Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative

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1 Artile Universl Hyperboli Geoetry, Sydpoints nd Finite Fields: A Projetive nd Algebri Alterntive Norn J. Wildberger Shool of thetis nd Sttistis, University of New South Wles, Sydney 05, Austrli; n.wildberger@unsw.edu.u; Tel.: Reeived: 6 Otober 07; Aepted: 6 eeber 07; Published: Jnury 08 Abstrt: Universl hyperboli geoetry gives purely lgebri pproh to the subjet tht onnets nturlly with Einstein s speil theory of reltivity. In this pper, we give n overview of soe spets of this theory relting to tringle geoetry nd in prtiulr the rerkble new nlogues of idpoints lled sydpoints. We lso disuss how the generlity llows us to onsider hyperboli geoetry over generl fields, in prtiulr over finite fields. Keywords: rtionl trigonoetry; universl hyperboli geoetry; sydpoints; finite fields. Two Fous Questions nd Projetive/Algebri ook t Hyperboli Geoetry While physiists hve long pondered the question of the physil nture of the ontinuu, thetiins hve struggled to siilrly understnd the orresponding thetil struture. In the lst dede, we hve seen the eergene of rtionl trigonoetry [,] s vible lterntive to trditionl geoetry, built not over ontinuu of rel nubers, but rther lgebrilly over generl field, so lso over the rtionl nubers, or over finite fields. Universl hyperboli geoetry (UHG) extends this understnding to the projetive setting, yielding new nd broder pproh to the yley Klein frework (see []) for the rerkble geoetry disovered now lost two enturies go by Bolyi, Guss nd obhevsky s in [4 6]. See lso [7,8] for the lssil nd odern use of projetive etril strutures in geoetry. In this pper, we will give n outline of this new pproh, whih onnets nturlly to the reltivisti geoetry of orentz, Einstein nd inkowski nd lso llows us to onsider hyperboli geoetries over generl fields, inluding finite fields. To void tehnilities nd ke the subjet essible to wider udiene, inluding physiists, we i to desribe things both geoetrilly in projetive visul fshion, s well s lgebrilly in liner lgebri setting.. The Polrity of oni isovered by Apollonius We ugent the projetive plne, whih we y regrd s two-diensionl ffine plne nd line t infinity, with fixed oni. This oni is lled the bsolute in yley Klein geoetry. In this universl hyperboli geoetry (UHG), developed in [9 ], we tke it to be irle, typilly in blue, nd ll it the null irle. The polrity ssoited with oni ws investigted by Apollonius nd gives dulity between points nd lines A = in the spe, whih we lso write s = A. Given point, onsider ny two lines through, whih eet the oni t two points eh s in Figure. The other two digonl points of this yli qudrilterl defines the dul line A. Rerkbly, this onstrution does not depend on the hoie of lines through, s Apollonius relized. Universe 08, 4, ; doi:0.90/universe

2 Universe 08, 4, of 0 When pprohes the oni, the dul A pprohes the tngent to the oni t tht point. If b lies on A =, then it turns out tht lies on B = b. A b K Figure. The dul of point is line = A. Fro this dulity, we y define the perpendiulrity, whih lies t the ore of hyperboli geoetry. Two points re perpendiulr when one lies on the dul of the other, nd siilrly two lines re perpendiulr when one psses through the dul of the other. We tret points nd lines syetrilly! Hyperboli geoetry is then the study of those spets of projetive geoetry tht re deterined by the fixed oni, with isoetries just those projetive trnsfortions, whih fix the null irle. This turns out to be essentilly the reltivisti group O (, ), with oeffiients in the bse field, s we shll see. However, to desribe the tul etril struture, we ove beyond the usul hyperboli distne nd ngle found in the lssil theory of Bolyi, Guss nd obhevsky; rther, we eploy hyperboli nlogues of the qudrne nd spred of rtionl trigonoetry.. Null Points, ines nd ight ones Null points re perpendiulr to theselves; these re the points lying on the originl null irle, suh s α nd β in Figure. Null lines re lso perpendiulr to theselves; these re just duls of null points or the tngents to the null irle, s shown in Figure. In lssil hyperboli geoetry, only the interior of the irle is usully onsidered, nd the bsolute irle is onsidered to be infinitely fr wy. Here, we re interested in the entire projetive plne, inluding lso the null oni itself nd its exterior, inluding tully points t infinity. A= b Figure. Null points nd null lines on the null oni. This orresponds to onsidering + reltivisti spe projetively: with the null oni orresponding to the light one; points inside the null oni to tie-like diretions; nd points outside the null oni to spe-like diretions. ike the physiist, we regrd the entire spe s of priry interest, not just the interior of our light one, even if this is our initil orienttion!

