1.4.5 Number Forms, Odd and Even Partition Geometry, and Minimal Elements

On the way to find­ing the number of per­mu­ta­tion el­e­ments $\in D_{3\mathbb{C}}^{\Delta}$, we made use of num­bers that were de­scribed as cone num­bers and axis num­bers. De­vel­op­ing an in­tu­ition for these num­bers comes by ex­pand­ing beyond a simple al­ge­bra­ic de­scrip­tion, to one that en­com­pass­es their geom­e­try. Our spa­tial abil­i­ty to con­sid­er higher-di­men­sion­al number forms may be hin­dered when trying to vi­su­al­ize them, but we can con­sid­er pro­ject­ing our equi­lat­er­al tri­an­gle $\in \mathbb{C}^{3}$ onto a disc $\in \mathbb{C}^{1}$ space. The con­sid­er­a­tions of arc length and areas on the com­plex unit circle, along with their spher­i­cal analogs, reveal that some num­bers and their area/vol­umes are sub­sets of other ones. For the real in­te­gers and ra­tio­nal num­bers, we are aware of the con­cept of even and odd num­bers, for which, the fol­low­ing hold:

The ex­am­ple con­sid­ered is a de­fined group $A$, which rep­re­sents the even nat­ur­al num­bers, with op­er­a­tions of ad­di­tion and mul­ti­pli­ca­tion. The values of the nat­ur­al num­bers re­sult­ing from these op­er­a­tions will remain even and con­sis­tent with the even sub­group of the nat­ur­al num­bers $A\subset \mathbb{N}$. Adding an odd number to any el­e­ment $a$, in the $A$ sub­group, gives us an odd-valued nat­ur­al number $\subset \mathbb{N}$. If we choose the odd nat­ur­al number $1 \subset \mathbb{N}$, then apply the ad­di­tion op­er­a­tion $+ 1$ to every el­e­ment $\in A$, the result will be the whole group of nat­ur­al num­bers

If we were to choose an odd nat­ur­al number such as $7,$we would be miss­ing some of the odd nat­ur­al num­bers in the lower valued bound of the set, seen as fol­lows

for $a_{n}$, the set of even nat­ur­al num­bers. In the above ex­pres­sion, we are miss­ing the odd nat­ur­al num­bers from $1$ to 5, given that the first odd nat­ur­al number was chosen to be $7$. As the number $1$ is re­quired to com­plete the set of nat­ur­al num­bers, we define this to be the min­i­mal el­e­ment of the set, since there is no nat­ur­al number in the set that is small­er than it. The de­f­i­n­i­tion of the min­i­mal el­e­ment can also be con­sid­ered for the groups we have been study­ing. We can start by ap­ply­ing this con­cept to spheres, and we will even­tu­al­ly find the con­cept holds for the other dis­cussed matrix group conic sec­tions. This is elu­ci­dat­ed by ex­pand­ing from the simple nat­ur­al num­bers to the com­plex num­bers and their rep­re­sen­ta­tion as ex­po­nen­tials in higher di­men­sions.

To start, we con­sid­er the in­fi­nite set of com­plex num­bers on the unit circle using $- 1,$raised to a vari­able power $\frac{n}{q}$, as $( - 1)^{\left( \frac{n}{q} \right)}$, for $n\in \mathbb{Z}$, $q \in \mathbb{Z}_{\geq 0}$, where the number zero is in­clud­ed for $n$ and $q$. For $n = 1$, the number $( - 1)^{\left( \frac{1}{q} \right)}$ is the ar­clength in ra­di­ans from the pos­i­tive real-axis, fit­ting the de­f­i­n­i­tion of the pri­ma­ry min­i­mal el­e­ment. The number $2q$ is the total number of el­e­ments in the ar­clength par­ti­tion set on the in­ter­val $\lbrack 0,2i\pi).$We define the number $1$, as

for $n = 0 \land q \in \mathbb{Z}_{\geq 0}$, sig­ni­fy­ing that the total number of el­e­ments in the par­ti­tioned set is zero, defin­ing the non-par­ti­tioned in­ter­val. Ex­press­ing com­plex num­bers on the unit circle up to a pri­ma­ry min­i­mal el­e­ment $( - 1)^{\left( \frac{1}{4} \right)}$, com­pris­es a set of min­i­mal pri­ma­ry el­e­ments $\left\{ ( - 1)^{0},( - 1)^{\left( \frac{1}{1} \right)},( - 1)^{\left( \frac{1}{2} \right)},( - 1)^{\left( \frac{1}{3} \right)},( - 1)^{\left( \frac{1}{4} \right)} \right\}$, for which the full sets are the fol­low­ing:

When con­sid­er­ing the unit circle for odd-valued el­e­ments for the ra­tio­nal ex­po­nent $\frac{n}{q}$ and the sets above, the powers of these el­e­ments will land on every member in the set when suc­ces­sive­ly it­er­at­ed. An ex­am­ple: for modulo $2\pi$, the number $( - 1)^{\left( \frac{3}{4} \right)}$ lands on every el­e­ment in the pri­ma­ry min­i­mal el­e­ment set $( - 1)^{\left( \frac{1}{4} \right)}$, seen by ex­pand­ing its se­quence

For geo­met­ric spaces like the hy­per­bo­la, this would not hold as the hy­per­bo­la is open and suc­ces­sive el­e­ments in the set would con­tin­ue to extend in­def­i­nite­ly out in the geom­e­try, seen by using the anal­o­gous ex­am­ple in the hy­per­bol­ic sense

