1.8.6 The Theory of Light

This sec­tion re­turns to some of the con­cepts cov­ered in Sec­tion 1.7.2, in­volv­ing as­pects of the di­men­sion­al trans­for­ma­tion map­ping $\mathbb{C}^{1}: \rightarrow \mathbb{R}^{2}$. This will help us better un­der­stand the nature of light and con­sid­er how the math­e­mat­i­cal frame­work we have been work­ing with is causal to the phys­i­cal phe­nom­e­nol­o­gy per­ceived around us. We start by look­ing at the first cone number in the pos­i­tive ori­en­ta­tion from the iden­ti­ty, $( - 1)^{\frac{1}{4}},$which has the fol­low­ing equiv­a­lence ex­pres­sion

and point on the com­plex unit circle is de­pict­ed in Figure 1.8.10. Based on the two pre­vi­ous sec­tions, this number should stand out as the pos­i­tive min­i­mal el­e­ment of the in­scribed oc­ta­gon, which, in math­e­mat­i­cal no­ta­tion, is

and this cone number be­longs to the set of Lie al­ge­bra gen­er­a­tors $Y\left( \mathbb{Z}_{8} \right)$

Each el­e­ment in the set above has an $8 \times 8$ matrix Lie al­ge­bra gen­er­a­tor rep­re­sen­ta­tion, which can be seen using the ex­am­ple for the al­ge­bra $Y\left( \mathbb{Z}_{8} \right)_{2} = ( - 1)^{\frac{1}{4}}$

where the set of el­e­ments $A\left( Y\left( \mathbb{Z}_{8} \right) \right)$ can be gen­er­at­ed by $\left( ( - 1)^{\frac{1}{4}} \right)^{k}$, given that by de­fault, the pos­i­tive and neg­a­tive-valued min­i­mal el­e­ments are pri­ma­ry gen­er­a­tors, $\mathcal{m}\mathcal{e}_{\left\lbrack \pm \frac{1}{8} \right\rbrack}.$The $Y\left( \mathbb{Z}_{8} \right)_{2}$ matrix has eigen­val­ues rep­re­sent­ing vertex po­si­tions in the com­plex unit circle, com­pris­ing the set

The pri­ma­ry gen­er­a­tor el­e­ment subset $\mathcal{pg}\left( Y\left( \mathbb{Z}_{8} \right) \right)$ rep­re­sents the cone num­bers $\in Y\left( \mathbb{Z}_{8} \right)$

and the re­main­ing el­e­ments of $A\left( Y\left( \mathbb{Z}_{8} \right) \right)$ are the axis num­bers, which define the in­scribed square geo­met­ric in­te­ger $Y\left( \mathbb{Z}_{4} \right)$,

The $8 \times 8$ matrix rep­re­sen­ta­tion of the $( - 1)^{\frac{1}{4}}$ pri­ma­ry gen­er­a­tor has a total of $8^{2} = 64$ en­tries, which can take on values of either $1$ or $0$, sig­ni­fy­ing bits of in­for­ma­tion. As a pri­ma­ry gen­er­a­tor, the spa­tial arrange­ment of the non-zero en­tries also en­codes the entire $Y\left( \mathbb{Z}_{8} \right)$ group by way of the mul­ti­pli­ca­tion op­er­a­tion $\left( ( - 1)^{\frac{1}{4}} \right)^{k}$. As a group, $Y\left( \mathbb{Z}_{8} \right)$ is en­cod­ed in $64$ bits of in­for­ma­tion, which can be seen when we con­sid­er the eight cyclic shifts of the Lie al­ge­bra gen­er­a­tors in binary

and given that there are eight non-zero el­e­ments in each cyclic shift matrix, there are $8^{2} = 64$ bits of in­for­ma­tion.

Pre­vi­ous­ly in the chap­ter, we con­sid­ered di­men­sion­al trans­forms of num­bers, re­call­ing the fol­low­ing ex­am­ple for the di­men­sion­al map­ping from a com­plex number to a real matrix

We apply the same di­men­sion­al trans­form map­ping in the op­po­site di­rec­tion for our min­i­mal el­e­ment $Y\left( \mathbb{Z}_{8} \right)_{2} = ( - 1)^{\frac{1}{4}}$, by using the map­ping no­ta­tion $Y\left( \mathbb{Z}_{\lbrack + 8\rbrack} \right)_{2}: \rightarrow Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2},$

We have gone from an $8 \times 8$ sized matrix to a $4 \times 4$ matrix and in­tro­duced a neg­a­tive-valued in­te­ger in the $a_{4,1}$ po­si­tion of the mapped result. As we know, the $Y\left( \mathbb{Z}_{4} \right)_{2}$ matrix rep­re­sents the in­scribed square, and a look at the eigen­val­ues of the $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2}$ matrix, re­veals the cone num­bers

The eigen­val­ues sig­ni­fy that the in­scribed square has been ro­tat­ed by $\frac{i\pi}{4}$ ra­di­ans when com­pared to $Y\left( \mathbb{Z}_{4} \right)_{2}$, where $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2}$ is seen in the fol­low­ing figure as the red square, and $Y\left( \mathbb{Z}_{4} \right)_{2}$ as the blue square.

The $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2}$ gen­er­a­tor forms the oc­ta­gon by way of its suc­ces­sive powers, $\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2} \right)^{k},$which gen­er­ates the fol­low­ing set of eight unique el­e­ments

In the set above the eigen­val­ues of the $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{1}$, $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{3},Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{5}$el­e­ments are

which, to­geth­er with the eigen­val­ues of the pri­ma­ry gen­er­a­tor

en­com­pass the $\lambda\left( \mathbb{Z}_{8} \right)$ eigen­val­ue set.

We show the in­scribed geom­e­try of the reg­u­lar oc­ta­gon, the result of the $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2}$ pri­ma­ry gen­er­a­tor's ro­ta­tions and rep­re­sent the el­e­ments in the $A\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2} \right)$ set. It may be coun­ter­in­tu­itive at first, but if we first con­sid­er the $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{3}$ and $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{4}$ el­e­ments

to rep­re­sent the vertex lo­ca­tions at $i$ and $( - 1)^{\frac{3}{4}}$ on the com­plex unit circle, the con­sis­ten­cy as vertex lo­ca­tions is ap­pre­ci­at­ed by squar­ing each

So, $\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{3} \right)^{2}$ is found to equal the neg­a­tive iden­ti­ty, which is the ex­pect­ed result given that $i^{2} = - 1,$and then we find $\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{4} \right)^{2} = - Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{3}$, rep­re­sent­ing $- i.$The $A\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right) \right)$ set cycles twice in terms of shifts before re­turn­ing to the iden­ti­ty el­e­ment. The eight Lie al­ge­bra gen­er­a­tors of the set $A\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right) \right)$ can also be rep­re­sent­ed as the fol­low­ing binary set

where the di­rec­tion­al arrow for $\overset{\rightharpoonup}{A}$ sig­ni­fies the above is a shift­ing set. Given that non-zero en­tries take on two values, $( - 1,1)$, there are eight bits of in­for­ma­tion in each Lie al­ge­bra gen­er­a­tor $\in A\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right) \right)$, which, when mul­ti­plied by the number of el­e­ments in $\overset{\rightharpoonup}{A}\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{j} \right)$, we find $8^{2} = 64$ bits of in­for­ma­tion are present in the group, re­veal­ing that a con­ser­va­tion of in­for­ma­tion exists in the map­ping $Y\left( \mathbb{Z}_{\lbrack + 8\rbrack} \right)_{2}: \rightarrow Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2}.$As ex­pect­ed, in the $A\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right) \right)$ set there are also four pri­ma­ry gen­er­a­tors

which differ from the non-gen­er­a­tors based on their lack of sym­me­try for non-zero en­tries about the di­ag­o­nal and the odd number dis­tri­b­u­tion of pos­i­tive to neg­a­tive values. These pri­ma­ry gen­er­a­tors rep­re­sent cone num­bers given that when squared, all the re­sult­ing el­e­ments are imag­i­nary-valued Lie al­ge­bras

ev­i­denced by the fact that when squared, the el­e­ments equal the neg­a­tive iden­ti­ty