3 Universe 08, 4, of 0 The usul geoetry of speil reltivity in + diensions, in vetor spe with inner produt: (x, y, z ) (x, y, z ) = x x + y y z z () when looked t projetively, gives us UHG, with the null one x + y z = 0. The usul hyperboloid of two sheets x + y z =, the top sheet of whih lssilly orresponds to the hyperboli plne, is Riennin sub-nifold of the orentzin three-diensionl spe. This vrint of Euliden struture holds in the interior of the null oni, but outside, we re in de Sitter-type spe s represented by the hyperboloid of one sheet x + y z =. This kind of universl hyperboli geoetry is no longer hoogeneous, s points outside the null oni behve differently fro points inside the oni, nd indeed over ore generl fields, the distintion between these two types of points is onsiderbly ore subtle. 4. Tringle nd ul To see, the iportne nd usefulness of onsidering both interior nd exterior points, let us look t tringle with sides the lines,,, nd its dul tringle l l l, with sides A, A nd A. These two tringles ply now syetril role: the duls of the points,, re the lines A, A, A, while the duls of the points l, l, l re the lines,,. The dul tringle plys nturl role in estblishing the existene of n orthoenter of generl tringle, whih is vlid theore in this for of hyperboli geoetry, lthough it is not in lssil hyperboli geoetry! In Figure, we see the three ltitudes of the tringle deterined by lines fro verties to dul verties, eeting t the orthoenter h. The reson tht this does not work in generl in lssil hyperboli geoetry is tht the eeting of the three ltitudes y well be outside the null oni even if ll three points of the tringle re inside, nd so it is invisible to the geoetry of Bolyi, Guss nd obhevsky! Sine this nedotlly ws one of Einstein s fvourite geoetry theores, it is definitely worthwhile hving it s prt of the piture. In ft, siilr disussion y be hd for the iruenter. This is ll prt of the rih nd ostly new subjet of hyperboli tringle geoetry; see for exple [,,4], whih reently hs lso been extended to inlude qudrilterl geoetry in [5]. A l l A h A l Figure. The dul tringle nd the orthoenter h. 5. Qudrne y Be efined Algebrilly In lssil hyperboli geoetry, the etril struture is introdued using differentil geoetry in the ontext of Riennin etris on sooth nifolds. In the ore projetive sitution, s in [6], the notions of projetive qudrne nd projetive spred n be introdued using the fundentl ide of ross-rtio of four points on line, s ppreited lso by [7].

4 Universe 08, 4, 4 of 0 If four olliner points hve projetive oordintes, b, nd d, whih n be either fro the given field or possibly hve the vlue infinity ( ), then their ross-rtio y be defined s: (, b :, d) ( ) (d b) ( b) (d ). Now, given two points nd in the hyperboli plne, they hve dul lines A nd A, whih eet the line in the onjugte points: b A ( ) nd b A ( ) giving four olliner points,, b nd b. Then, the (projetive) qudrne between nd is the ross-rtio: q (, ) (, b :, b ). In Figure 4, we see n exple of exterior nd interior, to ephsize the se tht this etril notion pplies very generlly to ll non-null points. B ( ) b A= B A = b Figure 4. onjugte points ke ross-rtio. efine the spred between lines dully, so tht: S (A, A ) q (, ). In this wy, the reltion between the points nd lines etrilly is opletely syetri. There is nturl onnetion with the usul lssil etril notions in the Beltri Klein odel (see [5]) when we restrit to interior points (inside the light one or null oni) nd lines tht eet lso t interior points; in these ses: 6. The Algebri Approh q (, ) = sinh (d (, )) nd S (, ) = sin (θ (, )). ue to the odern filirity with liner lgebr, it y be useful to refre the projetive setup bove using hoogeneous oordintes, where we follow: [9]. In three-diensionl vetor spe of row vetors (x, y, z), we y define (hyperboli) point [x : y : z] to be one-diensionl subspe through non-zero vetor (x, y, z). This orresponds to the plnr point [ x z, y z ] if z = 0. A (hyperboli) line (l : : n) y be defined to be two-diensionl subspe with eqution lx + y nz = 0. The inidene between these points nd lines is tht the point [x : y : z] lies on the line (l : : n), or equivlently psses through, preisely when: lx + y nz = 0.

