Neuberg cubics over finite fields

Neub erg cubics o v er finite fields N. J. Wildb erger Sc ho ol of Mathematics and Statistics UNSW Sydney 2052 Australia Octob er 27, 2018 Abstract The framew ork of universal geometry allow s us t o consider metrical prop erties of affine views of elliptic curve s, even o ver finite fields. W e sho w how the Neub erg cubic of triangle geometry extends to the fi nite field situation and provides in teresting p oten tial in va riants for elliptic curves, focussing on an explicit example o ver F 23 . W e also pro ve t hat tangen t conics for a W eierstrass cubic are identical or disjoin t. Metrical views of cubics This pap er lo ok s a t the connection b et ween mo dern Euclidea n triang le geo metry and the arithmetic of elliptic curves ov er finite fields using the fr amew o rk of universal geometry (see [8]), a metrical view of alge braic geometry based on the algebraic notions of quadr anc e and spr e ad ra ther than distanc e a nd angle. A go o d pa rt of triangle geo metr y app ears to e xtend to finite fields. In particular , the Neuber g cubic provides a rich or ganizational str ucture for many tria ngle centers and asso ciated lines throug h the gr oup law and rela ted Desmic (linking) structure, even in a finite field. It a nd other triangle cubics ha ve the p otent ia l to b e useful g eometrical to ols for understanding elliptic curves. See [1], [2], [3], [4], [5] and [6] for background o n tria ngle cubics. F or triang le geo meters, finite fields hold p otential applica tions to cryptog- raphy and a lso pr ovide a lab oratory for ex ploration that in many ways is mor e pleasant than the decimal num b ers. A price to be paid, ho wever, is that the usual tr i-linear co or dinate framework needs to b e replaced by Ca rtesian or barycentric co ordina tes. W e b egin with a br ief review of the relev ant no tions fro m ratio na l trigonom- etry , which a llows the set-up of metrical algebr a ic g e o metry . Then w e discuss the Neub erg cubic of a triangle a nd related centers, illustrated in a pa rticular example over F 23 . W e also prove that for affine c ubics in W eierstras s form the tangent conics a re a ll disjoint provided − 3 is not a square in the field. It should b e noted tha t there is also a pro jective version of universal g eometry (see [9]), but here w e s tic k to the affine situation. 1 La ws of Rational T rigonometry Fix a finite field F not of characteristic tw o, who se elements a re ca lle d n umbers . A p oint A is an o r dered pair [ x, y ] of num b ers, that is an element of F 2 . The quadrance Q ( A 1 , A 2 ) b et ween points A 1 ≡ [ x 1 , y 1 ] and A 2 ≡ [ x 2 , y 2 ] is the nu mber Q ( A 1 , A 2 ) ≡ ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . A line l is an order e d pro por tion h a : b : c i , wher e a and b are not b oth zero . This r epresents the equa tion ax + by + c = 0 . Such a line is null precisely when a 2 + b 2 = 