The $Y\left( \mathbb{Z}_{4} \right)$ geo­met­ric in­te­ger set is con­tained in the $A\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right) \right)$ set, $A\left( Y\left( \mathbb{Z}_{4} \right) \right) \in A\left( Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right) \right)$, where $A\left( Y\left( \mathbb{Z}_{4} \right) \right)$ is the subset

Geo­met­ri­cal­ly $A\left( Y\left( \mathbb{Z}_{4} \right) \right)$ is a subset since the ver­tices of the in­scribed square, $Y\left( \mathbb{Z}_{4} \right)$, are aligned on the four axes points of the com­plex unit circle and are lim­it­ed to vertex ro­ta­tions of $\frac{i\pi}{2}$ ra­di­ans, there­fore, $Y\left( \mathbb{Z}_{4} \right)$ can be ro­tat­ed in four dif­fer­ent ways to align its ver­tices with the in­scribed oc­ta­gon. Next, we reduce the size of the $Y\left( \mathbb{Z}_{\lbrack \pm 4\rbrack} \right)_{2}$ di­men­sion while con­serv­ing its in­for­ma­tion con­tent with the map­ping

where it is easy to verify that $\begin{pmatrix} 0 & 1 \\ i & 0 \end{pmatrix}$ matrix rep­re­sents a cone number by its square

and more specif­i­cal­ly, that

since the pos­i­tive ro­ta­tion $\left( ( - 1)^{\frac{1}{4}} \right)^{2} = i.$The $A\left( Y\left( \mathbb{Z}_{\lbrack \pm i2\rbrack} \right)_{j} \right)$ set of Lie al­ge­bra gen­er­a­tors is the fol­low­ing

which, like our other sets, con­tains a total of eight el­e­ments rep­re­sent­ing the oc­ta­gon. The pri­ma­ry gen­er­a­tors have the fol­low­ing cone number values for their eigen­val­ues

while the re­main­ing el­e­ments have eigen­val­ues that are equal to the axis values of the in­scribed square

where these axis num­bers have only one unique eigen­val­ue. This lack of unique­ness of eigen­val­ues is a math­e­mat­i­cal in­di­ca­tor that these are not pri­ma­ry gen­er­a­tors of the $A\left( Y\left( \mathbb{Z}_{\lbrack \pm 2i\rbrack} \right)_{j} \right)$ set. As the reader may have guessed, we can also fur­ther reduce the di­men­sions of the matrix rep­re­sen­ta­tion for the $A\left( Y\left( \mathbb{Z}_{\lbrack \pm i2\rbrack} \right)_{j} \right)$ set to the $1 \times 1$ form of the set $A\left( Y\left( \mathbb{Z}_{\left\lbrack \pm ( - 1)^{\left( \frac{1}{4} \right)}\epsilon \right\rbrack} \right)_{j} \right)$. This rep­re­sents the com­plete trans­for­ma­tion­al map­ping of the $\mathbb{Z}_{8}$ geo­met­ric in­te­ger to the com­plex plane and the set is com­prised of the fol­low­ing Lie al­ge­bra gen­er­a­tors

for which, each el­e­ment in the set is a point on the com­plex unit circle, rep­re­sent­ing the eigen­val­ues of the $\mathbb{Z}_{8}$ geo­met­ric in­te­ger. The $8 \times 8$ matrix rep­re­sen­ta­tion of the min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{1}{8} \right\rbrack}$ we began with, is a com­plex number, and the di­men­sion-re­duc­ing trans­form

makes this math­e­mat­i­cal­ly clear.

Pre­vi­ous­ly men­tioned was that the min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{1}{8} \right\rbrack} = \left( e^{2i\pi} \right)^{\frac{1}{8}}$ is phys­i­cal­ly as­so­ci­at­ed with light. Ex­pand­ing on this con­nec­tion is ap­pro­pri­ate with our back­ground from the pre­vi­ous sec­tion on the geo­met­ric in­te­gers and primes. In physics, light, specif­i­cal­ly elec­tro-mag­net­ism, is rep­re­sent­ed in the form of the $U(1)$ group, which is con­sis­tent with the result we ar­rived at using our di­men­sion­al trans­form $Y\left( \mathbb{Z}_{8} \right)_{1}: \rightarrow Y\left( \mathbb{Z}_{\left\lbrack \pm ( - 1)^{\left( \frac{1}{4} \right)}\epsilon \right\rbrack} \right)_{1}.$We have seen that there is a fun­da­men­tal geom­e­try that is cen­tral to the char­ac­ter­is­tics of groups, and the case for $A\left( Y\left( \mathbb{Z}_{\left\lbrack \pm ( - 1)^{\left( \frac{1}{4} \right)}\epsilon \right\rbrack} \right)_{j} \right)$ is no ex­cep­tion. Light is based on the first prime number $p_{1} = 2$, and the in­scribed geom­e­try of the $gp_{1}$ geo­met­ric prime. Early in the book, we of­fered a sim­i­lar rep­re­sen­ta­tion to Figure 1.8.13 below, where the geo­met­ric forms of the com­plex circle, the hy­per­bo­la, and the hy­per­bo­la's as­ymp­totes are pre­sent­ed. In the figure, we have la­beled points on the unit circle using the no­ta­tion $c_{j}$, rep­re­sent­ing the or­dered $A\left( c_{j} \right)$ set

which can be seen to in­clude points on the as­ymp­totes and the axes. Also present in the figure is cur­va­ture, in the form of pos­i­tive cur­va­ture for the unit circle, neg­a­tive cur­va­ture for the hy­per­bo­la, and light having flat cur­va­ture as the per­fect bal­ance be­tween the other two.

In 1905, Albert Ein­stein pub­lished a paper named "On the Elec­tro­dy­nam­ics of Moving Bodies"¹, in which he pro­posed his theory of spe­cial rel­a­tiv­i­ty. The theory is based on the pos­tu­lates that the laws of physics are in­vari­ant, and the speed of light is in­vari­ant for all ob­servers. The im­pe­tus behind the theory stemmed from the com­pat­i­bil­i­ty issues that New­ton­ian me­chan­ics had with Maxwell's equa­tions, and the theory cor­rects me­chan­ics for bodies moving at close to the speed of light, often re­ferred to as rel­a­tivis­tic ve­loc­i­ties.

