ni­fy­ing $\mathbb{Z}_{6}$ is com­posed of the geo­met­ric prime num­bers $gp_{1}$ and $gp_{2}$.

For the geo­met­ric prime set $A\left( gp_{2} \right),$

we now return to the anti-di­ag­o­nal sym­me­try of the Cartan al­ge­bra dis­cus­sion in Sec­tion 1.2.9, where each Lie al­ge­bra gen­er­a­tor in the set $A\left( gp_{2} \right)$ has sym­me­try about the anti-di­ag­o­nal axis. The iden­ti­ty matrix rep­re­sents the di­ag­o­nal axis, and, when taken as a prod­uct with an­oth­er matrix, acts to scale the values of that matrix's en­tries. The anti-iden­ti­ty matrix can be thought of as the analog of the re­la­tion­ship the iden­ti­ty matrix has with the di­ag­o­nal, and this anti-iden­ti­ty matrix we will define as

which is the same as the $X_{2}$-type Lie al­ge­bra that we have de­fined pre­vi­ous­ly. Sim­i­lar­ly, we can define that

which we arrive at by ro­tat­ing the $Y_{2}$ and $Y_{3}$ Lie al­ge­bras by 90 de­grees in the clock­wise di­rec­tion, and the reader should con­sid­er these trans­forms as higher-di­men­sion­al ro­ta­tions of the $gp_{2}$ geo­met­ric prime. From the stand­point of matrix rep­re­sen­ta­tion theory, these di­men­sion­al trans­forms are the result of the prod­uct of the ro­ta­tion matrix $\epsilon^{*}$, with the el­e­ments of the set $A\left( gp_{2} \right)$, seen as fol­lows

It is im­por­tant to note that any of the $X$-type ma­tri­ces can gen­er­ate the same matrix-matrix prod­ucts above, and our choice to use the anti-iden­ti­ty matrix, $\epsilon^{*}$, has its basis in main­tain­ing con­sis­ten­cy.

To better un­der­stand the higher-di­men­sion­al ro­ta­tions of the $gp_{2}$ geo­met­ric prime al­ge­bras, we show in the figure below the arrange­ment of the $Y$ al­ge­bras along the edges of the equi­lat­er­al tri­an­gle, where each al­ge­bra is po­si­tioned on the edge co­in­cid­ing with its vertex sym­me­try (yellow arrows). Ad­di­tion­al­ly, the el­e­ments of the al­ge­bras are rep­re­sent­ed as red cir­cles, in­stead of $1$ s, in each of the squares.

Next, we look at the set of Lie al­ge­bra gen­er­a­tor ro­ta­tions $\epsilon^{*} \times A\left( gp_{2} \right) = \left\{ X_{1},X_{3},X_{2} \right\}$ in matrix form as

and con­sid­er the ex­ten­sion of the geo­met­ric prime, $gp_{2},$into the higher-di­men­sion­al $\mathbb{C}^{3}$ space, which is pre­sent­ed in Figure 1.7.10. We can see in the figure that each $gp_{2}$ al­ge­bra projects into the higher di­men­sion of its spe­cif­ic $X$ al­ge­bra, which is onto a face of the tri­an­gu­lar prism rising from the $gp_{2}$ geo­met­ric prime's equi­lat­er­al tri­an­gle.

Just as the prime num­bers are the build­ing blocks for the in­te­gers, the geo­met­ric primes are the sine qua non of the Lie al­ge­bras we have stud­ied. In the higher-di­men­sion­al com­plex spaces, the geo­met­ric primes are the build­ing blocks of the com­pos­ite $n$-gons. Still, we will find they are re­spon­si­ble for build­ing the al­ge­bra's di­men­sions, and we will touch upon this sub­ject in the next sec­tion.