0 . Null lines o ccur precisely when − 1 is a s q uare. F or distinct p oints A 1 = [ x 1 , y 1 ] and A 2 = [ x 2 , y 2 ] the line passing through them b oth is A 1 A 2 = h y 1 − y 2 : x 2 − x 1 : x 1 y 2 − x 2 y 1 i . Two lines l 1 ≡ h a 1 : b 1 : c 1 i a nd l 2 ≡ h a 2 : b 2 : c 2 i a re p erp endicular precis e ly when a 1 a 2 + b 1 b 2 = 0 . F or any fixed line l and an y p oint A, there is a unique line n passing through A and p erpe ndicular to l , called the altitude from A to l . If l is a non-null line then the altitude n meets l at a unique p oint F , ca lled the fo ot of the altitude. In this case we may define the reflection of A in l to b e the p oint σ l ( A ) such that F is the midpoint of the side Aσ l ( A ). If m is ano ther line, then the reflection of m in l is the line Σ l ( m ) with the prop erty that the reflection in l of a n y p oint A on m lies on Σ l ( m ). The spread s ( l 1 , l 2 ) b et ween non- null lines l 1 ≡ h a 1 : b 1 : c 1 i and l 2 ≡ h a 2 : b 2 : c 2 i is the num b er s ( l 1 , l 2 ) ≡ ( a 1 b 2 − a 2 b 1 ) 2 ( a 2 1 + b 2 1 ) ( a 2 2 + b 2 2 ) = 1 − ( a 1 a 2 + b 1 b 2 ) 2 ( a 2 1 + b 2 1 ) ( a 2 2 + b 2 2 ) . This num b er s = s ( l 1 , l 2 ) is 0 precisely when the lines ar e parallel, and 1 precisely when the lines ar e p erp endicular. I t ha s the pr oper t y that s (1 − s ) is a square in the field, and every such spread n um b e r s can b e shown to b e the spread b et ween some tw o lines. The spr ead b et ween lines may alterna tiv ely be express ed a s a ra tio of qua d- rances: if l 1 and l 2 int er sect at a p oin t A, cho ose any o ther p oin t B on l 1 , and let C o n l 2 be the fo ot of the a ltitude line from B to l 2 , then s ( l 1 , l 2 ) = Q ( B , C ) Q ( A , B ) . Reflection in a line pr eserves quadrance betw een p o in ts and spread betw ee n lines. Given three distinct points A 1 , A 2 and A 3 , we use the notation Q 1 ≡ Q ( A 2 , A 3 ) Q 2 ≡ Q ( A 1 , A 3 ) Q 3 ≡ Q ( A 1 , A 2 ) 2 and s 1 ≡ s ( A 1 A 2 , A 1 A 3 ) s 2 ≡ s ( A 2 A 1 , A 2 A 3 ) s 3 ≡ s ( A 3 A 1 , A 3 A 2 ) . A triangle A 1 A 2 A 3 is a se t of three non-co llinear p oints, and is non-null precisely when its three lines A 1 A 2 , A 2 A 3 and A 1 A 3 are non-null. Her e a re the five main laws of r ational tr ig onometry , which may b e v iew ed a s pur ely a lgebraic ident ities in volving only ratio na l functions. T riple quad form ula The p oints A 1 , A 2 and A 3 are collinea r precis ely when ( Q 1 + Q 2 + Q 3 ) 2 = 2  Q 2 1 + Q 2 2 + Q 2 3  . Pythagoras’ theorem The lines A 1 A 3 and A 2 A 3 are pe rpendicula r precisely when Q 1 + Q 2 = Q 3 . Spread law F or a non- null triangle A 1 A 2 A 3 s 1 Q 1 = s 2 Q 2 = s 3 Q 3 . Cross l a w F or a non-null triangle A 1 A 2 A 3 define the cross c 3 ≡ 1 − s 3 . Then ( Q 1 + Q 2 − Q 3 ) 2 = 4 Q 1 Q 2 c 3 . T riple spread form ula F or a non-null tria ngle A 1 A 2 A 3 ( s 1 + s 2 + s 3 ) 2 = 2  s 2 1 + s 2 2 + s 2 3  + 4 s 1 s 2 s 3 . See [8] for pro ofs, and ma ny mor e facts ab out geo metry in such a purely algebraic setting. Neub erg cub ics Many in ter e sting po in ts, lines, circles, parab olas, h yp erb olas a nd cubics hav e bee n a sso ciated to a triangle in the plane, such a s the centroid G, o rtho c en ter O , cir cumcen ter C, incen ter I , E uler line e (which pa sses throug h O, G and C ), nine-p oin t circle and so on. Perhaps the most re mark able of these is the N eub er g cubic , which in the earlier literature was called the 32 p oint cubic, but these days is known to pas s through many mor e triangle centers (see [4], [5], [6]). Most such ob jects dep end crucially on a metr ical structure on the affine pla ne. Figure 1 shows the Neub erg cubic for the triangle A 1 A 2 A 3 with v er tices A 1 = [0 , 0], A 2 = [1 , 0] and A 3 = [3 / 4 , 3 / 4 ] . F o r this tr iangle the Euler line, which pas ses through the ortho cent er O and the circumcen ter C, is horiz o n tal. V arious incenters I i are shown, as w ell as reflectio ns o f the vertices in the sides. These are just a few of the man y po in ts on the Neub erg cubic. Note that the 3 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 -1.5 -1.75 -2 1 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 x y Eulerline 1 1 2 2 1 2 0 3 3 3 A A A A I I I I A A O C 8 8 e e Figure 1: Neub erg cubic for A 1 = [0 , 0], A 2 = [1 , 0] and A 3 = [3 / 4 , 3 / 4] tangents to the four incenters ar e also horizo n tal, and it turns out that the asymptote of the cubic is also . With the completely algebr aic languag e of r ational trigo nometry , we consider such a picture for triangles in finite fields . F or this it will b e convenien t to alter the usual p oin t of view somewhat, elev ating the qua dr angle I 0 I 1 I 2 I 3 of inc enters to a primary p osition. This ensures that the reference triang le A 1 A 2 A 3 actually has vertex bisector s. F or a triangle I 1 I 2 I 3 the altitudes from each p o in t to the opp osite side in- tersect at the o r thocenter, which we here denote I 0 . In ter ms of Car tesian co ordinates I j = [ x j, y j ], x 0 ≡  x 1 x 2 y 2 − x 1 x 3 y 3 + x 2 x 3 y 3 − x 3 x 2 y 2 + x 3 x 1 y 1 − x 2 x 1 y 1 + y 1 y 2 2 − y 1 y 2 3 + y 2 y 2 3 − y 3 y 2 2 + y 3 y 2 1 − y 2 y 2 1  x 1 y 2 − x 1 y 3 + x 2 y 3 − x 3 y 2 + x 3 y 1 − x 2 y 1 and y 0 ≡  x 1 y 1 y 2 − x 1 y 1 y 3 + x 2 y 2 y 3 − x 3 y 3 y 2 + x 3 y 3 y 1 − x 2 y 2 y 1 + x 2 1 x 2 − x 2 1 x 3 + x 2 2 x 3 − x 2 3 x 2 + x 2 3 x 1 − x 2 2 x 1  x 1 y 2 − x 1 y 3 + x 2 y 3 − x 3 y 2 + x 3 y 1 − x 2 y 1 . In terms of barycentric co ordinates, if the quadra nces of the triangle I 1 I 2 I 3 are R 1 , R 2 and R 3 and A = ( R 1 + R 2 + R 3 ) 2 − 2  R 2 1 + R 2 2 + R 3 3  = 4 ( x 1 y 2 − x 1 y 3 + x 2 y 3 − x 3 y 2 + x 3 y 1 − x 2 y 1 ) 2 4 is the quadrea o f the triangle (sixteen times the squa re of the area in the decimal num b e r situation), then I 0 = β 1 I 1 + β 2 I 2 + β 3 I 3 where β 1 ≡ ( R 3 + R 1 − R 2 ) ( R 1 + R 2 − R 3 ) / A β 2 ≡ ( R 1 + R 2 − R 3 ) ( R 2 + R 3 − R 1 ) / A β 3 ≡ ( R 2 + R 3 − R 1 ) ( R 3 + R 1 − R 2 ) / A . If I 1 I 2 I 3 is non-null, whic h we hencefor th ass ume, then the feet of its al- titudes e x ist and we call them resp ectively A 1 , A 2 and A 3 . Th us for example I 1 , I 0 and A 1 are c ollinear p oint s which lie on a line p erp endicular to I 2 I 3 . In the quadrangle I 1 I 2 I 3 I 4 we have a complete symmetry b et ween the four p oints I 0 , I 1 , I 2 and I 3 . So w e could hav e s ta rted with any three of these p oints, and the o rthoc e nter of such a triangle would ha ve b een the fourth p oin t, with the orthic triangl e A 1 A 2 A 3 obtained alwa ys the s ame. Theorem 1 (Orthic triangle) The lines I 0 I 1 and I 2 I 3 ar e bise ctors of the vertex of A 1 A 2 A 3 at A 1 , in the sense that s ( I 0 I 1 , A 1 A 2 ) = s ( I 0 I 1 , A 1 A 3 ) s ( I 2 I 3 , A 1 A 2 ) = s ( I 2 I 3 , A 1 A 3 ) . The pro of use s a co mputer, a nd one can further v er ify that the for mer spread is  x 2 x 3 − x 1 x 3 − x 1 x 2 − y 1 y 2 − y 1 y 3 + y 2 y 3 + x 2 1 + y 2 1  2 R 2 R 3 while the latter spread is ( x 1 y 2 − x 1 y 3 + x 2 y 3 − x 3 y 2 + x 3 y 1 − x 2 y 1 + x 3 y 2 ) 2 R 2 R 3 . These tw o spreads sum to 1 , as they must since I 0 I 1 and I 2 I 3 are per pendicular . So the triangle A 1 A 2 A 3 has a sp ecial pr oper t y: each o f its vertices has bise ctors. Over the dec imal num b ers, every tria ng le has vertex bisector s, but in [8] it is shown that in genera l this amounts to the condition that the sprea ds o f the triangle are squar es . F or a p oint P , let P 1 , P 2 and P 3 denote its reflectio ns in the s ides A 2 A 3 , A 1 A 3 and A 1 A 2 resp ectively , and define the Neub e rg cubic N c of A 1 A 2 A 3 to be the locus of p oints P such that P 1 , P 2 and P 3 are p ersp e c tiv e with A 1 , A 2 and A 3 resp ectively: in other words that P 1 A 1 , P 2 A 2 and P 3 A 3 are concur ren t lines. An example o v er F 23 W e work in the pr ime field F 23 , in which the squa r es a re 1 , 4 , 9 , 1 6 , 2 , 13 , 3 , 1 8 , 12 , 8 and 6 . Note that − 1 is no t a square , but tha t 3 = 7 2 is a square. The latter fact implies that equilateral triangle s exist in F 2 23 . Let I 1 = [6 , 4] I 2 = [22 , 22] I 3 = [21 , 12] . 5 These p oin ts hav e b een chosen so that the ortho center of I 1 I 2 I 3 is I 0 = [0 , 0]. The feet o f the altitudes are A 1 = [13 , 1] A 2 = [5 , 5] A 3 = [2 , 11] . The lines of A 1 A 2 A 3 are A 1 A 2 = h 3 : 6 : 1 i A 2 A 3 = h 6 : 3 : 1 i A 1 A 2 = h 12 : 4 : 1 i and the sprea ds of the triangle A 1 A 2 A 3 are s 1 = 12 s 2 = 16 s 3 = 6 . Note that as ex p ected these num ber s are squares, a nd o ne c an c heck that s ( A 1 I 0 , A 1 A 2 ) = s ( A 1 I 0 , A 1 A 3 ) = 5 s ( A 2 I 0 , A 2 A 1 ) = s ( A 2 I 0 , A 2 A 3 ) = − 6 s ( A 3 I 0 , A 3 A 1 ) = s ( A 3 I 0 , A 3 A 2 ) = − 7 . The connection b et ween for example the spread s = 5 a nd the spread