Spe­cial rel­a­tiv­i­ty is a con­se­quence of hy­per­bol­ic geom­e­try. We will touch upon the sub­ject briefly and will ul­ti­mate­ly go more in-depth into the sub­ject matter in the en­su­ing books on physics in the Arith­moí series. The most common form of the Lorentz trans­for­ma­tion rep­re­sent­ing ve­loc­i­ty unique to the $x$-di­rec­tion is

where $F = (t,x,y,z)$ and $F^{'} = \left( t^{'},x^{'},y^{'},z' \right)$, rep­re­sent the two co­or­di­nate frames of an ob­served event, for the sta­tion­ary frame $F$, the moving frame $F'$, the speed of light $c$, and the Lorentz factor $\gamma$

Space and time are one and the same, and by choos­ing to mea­sure them with the same units, $ct$, we re­place the $t$ time vari­able in the Lorentz trans­for­ma­tion to mea­sure the co­or­di­nate frames of our ob­served result as $(ct,x)$ and $\left( {ct}^{'},x' \right)$, where the trans­for­ma­tions become ²

The sym­me­try be­tween the space­time and ve­loc­i­ty trans­forms be­comes more ev­i­dent in the ex­pres­sions above. There are also in­verse Lorentz trans­for­ma­tions, given that no priv­i­leged ref­er­ence frame exists in the spe­cial rel­a­tiv­i­ty theory. These in­verse trans­for­ma­tions are found simply by solv­ing the trans­for­ma­tions above for $ct$ and $x$

and these rep­re­sent the frame $F$ moving with a neg­a­tive ve­loc­i­ty $- v$ that is equal in mag­ni­tude rel­a­tive to the $F'$frame. We sub­sti­tute the term, $\tanh(\beta)$, for$\ \frac{v}{c}$, where $\beta$ is the ra­pid­i­ty term, and for the unit hy­per­bol­ic equa­tion

we solve for $\cosh(\beta)$ as fol­lows

for which $\cosh(\beta)$ is the hy­per­bol­ic Lorentz factor, $\gamma$, and we set the equiv­a­lence

The fol­low­ing equiv­a­lence is then the case for the $\sinh(\beta)$ func­tion

and now, we can sub­sti­tute our hy­per­bol­ic equiv­a­lents back into the trans­for­ma­tions

then put the hy­per­bol­ic trans­for­ma­tions into matrix form, giving us

We take the ex­am­ple for a ra­pid­i­ty value of $\beta = \frac{\pi}{6},$for $\tanh(\beta)$, then being ap­prox­i­mate­ly equal to

and given that $\tanh(\beta) = \frac{v}{c}$ rep­re­sents a frame $F'$that is moving at $\approx 0.4805$ the speed of light rel­a­tive to the ob­serv­er's frame. We insert the value $\beta = \frac{\pi}{6}$ into the matrix form, to find

which sig­ni­fies the values of the space­time co­or­di­nates, dis­tance, and elapsed time that the moving ob­serv­er would record com­pared to the sta­tion­ary ob­serv­er. The in­verse trans­for­ma­tion can be found by left mul­ti­ply­ing by the in­verse of the trans­for­ma­tion matrix

Fig­ures 1.8.14,15 are plots of hy­per­bol­ic and Eu­clid­ean ro­ta­tions, which are both or­thog­o­nal ro­ta­tions in their re­spec­tive geome­tries. In the hy­per­bol­ic case, two lines are or­thog­o­nal when they are re­flec­tions across the as­ymp­tote of the spe­cif­ic hy­per­bo­la. The as­ymp­tote found in Figure 1.8.14 is la­beled $c_{1}$, and rep­re­sents the cone number $( - 1)^{\frac{1}{4}}$ as the speed of light. Hy­per­bol­ic or­thog­o­nal­i­ty is cen­tral to spe­cial rel­a­tiv­i­ty for defin­ing si­mul­ta­ne­ous events, where two events will be si­mul­ta­ne­ous when or­thog­o­nal to a spe­cif­ic time­line.

The hy­per­bol­ic ro­ta­tion in Figure 1.8.14 rep­re­sents the in­verse trans­for­ma­tion for values of $ct'$and $x'$, which are of equal mag­ni­tude. By choos­ing $ct^{'},x^{'} = 1$, this im­plies that elapsed time and dis­tance are equal to $1$, and cal­cu­lat­ing its matrix-vector, we find

which sig­ni­fies the space­time co­or­di­nates for an event seen by an ob­serv­er in the frame $F^{'} = \left( 1,1,y^{'} = y,z^{'} = z \right)$ was seen for the same by the ob­serv­er in frame $F$ for co­or­di­nates $F \approx \left( 1.6881,1.6881,{y = y}^{'},{z = z}^{'} \right).$

More ev­i­dence that hy­per­bol­ic geom­e­try truly un­der­lies spe­cial rel­a­tiv­i­ty is found by look­ing at the Lorentz factor

for which, values of $v$ are ex­pect­ed to be en­tered for $v < c.$If the value for the ve­loc­i­ty is en­tered as equal to the speed of light, $v = c$, the Lorentz factor is in­fi­nite

and for su­per­lu­mi­nal values where $v > c$, the Lorentz factor be­comes imag­i­nary valued

The hy­per­bol­ic ro­ta­tion in Figure 1.8.15 rep­re­sents the in­verse trans­for­ma­tion for values of $ct'$and $x'$, which are of equal mag­ni­tude. By choos­ing $ct^{'},x^{'} = 1$, this im­plies that elapsed time and dis­tance are equal to $1$, and cal­cu­lat­ing its matrix-vector, we find

which sig­ni­fies the space­time co­or­di­nates for an event seen by an ob­serv­er in the frame $F^{'} = \left( 1,1,y^{'} = y,z^{'} = z \right)$ was seen for the same by the ob­serv­er in frame $F$ for co­or­di­nates $F \approx \left( 1.6881,1.6881,{y = y}^{'},{z = z}^{'} \right).$Ev­i­dence that the hy­per­bol­ic geom­e­try is truly what un­der­lies spe­cial rel­a­tiv­i­ty is found by look­ing at the Lorentz factor

into which values of $v$ are ex­pect­ed to be en­tered for $v < c.$If the value for the ve­loc­i­ty is en­tered as equal to the speed of light, $v = c$, the Lorentz factor is in­fi­nite

and for su­per­lu­mi­nal values where $v > c$, the Lorentz factor be­comes imag­i­nary valued

The $\tanh(\beta)$ func­tion rep­re­sents the ratio of the hy­per­bol­ic func­tions $\frac{\sinh(\beta)}{\cosh(\beta)}$, and in­her­ent in its math­e­mat­i­cal form, it only ap­proach­es the value of one as $\beta \rightarrow + \infty$

and neg­a­tive one as $\beta \rightarrow - \infty$

where these limits can be seen clear­ly in the graph of the $\tanh(\beta)$ func­tion in the fol­low­ing figure.

So, the hy­per­bol­ic form of the Lorentz factor

is not pred­i­cat­ed upon en­ter­ing a ve­loc­i­ty less than the speed of light, and in­stead, we can see that $\gamma$ is the result of hy­per­bol­ic geom­e­try, and more pre­cise­ly, it is the $\cosh(\beta)$ func­tion. Geo­met­ri­cal­ly, as the ra­pid­i­ty $\beta$ be­comes larger, the degree of neg­a­tive cur­va­ture de­creas­es, and as $\beta \rightarrow \pm \infty$, the cur­va­ture be­comes flat, like that of the light as­ymp­tote.

