1.3.11 Summary

The im­por­tance of fa­mil­iar­iz­ing one­self with math­e­mat­i­cal cur­va­ture cannot be over-em­pha­sized. We will find it cen­tral in gain­ing a geo­met­ric un­der­stand­ing of what grav­i­ty is. Spher­i­cal forms have pos­i­tive cur­va­ture, hy­per­bol­ic forms are neg­a­tive, and with­out a doubt, as­ymp­tot­ic light forms have flat cur­va­ture. The reader should not be con­fused that signed cur­va­ture is the same as the de­f­i­n­i­tions of these geo­met­ric cur­va­tures. Signed cur­va­ture de­pends on the plane's ori­en­ta­tion and the di­rec­tion of the curve in that plane. We have pur­pose­ful­ly been biased to signed cur­va­ture over cur­va­ture, given that ori­en­ta­tion in­for­ma­tion is present in its form, com­pared to only the mag­ni­tude value of its coun­ter­part. The cur­va­ture char­ac­ter­is­tics are easily ex­tend­ed into the sphere of com­plex num­bers and func­tions, seen by the work of Need­ham. Need­ham's com­plex cur­va­ture pre­sen­ta­tion re­volves around the polar form of com­plex num­bers, and we ex­tend­ed this to en­com­pass cur­va­ture for com­plex-valued para­met­ric forms. Para­met­ric forms are es­sen­tial to how Lie theory and its groups are pre­sent­ed. We will need a tool of this kind to de­vel­op the con­cepts of cur­va­ture in the higher-di­men­sion­al matrix group forms.

Ori­en­ta­tion is in­her­ent in the bivec­tors that arise in geo­met­ric al­ge­bra, which led to the con­cept of an ori­ent­ed torque when ap­plied to the light func­tions. Torque is ro­ta­tion­al work, where a force acts on the end­point of an ori­ent­ed vector. We pushed the limit of the torque de­f­i­n­i­tion fur­ther by dis­card­ing the point con­straint and in­stead saw torque as a force acting on a func­tion over some given dis­tance in space and/or time. For the case of torque and the light func­tions, the force from an iso­lat­ed math­e­mat­i­cal light system comes from the com­plex angle space, which is syn­ony­mous with the tan­gent space and the Lie al­ge­bras of the con­sid­ered group. As dis­cussed in Chap­ter 2, the tan­gent space de­pends on the sub­tan­gent space. The sub­tan­gent space de­pends on its phase space, and this phase-nest­ing process is con­tin­u­ous through the log­a­rithm func­tion. The forms are all con­nect­ed.

The in­tro­duc­tion of math­e­mat­i­cal physics this early in the book shows a pur­pose­ful method­ol­o­gy to the math­e­mat­ics being pre­sent­ed. In this chap­ter, we in­tro­duced the con­cept that cir­cu­lar cur­va­ture has a sim­i­lar means to be de­fined, which is as the ar­clength dis­tance from the radius of cur­va­ture's cir­cu­lar arc, com­pared to the unit circle with a stan­dard an­gu­lar fre­quen­cy. Doing so, en­ables us to expand the con­cept of cur­va­ture into wave me­chan­ics and make the link be­tween the energy of a system and how it re­lates to cur­va­ture. Cur­va­ture is de­fined by how a con­tin­u­ous curve de­vi­ates from the line or the plane at a point on the said curve. The litmus de­ter­mines that cur­va­ture cen­ters around match­ing that curve to that of a circle with a cor­re­lat­ing radius. This radius rep­re­sents the radius of con­ver­gence, so it should not be a sur­prise that the math­e­mat­ics of cur­va­ture can seam­less­ly be in­ter­ject­ed with wavenum­bers, wave­lengths, an­gu­lar fre­quen­cies, and pe­ri­ods. Re­al­iz­ing this con­nec­tion led us to de­scribe cur­va­ture as the wave func­tion. As­ymp­tot­ic light func­tions are flat and know­ing this en­ables us to see that cur­va­ture in space­time mea­sures how much it de­vi­ates from the flat line/plane of light.

By con­sid­er­ing the com­plex angle space and its arrange­ment of hy­per­bol­ic, cir­cu­lar, and as­ymp­tot­ic light geome­tries, the reader should en­vi­sion this as a sym­met­ric slice of the conic sec­tion. This conic sec­tion de­parts from the real-valued conic sec­tions taught by typ­i­cal geom­e­try treat­ments since it is com­plex-valued. It pro­vides the nec­es­sary foot­ing for the sub­ject matter of com­plex conic sec­tions con­cern­ing Lie theory pre­sent­ed in the next chap­ter.

The open mind is not lim­it­ed by the con­straints of the ex­per­i­ment bound in space­time. So far, In the short span of the book, we have al­ready been able to piece to­geth­er what light func­tions are, where they sit in re­la­tion­ship to hy­per­bol­ic and spher­i­cal/cir­cu­lar geom­e­try in the com­plex angle space, and know that as­ymp­tot­ic light has flat cur­va­ture. The uni­verse re­veals her­self and what she gov­erns by, which is math­e­mat­ics. She is math­e­mat­ics, and every­thing is of her womb.