5 Universe 08, 4, 5 of 0 In trix ters, this is the reltion: ( ) 0 0 l x y z 0 0 = n The point [x : y : z] is then dul to the line (l : : n) preisely when: x : y : z = l : : n. In this se, we write = or =. This lgebri struture ensures tht these definitions work over generl field. The etril struture oes bout fro the syetri biliner for () of Einstein, orentz nd inkowski of the bient three-diensionl spe. It y be used to define reltion between one-diensionl subspes s follows: the qudrne between points [x : y : z ] nd [x : y : z ] is: (x q (, ) x + y y z z ) ( x + y ) ( z x + y. ) z ully, the spred between lines (l : : n ) nd (l : : n ) is: S (, ) (l l + n n ) ( l + ) ( n l +. ) n For three points, nd, the three qudrnes will be: q = q (, ) q = q (, ) q = q (, ) nd for three lines, nd, the three spreds will be: 7. The in Trigonoetri ws of UHG S = S (, ) S = S (, ) S = S (, ). Here re the in trigonoetri lws in the subjet, estblished first in [9]. We begin with essentilly the one-diensionl situtions: Theore (Triple qud forul). If, nd re olliner points, then: ( ) (q + q + q ) = q + q + q + 4q q q. Theore (Triple spred forul). If, nd re onurrent lines, then: ( ) (S + S + S ) = S + S + S + 4S S S. Now, for the qudrnes nd spreds of tringle s in Figure 5:

6 Universe 08, 4, 6 of 0 S q S q q S Figure 5. Qudrne nd spreds in hyperboli tringle. Theore (Pythgors). If nd re perpendiulr lines, then: q = q + q q q. Theore 4 (Pythgors dul). If nd re perpendiulr points, then: S = S + S S S. Theore 5 (Spred lw). Theore 6 (Spred dul lw). S q = S q = S q. q S = q S = q S. Theore 7 (ross lw). Theore 8 (ross dul lw). (q q S (q + q + q ) + ) = 4 ( q ) ( q ) ( q ). (S S q (S + S + S ) + ) = 4 ( S ) ( S ) ( S ). There re three syetril fors of Pythgors s theore, the ross lw nd their duls, obtined by rotting indies. These vrious lws reple the trnsendentl hyperboli Pythgors theore, the sine lw nd osine lw of both kinds. They work over generl field, both inside nd outside the null irle, nd tully even with ore generl biliner fors. They re rgubly ore nturl nd onvenient for physiists. The quntity: A q q S = q q S = q q S is the qudre of the tringle nd is soewht nlogous to the hyperboli re of the tringle, but it is deidedly of different hrter. It is big step to ke the trnsition fro trnsendentl to purely lgebri onepts here: oputtions n tully now be exhibited opletely nd lerly. 8. irles, idpoints nd iruenters A hyperboli irle with entre nd qudrne k is the lous of points x, whih stisfy q (, x) = k. This is oni, whih inludes wht in the lssil literture re lled equi-distnt urves in the se of n externl entre. A idpoint of side is point lying on the line tht stisfies q (, ) = q (, ). idpoints exist preisely when q (, ) is squre in the field. There re in generl two idpoints

7 Universe 08, 4, 7 of 0 if they exist t ll, nd they re perpendiulr. A idline is the dul of idpoint, or equivlently line through idpoint perpendiulr to the line joining the two originl points; in other words, the hyperboli version of perpendiulr bisetor. The following is illustrted in Figure 6. Theore 9 (iruenters). Assue tht the six idpoints of tringle exist. Then, they re olliner three t tie, lying on four distint irulines. The six idlines of re onurrent three t tie, eeting t four distint iruenters tht re dul to the irulines. The iruenters re the entres of in generl four hyperboli irles tht pss through the points of the tringle. Figure 6. iruenters nd irulines of tringle. 9. Sydpoints Augent idpoints While idpoints hve been studied sine the erly dys of the subjet, n iportnt relted notion ws only introdued very reently in []. A sydpoint of side is point s lying on tht stisfies q(, s) = q(, s). Sydpoints exist preisely when q (, ) is squre in the field. There re in generl two sydpoints, if they exist t ll, but they re not perpendiulr. A onstrution of sydpoints r nd s of b y be dedued fro Figure 7. l w s r n b z k x y Figure 7. onstrution of sydpoints.

8 Universe 08, 4, 8 of 0 First onstrut = (b), then the idpoints nd n of nd then use the null points x nd y lying on b s shown. Sydpoints work with idpoints to extend tringle geoetry to tringles with verties both inside nd outside the null oni. In Figure 8, we see entroids g nd irulines of suh tringle. For the rerkble onnetion of the iruenters to irles through the three points, see []. g s g s g s s Figure 8. entroids nd irulines of tringle. 0. Sydpoints nd the Prbol In [8,9], we defined the hyperboli prbol P 0 to be the lous of point p 0 (tully oni) stisfying: q(p 0, f ) + q(p 0, f ) = for fixed points f, f lled the foi. Equivlently: q (p 0, f ) = q (p 0, F ) or q (p 0, f ) = q (p 0, F ), where F f, F f re the diretries. Note tht the qudrne between point nd line is defined in ters of the perpendiulr trnsversl. Suh prbol is shown in red in Figure 9. V V f f b v b f b f v B F B F B F B P0 Figure 9. A prbol with foi f nd f. In generl, if we tke duls of the tngents of oni, we get dul oni. It turns out tht the dul of prbol P 0 is nother prbol: the twin prbol P 0 whose foi f, f re the sydpoints of