of its ‘double’ r = 12 is given by the se c ond spr e ad p olynomial , r = S 2 ( s ) = 4 s (1 − s ) which in chaos theory is known as the lo gistic map . The spread p olynomials hav e many remar k a ble prop erties that hold also ov er finite fields , se e [8]. W e need the following formula for a reflection. Theorem 2 (Reflection of a p oint in a line) If l ≡ h a : b : c i is a non-n ul l line and A ≡ [ x, y ] , then σ l ( A ) = "  b 2 − a 2  x − 2 aby − 2 ac a 2 + b 2 , − 2 abx +  a 2 − b 2  y − 2 bc a 2 + b 2 # . Using this, the reflections of P = [ x, y ] in the sides o f A 1 A 2 A 3 are P 1 = [4 x + 13 y + 12 , 13 x + 19 y + 6] P 2 = [13 x + 4 y + 1 , 4 x + 10 y + 8 ] P 3 = [19 x + 13 y + 6 , 13 x + 4 y + 12 ] . The lines P 1 A 1 , P 2 A 2 and P 3 A 3 are then h 13 x + 1 9 y + 5 : 19 x + 10 y + 1 : 19 x + 19 y + 3 i h 4 x + 10 y + 3; 1 0 x + 19 y + 4 : 22 x + 1 6 y + 11 i h 13 x + 4 y + 1 : 4 x + 10 y + 19 : 2 2 x + 20 y + 19 i 6 and these ar e concur rent precisely when       13 x + 1 9 y + 5 19 x + 1 0 y + 1 19 x + 1 9 y + 3 4 x + 10 y + 3 10 x + 19 y + 4 22 x + 16 y + 11 13 x + 4 y + 1 4 x + 1 0 y + 19 22 x + 20 y + 19       = 0 . Expanding gives the Neuber g cubic A 1 A 2 A 3 : an affine cur ve over F 23 with equation y 3 + x 2 y + 22 y 2 + 7 xy + 9 x 2 + 13 y = 0 . (1) The tangent line to a p oint [ a, b ] on the curve has equa tion x (18 a + 7 b + 2 ab ) + y  7 a + 2 1 b + a 2 + 3 b 2 + 13  + 3 b + 7 ab + 9 a 2 + 22 b 2 = 0 . There is a nother r ev ea ling wa y to obtain the Neub erg cubic. Theorem 3 (Reflection of a line i n a line) The r efle ction in the non-nul l line l ≡ h a : b : c i sends h a 1 : b 1 : c 1 i to  a 2 − b 2  a 1 + 2 abb 1 : 2 aba 1 −  a 2 − b 2  b 1 : 2 aca 1 + 2 bcb 1 −  a 2 + b 2  c 1  . In a tria ngle with vertex bisectors , the reflectio n of a line thr o ugh a given vertex in either of the vertex bisectors at that v er tex is the same. Theorem 4 (Isogonal conjugates) If a triangle A 1 A 2 A 3 has vertex bise ctors at e ach vert ex , then for any p oint P the r efle ctions of A 1 P, A 2 P and A 3 P in the vertex bise ctors at A 1 , A 2 and A 3 r esp e ctively ar e c oncurre n t . The p oin t of co ncurrence o f these lines is P ∗ , the iso gonal conjugate of P = [ x, y ] . The pro of again is a calcula tion using co or dina tes. W e may use the reflection of a line in a line theorem to esta blish a precise formula for P ∗ in the sp ecial case of our e x ample reference triang le A 1 A 2 A 3 : P ∗ =  2 x + 22 xy + 2 x 2 + 17 y 2 4 x + 20 y + 5 x 2 + 5 y 2 + 21 , 2 y + 15 xy + x 2 + 2 y 2 4 x + 20 y + 5 x 2 + 5 y 2 + 21  . Over the decimal n umbers , the Neub erg cubic is also the lo cus of tho se P = [ x, y ] such that P P ∗ is par a llel to the Euler line. W e can verify this also in our finite example, since this co ndition a moun ts to y = 2 y + 1 5 xy + x 2 + 2 y 2 4 x + 20 y + 5 x 2 + 5 y 2 + 21 which in turn is equiv alent to the equation (1) of N c . It follows that if P lies on N c , then so do es P ∗ —this is a useful w ay to obtain new p oints from o ld ones. In fact the E uler line is para llel to the tangent to N c at the infinite point ∞ e . 