We return to the in­verse tri­an­gu­lar number set $A\left( \left( \mathcal{T}_{n} \right)^{- 1} \right)$

to fur­ther con­sid­er how it is re­lat­ed to the min­i­mal el­e­ment, $\mathcal{m}\mathcal{e}_{\lbrack 1/n\rbrack}$, which we recall is de­fined as

When as­sum­ing a pos­i­tive-valued $\mathcal{e}_{\lbrack + 1/n\rbrack}$, the first vertex of the rep­re­sen­ta­tive $n$-gon will be ori­ent­ed at a counter-clock­wise dis­tance of $\frac{2i\pi}{n}$ ra­di­ans from the iden­ti­ty. We take the $A\left( \left( \mathcal{T}_{n} \right)^{- 1} \right)$ set and con­sid­er its values to be on the com­plex unit circle and in ra­di­ans, where ${i\pi\left( \mathcal{T}_{n} \right)}^{- 1}$ is equal to

and the set $A\left( {i\pi\left( \mathcal{T}_{n} \right)}^{- 1} \right)$ would then equal

Each value in the $A\left( {i\pi\left( \mathcal{T}_{n} \right)}^{- 1} \right)$ rep­re­sents the first vertex of the rep­re­sen­ta­tive $n$-gon from the iden­ti­ty in the clock­wise ori­en­ta­tion and rep­re­sents the ar­gu­ment that de­fines the neg­a­tive-valued min­i­mal el­e­ment, $\mathcal{m}\mathcal{e}_{\lbrack - 1/n\rbrack}$. In the case of the $gp_{2}$ geo­met­ric prime, $\mathcal{m}\mathcal{e}_{\lbrack - 1/3\rbrack}$ is equal to

which has the equiv­a­lent form with a pos­i­tive­ly valued ar­gu­ment

By con­sid­er­ing the in­verse tri­an­gu­lar sum ${i\pi\left( \mathcal{T}_{3} \right)}^{- 1} = 2i\pi\left( 1 - \frac{1}{3} \right)$, we find it equals the same ar­gu­ment above, given that

and the same is true for all el­e­ments in the set $A\left( {i\pi\left( \mathcal{T}_{n} \right)}^{- 1} \right),$where the neg­a­tive-valued min­i­mal el­e­ment, $\mathcal{m}\mathcal{e}_{- \left\lbrack \frac{1}{n} \right\rbrack}$, is equal to

Going back to the Euler prod­uct we de­scribed,

where the con­tour in­te­gral $\oint_{}^{}{z^{- 1}\ dz}$, we know is equal to $2i\pi$, it be­comes clear that the terms con­tain­ing the rec­i­p­ro­cal primes $\left( 1 - \frac{1}{p_{n}} \right)$ are equal to the rec­i­p­ro­cal primes of the set $A\left( {i\pi\left( \mathcal{T}_{n} \right)}^{- 1} \right)$, which we define as $A\left( {i\pi\left( \mathcal{T}_{p_{n}} \right)}^{- 1} \right)$

and this set also rep­re­sents the radian dis­tance of the neg­a­tive min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{- \left\lbrack \frac{1}{n} \right\rbrack}$, for rec­i­p­ro­cal prime values, de­fined as $A\left( \mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{p_{n}} \right\rbrack} \right),$

Sim­i­lar­ly, the pos­i­tive-valued min­i­mal el­e­ments with rec­i­p­ro­cal prime values, are de­fined as

and el­e­ments be­tween the sets with like rec­i­p­ro­cal prime number basis are re­lat­ed as con­ju­gate pairs, where their prod­ucts are equal to the iden­ti­ty

For any $n$-gon in­scribed in the com­plex unit circle, the value $n$ is the sum of con­ju­gate prod­uct pairs, given that

and the par­tial sum of $1\text{s}$ is simply equal to $n$

We now have a geo­met­ric in­ter­pre­ta­tion of the prime num­bers, rep­re­sent­ing a solid foun­da­tion we will build upon in the next book in the Arith­moí series. In this sec­tion, we have found sev­er­al con­nec­tions prime num­bers have with group theory, higher di­men­sion­al com­plex spaces, and the tri­an­gu­lar num­bers, made pos­si­ble by putting the min­i­mal el­e­ment into its proper per­spec­tive.