Re­turn­ing to the $A\left( c_{j} \right)$ set of light el­e­ments

we con­sid­er the pos­i­tive­ly ori­ent­ed min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{1}{4} \right\rbrack} = ( - 1)^{\frac{1}{4}},$where its suc­ces­sive powers can be seen to move along the axes and light as­ymp­totes, seen by its abil­i­ty to gen­er­ate the $A\left( c_{j} \right)$ set

This axis-as­ymp­tote prop­a­ga­tion char­ac­ter­is­tic is how light moves math­e­mat­i­cal­ly and phys­i­cal­ly, which dif­fers sig­nif­i­cant­ly from the phe­nom­e­nol­o­gy of mas­sive ob­jects bound by the rel­a­tiv­i­ty in­her­ent to hy­per­bol­ic geom­e­try. Light is out­side of the geo­met­ric con­fines of space­time, but, at the same time, it moves freely through the space­time geom­e­try.

Light is cen­tral to gain­ing a rea­son­able un­der­stand­ing of the uni­verse, and any theory that does not con­sid­er its im­por­tance will be woe­ful­ly in­com­plete. In Chap­ter 4, we con­sid­er conic sec­tions, no­tably the light cone, which was sig­nif­i­cant be­cause the light cone gen­er­at­ed the $SU(2)$ sub­group. This sub­group stands out, given its dis­tance metric is Eu­clid­ean, and in real space at non-rel­a­tivis­tic ve­loc­i­ties the world around us ap­pears dis­tinct­ly Eu­clid­ean. Light does not just rep­re­sent a ve­loc­i­ty limit for mas­sive ob­jects in space or pho­tons/elec­tro­mag­net­ic waves; it is math­e­mat­i­cal­ly re­spon­si­ble for cre­at­ing the phys­i­cal spaces in which we exist. This not only in­cludes the real space we move around in, but also the imag­i­nary space in which we move through time.

To un­der­stand the math­e­mat­ics behind light, we return to the geo­met­ric in­te­gers to con­sid­er the do­decagon formed by cyclic ro­ta­tions of the equi­lat­er­al tri­an­gle. The $gp_{2}$ geo­met­ric prime set $A\left( Y\left( gp_{2} \right)_{j} \right)$ had three unique el­e­ments, which co­or­di­nat­ed with the three cyclic shifts of its ver­tices in­scribed in the com­plex unit sphere

The in­scribed do­dec­a­he­dron is rep­re­sent­ed by ro­tat­ing the equi­lat­er­al tri­an­gle by $\frac{i\pi}{2}$ ra­di­ans, seen as the trans­for­ma­tion in the image below.

The first matrix rep­re­sen­ta­tion of the ro­tat­ed min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}$ from the $A\left( Y\left( gp_{2} \right)_{j} \right)$ set, we denote as fol­lows

which has one imag­i­nary entry, and the other two are cone num­bers sit­u­at­ed $\pm \frac{i\pi}{4}$ ra­di­ans away. By find­ing the eigen­val­ues of the matrix, we find con­sis­ten­cy with the vertex po­si­tion­ing de­pict­ed in the figure above

which are all so­lu­tions to its char­ac­ter­is­tic poly­no­mi­al

To see the in­tu­ition behind the result for the $\mathcal{cp}\left( \mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack} \right)$ eigen­val­ues, each of these are so­lu­tions to the equa­tion $\left| \mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack} - I\lambda \right| = 0$. The eigen­val­ues are cal­cu­lat­ed as fol­lows

for which, taking the de­ter­mi­nant of the matrix yields

and the eigen­val­ues are the said so­lu­tions to this char­ac­ter­is­tic poly­no­mi­al in the com­plex plane. As we have seen pre­vi­ous­ly, the matrix $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}$ is also a so­lu­tion to the char­ac­ter­is­tic poly­no­mi­al, given that

We gen­er­ate the sub­group by taking suc­ces­sive powers of the min­i­mal el­e­ment, $\left( \mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack} \right)^{k},$to find the fol­low­ing set of twelve el­e­ments, where $( \nwarrow )$ presents a pos­i­tive­ly ori­ent­ed ro­ta­tion,

Geo­met­ri­cal­ly, these are rep­re­sent­ed in the figure below, where each vertex may orient in one of twelve pos­si­ble po­si­tions on the com­plex unit circle.

equi­lat­er­al tri­an­gles in­scribed in the do­decagon.

The group char­ac­ter­is­tic poly­no­mi­al has re­mark­able sym­me­try, seen by the fol­low­ing

which rep­re­sents the com­po­si­tion of the fol­low­ing sub­groups of the $gp_{2}$ geo­met­ric prime

The el­e­ments in the dis­tinct sub­sets have de­ter­mi­nants equal to $k$ in the ex­pres­sion $A\left( kgp_{2} \right)$. By its sym­me­try, the $\mathcal{CP}\left( Y\left( {gp}_{\lbrack 2i\rbrack} \right)( \nwarrow ) \right)$ group char­ac­ter­is­tic poly­no­mi­al re­duces to the sim­pli­fied form

show­ing its do­dec­a­he­dron char­ac­ter­is­tic form. Only the triv­ial so­lu­tion exists for the char­ac­ter­is­tic poly­no­mi­al equa­tion $12\lambda^{3} = 0,$sig­ni­fy­ing that the func­tion cross­es the real-axis only at the origin, as seen in the figure below.

The net-zero char­ac­ter­is­tic is present in the graph above, mean­ing that for any point on the pos­i­tive por­tion of the curve, there is an­oth­er point on the neg­a­tive por­tion that, when added to it, will equal zero. In other words, the char­ac­ter­is­tic poly­no­mi­al is sym­met­ric about the hor­i­zon­tal axis. Net-zero is es­sen­tial for any theory of every­thing to be con­sis­tent math­e­mat­i­cal­ly and philo­soph­i­cal­ly. The­o­ries pred­i­cat­ed upon a uni­verse spring­ing from the zero-point with­out a net-zero bal­ance of its parts equates to the free-lunch model, for which any claim of causal­i­ty is void. The cyclic group nature is in­her­ent in the el­e­ments of the $A\left( Y\left( {gp}_{\lbrack i2\rbrack} \right)( \nwarrow ) \right)$ set, ap­pre­ci­at­ed by first defin­ing its sub­sets of el­e­ments

These can be sit­u­at­ed into an array as

where the el­e­ments of each sub­group are arranged in three-el­e­ment cyclic di­ag­o­nals and the $A_{k}^{(1)}$ define the min­i­mal el­e­ments, or, in other words, the pri­ma­ry gen­er­a­tors. For clar­i­ty, we note as an ex­am­ple that $A_{k}^{(2)} = {{(A}_{k}^{(1)})}^{2}.$Now, we take the $A\left( igp_{2} \right)$ sub­group el­e­ments found in the $3 \times 3$ upper-left sub-matrix and define the min­i­mal el­e­ment sub­group, $\mathcal{A}\left( igp_{2} \right),$as

Suc­ces­sive powers of the min­i­mal el­e­ment sub­group $\mathcal{A}\left( igp_{2} \right)$ will form a co­he­sive twelve el­e­ment su­per­group, $\mathcal{A}\left( Y\left( {gp}_{\lbrack i2\rbrack} \right)( \nwarrow ) \right),$com­posed of the fol­low­ing el­e­ments

This has a very sim­i­lar struc­ture to the $A\left( Y\left( {gp}_{\lbrack i2\rbrack} \right)( \nwarrow ) \right)$ set it was de­rived from, ev­i­denced by the re­peat­ing three-el­e­ment cyclic di­ag­o­nals, the pres­ence in the bottom half of the neg­a­tive-valued el­e­ments co­in­cid­ing with re­spec­tive el­e­ments in the top half, and the pres­ence of axes iden­ti­ties in the right-most column.