7 The cubic (1) is nonsingular, has 27 p oints lying on it, and its pro jectiv e extension has one more p oint at infinit y , namely ∞ e = [1 : 0 : 0 ]. Here are a ll the p oints, the nota tion will be explained more fully below: [0 , 0 ] = I 0 [0 , 8] = E ′ 3 [0 , 1 6] = S ′ [2 , 11] = A 3 [3 , 1 3] = S = A 3 [4 , 5] [5 , 5 ] = A 2 [5 , 14] = ∞ ∗ e [6 , 4 ] = I 1 [7 , 1] [7 , 2 ] = E ′ 2 [7 , 21] = O [8 , 1 0] = A 1 = E 2 [13 , 1] = A 1 [13 , 7] [13 , 16] = F ′ [14 , 9] [16 , 11] [17 , 7] = E ′ 1 [17 , 8] [17 , 9] = A 2 = E 1 [18 , 21] = C = E 3 [19 , 13] = F [21 , 2] [21 , 10] [21 , 12] = I 3 [22 , 22] = I 2 [1 : 0 : 0] = ∞ e 0 1 -1 2 - 2 3 - 3 4 - 4 5 - 5 0 1 1 1 1 2 3 2 3 2 1 2 2 1 2 0 3 3 3 -1 2 -2 3 -3 4 - -4 5 - -5 6 7 8 9 10 11 -6 -7 -8 -9 -10 -11 -6 -7 -8 -9 -10 -11 6 7 8 9 10 11 A =E =S E E E =E =E E A A A I I I I A A O C S F F * Figure 2: The Neub erg cubic for A 1 = [13 , 1], A 2 = [5 , 5 , ] and A 3 = [2 , 11] T o define the group structure on the cubic, define X ⋆ Y to be the (third) int er section of the line X Y with the (pro jectiv e) cubic, so that for example I 0 ⋆ I 0 = ∞ e . W e choose the base p oint of the g roup s tructure to b e the p oint I 0 . Then define X · Y = ( X ⋆ Y ) ⋆ I 0 8 with inv erse X − 1 = X ⋆ ( I 0 ⋆ I 0 ) = X ⋆ ∞ e = X ∗ . So w e ar e wr iting the group m ultiplicatively , and note that I 0 is not a flex, so that X , Y and Z collinear is equiv alent to X · Y · Z = ∞ e , no t X · Y · Z = I 0 . F urthermore X is o f or der tw o whe n X ⋆ X = ∞ e and X is not I 0 , which ha pp ens when X is I 1 , I 2 or I 3 , and the four incenters fo r m a Klein 4-g r oup. The triangle A 1 A 2 A 3 can be recov er ed from the Neub erg c ubic by first find- ing the asymptote (tangent to the p oint at infinit y), then finding the four p oints I 0 I 1 I 2 I 3 on the cubic with a ta ng en t pa rallel to this a symptote, and then taking the orthic triang le of any three of them. This can all be done algebr aically using the gro up law, since I j ⋆ I j = ∞ e and A 1 = I 2 ⋆ I 3 etc. P oin ts on the Neub erg cub ic Not only is the Neub erg cubic defined metrically , but it a lso has many p oints on it that ar e metrical in nature. More than a h undred are known, we will illustrate so me of these for our example. The Neuber g cubic N c of A 1 A 2 A 3 first of all pass e s through A 1 , A 2 and A 3 . It also passes thro ug h the reflections of these p o in ts in the sides o f the triangle, in this case A 1 = [8 , 10] A 2 = [17 , 9] A 3 = [3 , 13] . It also passes through the four incenters I 0 , I 1 , I 2 and I 3 of A 1 A 2 A 3 . In a general field there is no no tion of ‘int er ior p oint’, so these four incenters should be regar de d s ymmetrically . The Neuber g cubic passes throug h the ortho cen ter O = [7 , 21] and the cir cumcen ter C = [18 , 21] of A 1 A 2 A 3 , and these ar e is ogonal conjugates, that is O ∗ = C. The line OC is the Euler line e of A 1 A 2 A 3 and it ha s equation y = 21 , s o that it is hor izont a l and passes through the infinite p oint ∞ e = [1 : 0 : 0 ] . No te that ∞ ∗ e = [5 , 14] . Since 3 = 7 2 is a squa re, on any side of A 1 A 2 A 3 we may c reate tw o equila teral triangles, for example on the side containing A 1 = [13 , 1] and A 3 = [2 , 1 1] we can choose a third point [13 + 2 , 1 + 11] / 2 ± 7 2 [1 − 1 1 , 2 − 1 3 ] namely E 2 = [8 , 1 0] or E ′ 2 = [7 , 2] . Thu s A 1 A 3 E 2 and A 1 A 3 E ′ 2 are equilater al tria ngles with Q ( A 1 , A 3 ) = Q ( A 1 , E 2 ) = Q ( A 3 , E 2 ) = Q ( A 1 , E ′ 2 ) = Q ( A 3 , E ′ 2 ) = 1 4 . 9 As opp osed to the case ov er the decimal num b ers, there seems to b e no obvious no tion of these tr iangles b eing either ‘exter ior’ or ‘int er ior’ to A 1 A 2 A 3 . Using all three sides gives the six po ints E 1 = [17 , 9] E 2 = [8 , 10] E 3 = [18 , 21] E ′ 1 = [13 , 7] E ′ 2 = [7 , 2] E ′ 3 = [0 , 8] and their iso gonal conjuga tes E ∗ 1 = [14 , 9] E ∗ 2 = [21 , 10] E ∗ 3 = [7 , 21] E ′∗ 1 = [17 , 7] E ′∗ 2 = [21 , 2] E ′∗ 3 = [17 , 8] . All t welve of these po in ts lie on the Neub erg cubic. Y et the centroids o f the six equilateral triangles th us formed a re G 1 = [8 , 16] G 2 = [0 , 15] G 3 = [12 , 9] G ′ 1 = [22 , 0] G ′ 2 = [15 , 20] G ′ 3 = [6 , 20] and you may chec k that Q ( G 1 , G 2 ) = Q ( G 2 , G 3 ) = Q ( G 1 , G 3 ) = 1 9 Q ( G ′ 1 , G ′ 2 ) = Q ( G ′ 2 , G ′ 3 ) = Q ( G ′ 1 , G ′ 3 ) = 1 2 so that Nap olean’s theo rem that the cen tro ids of b oth ‘external’ and ‘in ternal’ equilateral triangles themselves form an equila teral triang le seems to hold. It seems cur ious that the six p oints E i and E ′ j are thereby divided naturally into t wo groups. The F e r mat p oints of a tria ngle may b e defined over the decima l num b ers as the p ersp ectors o f the ‘external and internal equila teral triang les’, and with the ab ove interpretation, these p oin ts exist also in this field. Ther e is another approach to their definitio n. The vertex bisectors at A 1 of A 1 A 2 A 3 int er sect A 2 A 3 at the points X 1 = [20 , 21] and Y 1 = [12 , 14]. The circle through these po in ts with cen ter the midpoint of X 1 Y 1 has equation ( x − 16) 2 + ( y − 6 ) 2 = 11 and is called an Ap ol lonius circle of A 1 A 2 A 3 . Ther e is also such a circle starting with A 2 , with equa tion ( x − 1 ) 2 + ( y − 14) 2 = 5 a nd o ne s ta rting with A 3 , with equatio n ( x − 4 ) 2 + ( y − 17) 2 = 1 7 . These three App ollonius circles int er sect at t wo p oin ts, called the iso dy nami c p oin ts of A 1 A 2 A 3 , given by S = [3 , 13] and S ′ = [0 , 16] . The Neub erg cubic passes throug h the tw o iso dynamic p oint s . The centres of the three Appollonius circle s are co lline a r, and lie on the Lemoine li ne with equation 16 x + 7 y + 1 = 0 . The isog onal co njugates of the iso dynamic p oint s are the F ermat p o in ts F = S ∗ = [19 , 13] and F ′ = ( S ′ ) ∗ = [13 , 16] . It may be chec ked that F is als o the cen tr e of p ersp ectivity b et ween A 1 A 2 A 3 and E 1 E 2 E 3 , while F ′ is the centre of p ersp ectivity b et ween A 1 A 2 A 3 and E ′ 1 E ′ 2 E ′ 3 . 