So, in the case of suc­ces­sive ro­ta­tions on a com­plex sphere, where we want to track non-modulo powers, our pri­ma­ry min­i­mal el­e­ment is like the case of our nat­ur­al num­bers for $1$, and is the only el­e­ment that when mul­ti­plied by itself will yield every el­e­ment in the set. This should sound con­sis­tent with a cyclic group, at least for the two-di­men­sion­al case on the com­plex unit circle. If our goal is to find a sym­met­ric group, we need to think along the lines of taking ad­van­tage of min­i­mal el­e­ments to arrive at all the el­e­ments in the group. Min­i­mal el­e­ments, and any con­gru­ent modulo $2\pi$ el­e­ments for the ex­po­nent $\frac{n}{q}$ equal­ing an odd ra­tio­nal, is anal­o­gous to the odd num­bers from the case of the nat­ur­al num­bers with the de­fined mul­ti­pli­ca­tion op­er­a­tion, where ex­po­nents will add and the fol­low­ing hold,

To find all the gen­er­a­tors en­com­pass­ing the sym­met­ric conic group in com­plex space, we remain mind­ful of the de­ter­mi­nant re­sults that occur when the de­fined op­er­a­tion is ap­plied on odd and even per­mu­ta­tions. For ex­am­ple, the op­er­a­tion of mul­ti­pli­ca­tion, with only even de­ter­mi­nant per­mu­ta­tions, will be bound to even de­ter­mi­nants. A way to po­ten­tial­ly de­scribe the min­i­mal el­e­ments of our ma­tri­ces $\in \mathbb{C}^{3}$, is by pro­ject­ing them onto a two-di­men­sion­al com­plex disc. We con­sid­er el­e­ments of the $X_{1}$-type for hy­per­bol­ic forms and spher­i­cal forms, by using the ex­am­ples

We can see the $a_{1,1}$ en­tries are not only the min­i­mal el­e­ments, but they are also the axes of ro­ta­tion. In the bottom-right $2 \times 2$ sub­ma­tri­ces are the radial end­points of the min­i­mal el­e­ment mod­u­lus coun­ter­clock­wise $\left( a_{2,3} \right)$, and clock­wise $\left( a_{3,2} \right)$, from the min­i­mal el­e­ment. It should also be noted that the bottom-right sub­ma­trix is the imag­i­nary hy­per­bol­ic analog and the real unit circle analog. Most are fa­mil­iar with the real num­bers rep­re­sent­ing the sub­ma­trix en­tries in the anti-di­ag­o­nal for $\mathbb{C} \rightarrow\mathbb{R}^{2}$, but may not be aware that there is a sim­i­lar analog for the split-com­plex num­bers. For these gen­er­a­tors alone, we can raise them to in­te­ger powers, let­ting the min­i­mal el­e­ment cycle through the sub­group el­e­ments. The case for $X_{Re1}$ is straight­for­ward

for which, we are bound by the pos­i­tive hy­per­bol­ic tan­gent space. For $X_{Imag1}$, the min­i­mal el­e­ment cycles the group through all four axes el­e­ments:

These el­e­ments, as two-di­men­sion­al pro­jec­tions, are shown graph­i­cal­ly in Figure 1.4.13, for the iden­ti­ty el­e­ment, the pos­i­tive and neg­a­tive $X_{1}$ hy­per­bol­ic tan­gent space, and the tan­gent spaces for the com­plex sphere $X_{1}$-com­po­nent. For all the el­e­ments above, the matrix de­ter­mi­nants are even and bound to the axis num­bers.

4- pos­i­tive spher­i­cal,5- neg­a­tive spher­i­cal.

Turn­ing to the cone number el­e­ments of the $X_{1}$-form, we define the min­i­mal el­e­ment to be

where once again the entry $a_{1,1}$ rep­re­sents the min­i­mal par­ti­tion el­e­ment and the axis of ro­ta­tion. The min­i­mal par­ti­tion has an odd de­ter­mi­nant, given it is a cone number, and cycles through the entire al­ge­bra set

${- ( - 1)}^{\left( \frac{1}{4} \right)},\ - ( - 1)^{\left( \frac{3}{4} \right)}$, figs. 1,2,3,4, re­spec­tive­ly.

Figure 1.4.14 shows a pro­jec­tive two-di­men­sion­al graph­ic rep­re­sen­ta­tion for the $X_{1}$-type cone num­bers, as they cycle through the quad­rants.

As a final ex­am­ple, we take the min­i­mal el­e­ment $( - 1)^{\left( \frac{1}{8} \right)}$, which turns out to be like the $( - 1)^{\left( \frac{1}{4} \right)}$ ex­am­ple, having a total of 16 el­e­ments in the sub­group, having the $( - 1)^{\left( \frac{1}{8} \right)}$ min­i­mal el­e­ment matrix rep­re­sen­ta­tion

In Figure 1.4.15, shown are four of the el­e­ments on the upper half of the com­plex circle for $X_{1}$, raised to odd powers

As men­tioned pre­vi­ous­ly, having a better un­der­stand­ing of the min­i­mal el­e­ments of our group and the con­cept of odd and even de­ter­mi­nants give us better geo­met­ric clar­i­ty con­cern­ing the sym­met­ric conic group. This clar­i­ty is by way of the fact that we will need to start with gen­er­a­tors $\in \mathcal{X},\mathcal{Y}$, which have odd value de­ter­mi­nants and con­tain the min­i­mal el­e­ment for each.