The su­per­group process can be re­peat­ed in­def­i­nite­ly, con­sid­er­ing the $\mathcal{A}\left( Y\left( {gp}_{\lbrack i2\rbrack} \right)( \nwarrow ) \right)$ set can be arranged into the array

where the min­i­mal el­e­ment sub­group $\mathcal{A}_{\lbrack 2\rbrack}\left( igp_{2} \right)$ can also be de­fined as

whose suc­ces­sive powers $\mathcal{A}_{\lbrack 2\rbrack}\left( igp_{2} \right)^{k}$ also form the twelve el­e­ment $\mathcal{A}_{\lbrack 2\rbrack}\left( Y\left( {gp}_{\lbrack i2\rbrack} \right)( \nwarrow ) \right)$ set

The $\left( \mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack} \right)^{k}$ min­i­mal el­e­ment is ori­entable in the other di­men­sions of $\mathbb{C}^{3}$, and our ini­tial min­i­mal el­e­ment can be dis­tin­guished as

where $( \nwarrow )$ presents a pos­i­tive­ly ori­ent­ed ro­ta­tion. The min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nearrow )$ is the trans­pose of $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nwarrow )$

has the same eigen­val­ues as $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nwarrow )$

and the set of el­e­ments $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nearrow )$ gen­er­at­ing $A\left( Y\left( {gp}_{\lbrack 2i\rbrack} \right)( \nearrow ) \right)$, is the fol­low­ing

The re­main­ing com­plex di­men­sion has the fol­low­ing min­i­mal el­e­ment

which has the fol­low­ing re­la­tion­ships with the other com­plex di­men­sion­al min­i­mal el­e­ments

and the ro­ta­tion ma­tri­ces, $R( \nwarrow ),R( \nearrow )$ rep­re­sent ori­ent­ed tri­an­gles in $\mathbb{C}^{3}$, given the eigen­val­ues for both are $\lambda = \left\{ e^{\left( \frac{2i\pi}{3} \right)},e^{\left( \frac{4i\pi}{3} \right)},1 \right\}$.

Al­though $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \rightarrow )$ may appear dif­fer­ent from the other min­i­mal el­e­ments, $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nwarrow )$ and $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nearrow )$, its eigen­val­ues are the same

which is the case given that a di­ag­o­nal­ized matrix has its el­e­ments as eigen­val­ues. Graph­i­cal­ly we can rep­re­sent the two ma­tri­ces, $R( \nwarrow ),R( \nearrow )$, which rotate $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nwarrow )$ and $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nearrow )$ to $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \rightarrow )$, as points on the com­plex unit circle (Figure 1.8.20).

The set of el­e­ments $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \rightarrow )$ gen­er­at­ing $A\left( Y\left( {gp}_{\lbrack 2i\rbrack} \right)( \rightarrow ) \right)$, is the fol­low­ing

and the set of all sets $A\left( Y\left( {gp}_{\lbrack 2i\rbrack}( \nwarrow , \nearrow , \rightarrow ) \right) \right)$, all have the same group char­ac­ter­is­tic poly­no­mi­al

which, as we are aware, re­duces to

The di­men­sion­al trans­form ma­tri­ces are de­fined as

which, when map­ping sim­i­lar point­ing arrows, rotate the min­i­mal el­e­ments. The prod­uct of $A\left( Y\left( {gp}_{\lbrack 2i\rbrack}( \nwarrow , \nearrow , \rightarrow ) \right) \right)$ with matched ma­tri­ces $Y( \nwarrow , \nearrow , \rightarrow )$

will yield a set that in­cludes all the cone gen­er­a­tor Lie al­ge­bras, seen by pre­sent­ing the com­po­si­tion of el­e­ments for each subset

In the $A\left( Y\left( {gp}_{\lbrack 2i\rbrack}( \rightarrow ) \right) \right) \times Y( \rightarrow )$ subset, the square of the last el­e­ment in each row is as fol­lows:

where all are con­sis­tent with $3 \times 3$ matrix rep­re­sen­ta­tions of the com­plex num­bers $\left\{ - i, - 1,i,1 \right\}.$With the subset re­sults after the di­men­sion­al trans­for­ma­tion, a pri­ma­ry gen­er­a­tor, that is alone ca­pa­ble of gen­er­at­ing all the other el­e­ments, does not exist. Using the ex­am­ple of the trans­formed min­i­mal el­e­ment

the gen­er­at­ed set from the result $A\left( \mathcal{m}\mathcal{e}_{\left\lbrack + \frac{i}{gp_{2}} \right\rbrack}( \nwarrow ) \times Y( \nwarrow ) \right)$, has the subset rep­re­sen­ta­tion

which con­tains the two cone por­tions $\in \mathbb{C}^{3}$ ori­ent­ed with pos­i­tive and neg­a­tive imag­i­nary axes. Also present in the subset is the neg­a­tive min­i­mal el­e­ment

which is bound to the same subset, given that its gen­er­at­ed set is the same

Next, we take the trans­for­ma­tion of the $\mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{gp_{2}} \right\rbrack}( \nwarrow )$ min­i­mal el­e­ment

which is equal to the cone por­tion $\in \mathbb{C}^{3}$ ori­ent­ed with the neg­a­tive real-axis, and its gen­er­at­ed set, $A\left( \mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{gp_{2}} \right\rbrack}( \nwarrow ) \times Y( \nwarrow ) \right),$con­tains only two el­e­ments, itself and the iden­ti­ty

We find the same holds for trans­form­ing the $\mathcal{m}\mathcal{e}_{\left\lbrack \frac{1}{gp_{2}} \right\rbrack}( \nwarrow )$ min­i­mal el­e­ment

which also has itself and the iden­ti­ty in its gen­er­at­ed subset

So, a beau­ti­ful thing hap­pens in the di­men­sion­al trans­for­ma­tion process, where sub­spaces are formed with el­e­ments bound in the sub­space.

The light num­bers can take on many forms as matrix rep­re­sen­ta­tions, and we use the ex­am­ple for the min­i­mal el­e­ment $\mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{gp_{2}} \right\rbrack}$, having the matrix rep­re­sen­ta­tion

which has eigen­val­ues $\lambda\left( \mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{gp_{2}} \right\rbrack} \right)$ equal to $\left\{ - 1,e^{\left( \frac{2i\pi}{3} \right)},e^{\left( \frac{4i\pi}{3} \right)} \right\},$and its char­ac­ter­is­tic poly­no­mi­al is $\mathcal{cp}\left( \mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{gp_{2}} \right\rbrack} \right) = \lambda^{3} + 1$, all of which are con­sis­tent with its nomen­cla­ture $\mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{gp_{2}} \right\rbrack}$. Its gen­er­at­ed set is com­posed of the fol­low­ing six el­e­ments

and we take the di­men­sion­al trans­for­ma­tion of the set $A\left( \mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{gp_{2}} \right\rbrack} \times Y( \nwarrow ) \right),$to find