9 Universe 08, 4, 9 of 0 the originl foi pir f, f, shown in ornge in Figure 0. Therefore, sydpoints pper proinently in the geoetry of the prbol! f b F p0 P0 f f v b b v f 0 P F b F p0 P 0 V F P0 V Figure 0. The twin prbol with foi f nd f.. UHG over Finite Fields n idpoints nd sydpoints exist together? The onditions tht q (, b) is squre nd q (, b) is squre re not siultneously stisfied over our usul nuber syste, or over field F p where p od 4. However, in the se of F p where p od 4, then is squre, so both idpoints nd sydpoints n exist together. This is n entirely new spet of hyperboli geoetry tht is invisible to us usully. However, how n we visulize UHG over finite field? Fortuntely, new nd powerful progr being developed by ihel Reynolds t University of New South Wles (UNSW) Sydney oplishes extly this. This progr will hopefully be vilble for publi use in the ner future. In ft, hyperboli geoetry over finite fields hs been onsidered previously; see [0,]. With UHG, we get novel pproh tht unifies ll these geoetries over rbitrry fields, so freeing us fro the reline on prtiulr ideology with respet to the ontinuu. onflits of Interest: The uthors delre no onflits of interest. Abbrevitions The following bbrevitions re used in this nusript: UHG Referenes Universl hyperboli geoetry. Wildberger, N.J. ivine Proportions: Rtionl Trigonoetry to Universl Geoetry; Wild Egg Books: Sydney, Austrli, Wildberger, N.J. Affine nd Projetive Universl Geoetry. rxiv 006, rxiv:th/ Busenn, H.; Kelly, P.J. Projetive Geoetry nd Projetive etris; over Publitions: New York, NY, USA, 006; originlly published by Adei Press: New York, NY, USA, oolidge, J.. The Eleents of Non-Euliden Geoetry; Oxford lrendon Press: Oxford, UK, Greenberg,.J. Euliden nd Non-Euliden Geoetries: evelopent nd History, 4th ed.; W. H. Freen nd o.: Sn Frniso, A, USA, Rsy, A.; Rihtyer, R.. Introdution to Hyperboli Geoetry; Springer: New York, NY, USA, Bhnn, F. Aufbu der Geoetrie us de Spiegelungsbegriff; Springer: New York, NY, USA, 97.

10 Universe 08, 4, 0 of 0 8. olnár, E.; Prok, I.; Sziri, J. On xil hoogeneous -geoetries nd their visuliztion. Universe 07,, Wildberger, N.J. Universl Hyperboli Geoetry I: Trigonoetry. Geo. edi. 0, 6, Wildberger, N.J. Universl Hyperboli Geoetry II: A pitoril overview. KoG 00, 4, 4.. Wildberger, N.J. Universl Hyperboli Geoetry III: First steps in projetive tringle geoetry. KoG 0, 5, Wildberger, N.J.; Alkhldi, A. Universl Hyperboli Geoetry IV: Sydpoints nd Twin iruirles. KoG 0, 6, Ungr, A.A. Hyperboli trigonoetry in the Einstein reltivisti veloity odel of hyperboli geoetry. oput. th. Appl. 000, 40,. 4. Ungr, A. Hyperboli Tringle enters: The Speil Reltivisti Approh; Springer: New York, NY, USA, Blefri, S.; Wildberger, N.J. Qudrngle entroids in universl hyperboli geoetry. KoG 07, 0, Onishhik, A..; Sulnke, R. Projetive nd yley-klein Geoetries; Springer: New York, NY, USA, Bruner, H. Geoetrie Projektiver Räue I, II; Bibliogrphishes Institut: nnhei, Gerny, Alkhldi, A.; Wildberger, N.J. The Prbol in Universl Hyperboli Geoetry I. KoG 0, 7, Alkhldi, A.; Wildberger, N.J. The Prbol in Universl Hyperboli Geoetry II: nonil points nd the Y-oni. J. Geo. Grph. 06, 0,. 0. Angel, J. Finite upper hlf plnes over finite fields. Finite Fields Appl. 996,, Soto-Andrde, J. Geoetril Gel fnd odels, tensor quotients, nd Weil representtions. Pro. Syp. Pure th. 987, 47, by the uthors. iensee PI, Bsel, Switzerlnd. This rtile is n open ess rtile distributed under the ters nd onditions of the retive oons Attribution ( BY) liense (