The eigen­val­ues of four of the el­e­ments in the set above in­di­cate that these are ro­ta­tion ma­tri­ces,

where each of these ma­tri­ces gen­er­ates

Except for the iden­ti­ty ma­tri­ces, all of these gen­er­a­tors act to rotate/re­flect in the $X_{2}/X_{3}$ and $X_{1}/X_{2}$-planes by $\pm i$ and $- 1$. We select the two non-di­ag­o­nal ma­tri­ces from these sets with their axis ori­ent­ed along the real $X_{1}$-plane,

which are in­verse­ly re­lat­ed to one an­oth­er, given that

and the eigen­val­ues of the ma­tri­ces are

The light cone we first en­coun­tered in the chap­ter on conic sec­tions ori­ent­ed along the real $X_{1}$-axis, is the result of the prod­uct of these two ma­tri­ces, with the real-axis ori­ent­ed gen­er­a­tor from the $A\left( Y\left( {gp}_{\lbrack 2\rbrack}( \nwarrow ) \right) \right) \times Y( \nwarrow )$ set

We rotate this cone sec­tion gen­er­a­tor with the fol­low­ing matrix-matrix prod­uct

which re­sults in the other light cone Lie al­ge­bra gen­er­a­tor de­fined in the chap­ter on conic sec­tions. We can now define the pri­ma­ry el­e­ments which form the light cone ori­ent­ed in the $X_{1}$-axis $\in \mathbb{R}^{3 +}$, as

which to­geth­er form the fol­low­ing prod­ucts

The Lie al­ge­bra re­sults above are the $3 \times 3$ ver­sions of the $SU(2)$ gen­er­a­tors seen pre­vi­ous­ly in the book. Keep­ing the axis entry equal to the iden­ti­ty, we can rep­re­sent this group of el­e­ments with vari­ables $a,b,c,d$, as

whose de­ter­mi­nant rep­re­sents the dis­tance metric in the space

find­ing the Eu­clid­ean dis­tance metric of the space we per­ceive phe­nom­e­nal­ly. With the re­sult­ing ma­tri­ces formed by the light cone gen­er­a­tor prod­ucts, we define the $\mathcal{L}\mathcal{C}_{\left\lbrack + X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$ set, as

The light cone along the neg­a­tive real $X_{1}$-axis is formed sim­i­lar­ly, where we rotate the $A\left( \mathcal{L}{\mathcal{C}_{\left\lbrack + X_{1} \right\rbrack}}_{j} \right)$ gen­er­a­tors by $e^{i\pi}$

We define the pri­ma­ry el­e­ments which form the light cone ori­ent­ed in the $X_{1}$-axis $\in \mathbb{R}^{3 -}$, as

which to­geth­er form the fol­low­ing prod­ucts

We take these Lie al­ge­bra re­sults to define the $\mathcal{L}\mathcal{C}_{\left\lbrack - X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$ set as

which we find is equiv­a­lent to the $\mathcal{L}\mathcal{C}_{\left\lbrack X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$ set. It is im­por­tant to note el­e­ments from the $\mathcal{L}\mathcal{C}_{\left\lbrack + X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$ and the $\mathcal{L}\mathcal{C}_{\left\lbrack - X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$ can gen­er­ate other el­e­ments which depend on the order of the matrix prod­uct. For ex­am­ple, we con­sid­er the prod­uct of the fol­low­ing two el­e­ments $\in \mathcal{L}\mathcal{C}_{\left\lbrack + X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$

whose result has neg­a­tive values in the bottom-right $2 \times 2$ sub­ma­trix, com­pared with the pos­i­tive el­e­ment found

$\in \mathcal{L}\mathcal{C}_{\left\lbrack - X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack,\ \mathcal{L}\mathcal{C}_{\left\lbrack X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$. These neg­a­tive-valued el­e­ments rep­re­sent ro­ta­tions in the op­pos­ing di­rec­tions, and by taking these into ac­count, the light cone set $\mathcal{L}\mathcal{C}_{\left\lbrack X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$ is de­fined as the fol­low­ing set of Lie al­ge­bra gen­er­a­tors

This is a beau­ti­ful math­e­mat­i­cal result, given that the spaces cre­at­ed by two light cones, which are an­tipo­dal to one an­oth­er, create seam­less Eu­clid­ean space. So, moving in one space will feel like moving in the other space, and the two spaces will be in­dis­cernible to any­thing bound in the $\mathcal{L}\mathcal{C}_{\left\lbrack X_{1} \right\rbrack}\left\lbrack SU(2) \right\rbrack$ subset. Light is re­spon­si­ble for this, par­tic­u­lar­ly the cone num­bers of light. The other light cones that are ori­ent­ed in the $X_{2},X_{3}$-axes will yield sim­i­lar sets, and we will ded­i­cate a whole physics book on the sub­ject in Arith­moí -Volume II, Part 1: "Phe­nom­e­nol­o­gy and the Geom­e­try of Light".

To vi­su­al­ize the space along the $X_{1}$-axis that the min­i­mal el­e­ments

create, we plot in Figure 1.8.21 the eigen­val­ues of $\mathcal{m}\mathcal{e}_{\left\lbrack \pm \frac{1}{gp_{2}} \right\rbrack}( \nwarrow )$, as the ver­tices of the in­scribed equi­lat­er­al tri­an­gle. Em­a­nat­ing from the com­plex unit sphere in the pos­i­tive and neg­a­tive di­rec­tions are the light cones of the $X_{1}$-axis.

For com­plete­ness, we con­sid­er ro­tat­ing to the $A\left( \mathcal{L}{\mathcal{C}_{\left\lbrack + X_{2} \right\rbrack}}_{j} \right)$ and $A\left( \mathcal{L}{\mathcal{C}_{\left\lbrack + X_{3} \right\rbrack}}_{j} \right)$ light cone sets, ori­ent­ed along the $X_{2}$ and $X_{3}$- axes, using the start­ing point of the $A\left( \mathcal{L}{\mathcal{C}_{\left\lbrack + X_{1} \right\rbrack}}_{j} \right)$ light cone set

Using the simple case of an $X_{1}$-type Lie al­ge­bra

the $Y_{2}$ and $Y_{3}$-types of Lie al­ge­bra will rotate $X_{1}$ as fol­lows

where we see that $X_{1} \times Y_{3}$ ro­tates to the $X_{2}$-type al­ge­bra, and $X_{1} \times Y_{2}$ ro­tates to the $X_{3}$-type al­ge­bra. It should now be clear that the $Y_{2},Y_{3}$ gen­er­a­tors we have worked with through­out the book are the $gp_{2}$ geo­met­ric prime in its sim­plest in­scribed ori­en­ta­tions, having eigen­val­ues equal to $\lambda\left( Y_{2},Y_{3} \right) = \left\{ e^{\left( \frac{2i\pi}{3} \right)},e^{\left( \frac{4i\pi}{3} \right)},1 \right\}.$

We have seen before that chang­ing the order of each prod­uct will yield ro­ta­tions in op­pos­ing di­rec­tions

Given these ro­ta­tions, we know the right mul­ti­pli­ca­tion of the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{1} \right\rbrack}}_{1}$ gen­er­a­tor by some $Y_{3}$-type arrange­ment of el­e­ments will yield $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{2} \right\rbrack}}_{1}$, and right mul­ti­pli­ca­tion by a $Y_{2}$-type arrange­ment will result in the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{3} \right\rbrack}}_{1}$ gen­er­a­tor.

The $Y_{3}$-type gen­er­a­tor is found by taking the min­i­mal el­e­ment from the $A\left( Y\left( {gp}_{\lbrack 2i\rbrack}( \nwarrow ) \right) \right)$ set, $\mathcal{m}\mathcal{e}_{\left\lbrack \frac{i}{gp_{2}} \right\rbrack}( \nwarrow ),$which has the matrix rep­re­sen­ta­tion

then switch the values for the $a_{2,1}$ and $a_{1,3}$ en­tries, which gives us

As a pri­ma­ry gen­er­a­tor, $\mathcal{m}\mathcal{e}_{\left\lbrack \frac{i}{gp_{2}} \right\rbrack}( \nwarrow )\left( \overset{\leftrightarrow}{a_{2,1}:a_{1,3}} \right)$ yields the fol­low­ing set

which in­ter­changes $a_{2,1}$,$a_{1,3}$ en­tries for the ma­tri­ces in the left column ($Y_{3}$-type), and their trans­pose en­tries $a_{1,2}$,$a_{3,1}$ in the ma­tri­ces of the middle column ($Y_{2}$-type).

We define the fol­low­ing geo­met­ric prime ro­ta­tion ma­tri­ces using the no­ta­tion $R_{\lbrack j,k\rbrack}$

for which the $j,k$ values in the sub­script in­di­cate an ori­ent­ed ro­ta­tion from the $j$-axis to the $k$-axis. Each pair­ing of the ro­ta­tion ma­tri­ces $R_{\lbrack j,k\rbrack},R_{\lbrack k,j\rbrack}$ are in­verse­ly re­lat­ed to one an­oth­er, and these ro­ta­tion ma­tri­ces are de­rived from the prod­ucts of the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{1}$ light cone seg­ments as fol­lows

The $R_{\lbrack j,k\rbrack}$ ma­tri­ces rep­re­sent $gp_{2}$-type geo­met­ric primes, which result from the prod­ucts of the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{1}$ light cone seg­ments. All these ma­tri­ces have de­ter­mi­nants equal to one, which is con­sis­tent with the value of light being set to unity. Given these ma­tri­ces, we can define the fol­low­ing light cone seg­ment ro­ta­tions and re­spec­tive in­verse ro­ta­tions as

The math­e­mat­i­cal rea­son­ing as to how the ro­ta­tions work, is based upon the fact that each $\mathcal{L}{\mathcal{C}_{\left\lbrack X_{j} \right\rbrack}}_{1}$ light cone seg­ment will square to the iden­ti­ty, and the ro­ta­tor is de­fined as $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{1}\times \mathcal{L}{\mathcal{C}_{\left\lbrack + X_{k} \right\rbrack}}_{1} = R_{\lbrack j,k\rbrack}$. So, given that $\left( \mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{1} \right)^{2} = \epsilon$, the de­fined ro­ta­tions will all gen­er­al­ize as fol­lows

The straight­for­ward $R_{\lbrack j,\rbrack}$ matrix prod­ucts

yield the pos­i­tive and neg­a­tive $gp_{2}$ min­i­mal el­e­ments. There are sev­er­al ways to rotate the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{2}$ light cone sec­tions, and we can choose to use the $gp_{2}$ min­i­mal el­e­ments, along with the $R_{\lbrack 2,3\rbrack},R_{\lbrack 3,2\rbrack}$ ro­ta­tors to define the fol­low­ing bi­jec­tive ro­ta­tion prod­ucts

Note, as de­fined in the conic sec­tions chap­ter, $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{2} \right\rbrack}}_{2}$, $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{3} \right\rbrack}}_{2}$ have the same matrix rep­re­sen­ta­tions

With light cone sec­tion ro­ta­tions de­fined, we see that there is a very suc­cinct in­ter­play be­tween the in­scribed geo­met­ric primes and the light cone sec­tions $\left( \mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{1},\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{2} \right),$where the prod­ucts of light cone sec­tions yield in­scribed geo­met­ric prime re­sults, and these geo­met­ric primes also act to rotate the light cone sec­tions them­selves.

Up to this point, we have been con­sid­er­ing light number so­lu­tions to the char­ac­ter­is­tic poly­no­mi­al equal to axis num­bers. We let the $Y_{\mathcal{L}k}$ no­ta­tion define the fol­low­ing $Y$-type ma­tri­ces

for which all eigen­val­ue and eigen­vec­tor so­lu­tions will be as fol­lows

The $Y_{\mathcal{L}k}$ char­ac­ter­is­tic poly­no­mi­als are found to be the fol­low­ing

which sum to the group char­ac­ter­is­tic poly­no­mi­al

As stated ear­li­er in the sec­tion, the num­bers re­lat­ed to light will travel via as­ymp­totes and axes. More specif­i­cal­ly, the di­men­sion­al­ly trans­formed light num­bers move in this manner. The in­scribed geom­e­try of the as­ymp­tot­ic light num­bers will have Lie gen­er­a­tors of the form

where the $c_{a,b,c}$ vari­ables are the as­ymp­tote subset el­e­ments from the $A\left( c_{j} \right)$ light set, which was de­fined ear­li­er in the sec­tion, as

and the subset $A\left( c_{\otimes} \right)$ is

where the symbol $\otimes$ de­notes as­ymp­tot­ic and $\oplus$ rep­re­sents axes for the axis subset $A\left( c_{\oplus} \right)$

which were pic­tured as the $c_{j}$ values in Figure 1.8.13.

We define the fol­low­ing min­i­mal el­e­ments $\mathcal{m}\mathcal{e}_{\left\lbrack ( - 1)^{\frac{1}{4}}\left( \frac{1}{gp_{1}} \right) \right\rbrack}( \nwarrow )$, $\mathcal{m}\mathcal{e}_{\left\lbrack ( - 1)^{\frac{1}{4}}\left( \frac{1}{gp_{1}} \right) \right\rbrack}( \nearrow )$

which are both pri­ma­ry gen­er­a­tors. We cal­cu­late the set the $\mathcal{m}\mathcal{e}_{\left\lbrack ( - 1)^{\frac{1}{4}}\left( \frac{1}{gp_{1}} \right) \right\rbrack}( \nwarrow )$ el­e­ment, find­ing twenty-four el­e­ments in the set

The twenty-four el­e­ments are a due to the fact that the com­pos­ite number $\mathbb{Z}_{24}$ is the result of the prod­uct of the prime num­bers $gp_{1} \times gp_{1} \times gp_{1} \times gp_{2}$, and there is a sym­met­ric dis­tri­b­u­tion of the eight unique el­e­ments into sub­sets of the form

We con­sid­er the group char­ac­ter­is­tic poly­no­mi­al

and given that for each $Y_{\mathcal{L}k}\left( c_{j} \right)$ subset, we have $a = b = c = c_{j}$, the subset group char­ac­ter­is­tic poly­no­mi­al $\mathcal{CP}\left( Y_{\mathcal{L}k} \right)$ is then equal to

Each value for $c_{j}$ is the result of ro­ta­tion on the com­plex unit circle, and we can ex­press each $c_{j}$ value as equal­ing

and the group char­ac­ter­is­tic poly­no­mi­al can be found by the sum­ma­tion

We sub­sti­tute $c_{j} = \left( ( - 1)^{\frac{1}{4}} \right)^{j}$, and cal­cu­late the sum to find the group char­ac­ter­is­tic poly­no­mi­al re­duces to

which is very sim­i­lar to what we found for the group char­ac­ter­is­tic poly­no­mi­al for our light cone num­bers

This sim­i­lar­i­ty is based upon the same sym­me­try in the el­e­ments of the group, and the two func­tions have the same plots except for a scaled dif­fer­ence of two. So, once again, we find the pres­ence of the net-zero char­ac­ter­is­tic in the as­ymp­tot­ic light num­bers.

The hy­per­bol­ic geom­e­try of spe­cial rel­a­tiv­i­ty is also due to light num­bers. Each hy­per­bo­la along the $X_{1},X_{2},X_{3}$-axes is the di­men­sion­al trans­form of the $gp_{2}$ set el­e­ments con­sid­ered ear­li­er

which also play an im­por­tant role in $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{2}$ light cone ro­ta­tions. Using our ro­ta­tion set no­ta­tion, we can define the $gp_{2}$ set as

given that this is a pos­i­tive­ly ori­ent­ed ro­ta­tion of the in­scribed equi­lat­er­al tri­an­gle. The di­men­sion­al trans­form will yield the $X$-type Lie al­ge­bra gen­er­a­tors

all of which are hy­per­bol­ic gen­er­a­tors, seen by their vari­able-re­lat­ed matrix ex­po­nen­tials

So, the hy­per­bol­ic geom­e­try that un­der­scores spe­cial rel­a­tiv­i­ty is also a man­i­fes­ta­tion of light due to the $gp_{2}$ geo­met­ric prime and its re­sult­ing dy­nam­ics. We also saw the role that these play in cone sec­tion ro­ta­tions, and one way to think about this re­la­tion­ship is to con­sid­er the $R_{\lbrack j,k\rbrack}$ ro­ta­tions ma­tri­ces as a split­ting of the $gp_{2}$ geo­met­ric prime, which im­plies they are the result of light split­ting. This notion can be ex­pand­ed upon by con­sid­er­ing that each $R_{\lbrack j,k\rbrack}$ matrix is the in­verse of its $R_{\lbrack k,j\rbrack}$ pair, and the prod­uct

we see yields the iden­ti­ty of the $A\left( gp_{2} \right)$ set. The square for the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{1}$ seg­ments is also equal to the iden­ti­ty

rep­re­sent­ing a sym­met­ric split­ting of the iden­ti­ty from the $A\left( gp_{2} \right)$ set, and this should remind the reader of the split com­plex number $j$, whose square is equal to the iden­ti­ty as well. A closer look at the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{j} \right\rbrack}}_{1}$ num­bers using the $\mathcal{L}{\mathcal{C}_{\left\lbrack + X_{1} \right\rbrack}}_{1}$ ex­am­ple

has its axis rep­re­sent­ed by the $a_{1,1}$ entry, along with the $2 \times 2$ sub­ma­trix in the bottom right corner, rep­re­sent­ing the sym­met­ric iden­ti­ty split, man­i­fest­ed by ro­tat­ing equal dis­tances in ra­di­ans away from $\epsilon$ in pos­i­tive and neg­a­tive di­rec­tions. This sub­ma­trix is also an­oth­er form of a split com­plex number, given that

which is con­sis­tent with our de­f­i­n­i­tion that $j = e^{i\pi}$, nor­mal­ly taking the $2 \times 2$ matrix rep­re­sen­ta­tion form

This rep­re­sen­ta­tion ob­scures that it is also a sym­met­ric split­ting of the iden­ti­ty in pos­i­tive and neg­a­tive di­rec­tions and right­ful­ly sits in the $\mathbb{C}^{2}$ space as

Fur­ther ev­i­dence that both split com­plex num­bers are close­ly re­lat­ed is that they share the same de­ter­mi­nants and eigen­val­ues

and sub­se­quent­ly have the same char­ac­ter­is­tic poly­no­mi­al $\lambda^{2} = 1.$In the $A\left( gp_{2} \right)$ set, the non-iden­ti­ty el­e­ments should right­ful­ly be seen as sym­met­ric cyclic split­tings of the iden­ti­ty, given that

and more pre­cise­ly, both el­e­ments are pos­i­tive and neg­a­tive di­rec­tion­al splits of the iden­ti­ty into thirds, which is the math­e­mat­i­cal un­der­pin­nings for the fact that

The $R_{\lbrack j,k\rbrack}$ ro­ta­tion ma­tri­ces are close­ly re­lat­ed to the cyclic split­ting of the iden­ti­ty above, given that each will also equate to the iden­ti­ty when cubed

and by as­so­cia­tiv­i­ty, all $R_{\lbrack j,k\rbrack} \times R_{\lbrack l,m\rbrack}$ prod­ucts will as well when $j \neq l\ \text{and}\ k \neq m$,

This ap­plies to the prod­ucts we en­coun­tered

where a three-way-split­ting of the iden­ti­ty is causal to both

The log­i­cal con­clu­sion is that all on­to­log­i­cal­ly co­he­sive al­ge­bras are a split of the iden­ti­ty, known as the number one. This brings us full circle and back to the de­f­i­n­i­tion of the min­i­mal el­e­ment

which is ex­act­ly this split­ting of the iden­ti­ty, given that $e^{2i\pi} = 1$, and the net-zero re­quire­ment for co­he­sive­ness. For each $\mathcal{m}\mathcal{e}_{\left\lbrack + \frac{1}{k} \right\rbrack}$, when $k \geq 1$, there exists its split analog, $\mathcal{m}\mathcal{e}_{\left\lbrack - \frac{1}{k} \right\rbrack}$, for which the result of the fol­low­ing prod­uct always holds

How and why does the iden­ti­ty exist? The answer lies in the fact that it is the volume of the zero-di­men­sion­al point, which is indeed equal to $1$, and is based on the volume de­f­i­n­i­tion $V_{n}$ of the $n$-di­men­sion­al Eu­clid­ean ball

where $R$ is the radius, and $\Gamma$ is the gamma func­tion. By solv­ing for $n = 0$, we find the stated result

The net-zero split­ting of the zero-di­men­sion­al point is causal to all the Lie al­ge­bras and groups we have con­sid­ered in this book, and it is the origin of light in its raw and pri­mor­dial form. The zero-di­men­sion­al point is eter­nal and un­caused, and with the birth and death of it­er­a­tions of uni­vers­es, it per­sists intact.

Light orig­i­nates from inside of the sin­gu­lar­i­ty, where it is based on the dy­nam­ics of geo­met­ric primes and their polyg­o­nal forms, ro­tat­ing in the com­plex unit sphere. Through the di­men­sion­al trans­for­ma­tion, light man­i­fests itself as space­time, rel­a­tiv­i­ty, matter, forces, and all the other phe­nom­e­na per­ceived lo­cal­ly. Inside the sin­gu­lar­i­ty, as the geo­met­ric prime gains more angles of free­dom for its ver­tices, it takes on the com­pos­ite geom­e­try of the geo­met­ric in­te­ger and its poly­gon with that number of ver­tices.

This sec­tion con­sid­ers the first and second geo­met­ric primes and their com­pos­ite arrange­ments. The $gp_{2}$ geo­met­ric prime will di­men­sion­al­ly trans­form to the $gp_{1}$ prime and its com­pos­ites, with an axis point that varies from fixed to ge­o­des­ic on the com­plex unit sphere once ex­po­nen­ti­at­ed.

1. Albert Ein­stein (1905) "Zur Elek­tro­dy­namik be­wegter Körper", An­nalen der Physik 17: 891
2. Dray, Tevian, "The Geom­e­try of Spe­cial Rel­a­tiv­i­ty", 2012, CRC Press