10 It may b e remarked that in the decima l num b er plane, the F ermat p oints also hav e an interpretation in terms o f minimizing the sum o f the distanc e s to the vertices of the tr iangle, but this kind of s tatemen t cannot be exp ected to ha ve a simple a na log in universal geo metry . The Bro card line with equation 1 0 x + 10 y + 1 = 0 pa sses through the circumcenter C = [18 , 2 1] , the symmedi an p oint K = G ∗ = [10 , 6] a nd the t wo iso dynamic p oint s S and S ′ . It is p erp endicular to the Lemoine line. Quadrangles and Desmic structure Elliptic curves naturally give rise to interesting configur ations o f 12 p oints and 16 lines, called by John Conw ay Desmic (or linking) structure , where each line passes thro ugh three p oints and ea ch p oin t lies o n four lines (see [7]). T o desc r ibe this situation, b egin with a triangle AB C and tw o g eneric p oints P and Q. Then define A ′ = ( B P ) ( C Q ) B ′ = ( C P ) ( AQ ) C ′ = ( AP ) ( B Q ) A ′′ = ( B Q ) ( C P ) B ′′ = ( C Q ) ( AP ) C ′′ = ( AQ ) ( B P ) This insures that AB C and A ′ B ′ C ′ are pe r spective from so me p ersp ector D ′′ , that A ′ B ′ C ′ and A ′′ B ′′ C ′′ are per s pective fro m so me p ersp ector D and tha t A ′′ B ′′ C ′′ and AB C a re per spective from so me p ersp ector D ′ . F ur thermore the po in ts D , D ′ and D ′′ are collinear. Put ano ther way , tw o tria ng les which a re doubly p ersp ective a re triply p er- sp ectiv e (essentially a co nsequence of Pappus’ theor em). The v ar ious co llinea r- D D ' C ' ' B ' ' A ' ' D ' ' C ' B ' A ' A B C P Q Figure 3: Desmic 16 − 12 structur e 11 ities can b e r ecorded in terms of the array A B C D A ′ B ′ C ′ D ′ A ′′ B ′′ C ′′ D ′′ A triple of p oints from the array is collinear precisely w hen a) ea ch is fro m a different row, a nd b) if no D is inv olved each is from a different column, a nd c) if a D is involv ed, then the other tw o are b oth from the same column. An example of the for mer would b e A, B ′′ and C ′ , or C, B ′ and A ′′ . An example of the latter would be D ′ , B a nd B ′′ or D , D ′ and D ′′ . There a re then exactly 16 such collinear ities among these 12 p oin ts. Given a p o in t P on a cubic, there are in general four p oints X 1 , X 2 , X 3 and X 4 on the cubic, other than P , whose tang en ts pas s through P. Call these four po in ts a quadrangle of the Neub erg cubic, and more sp ecifically the quad- rangle to P . T o illustr ate this, we write X 1 , X 2 , X 3 , X 4 : P . Here ar e some q ua drangles for our cubic: A 1 , A 2 , A 3 , ∞ e : ∞ ∗ e I 1 , I 2 , I 3 , I o : ∞ e A 1 , A 2 , A 3 , C : [0 , 8] A 1 ∗ , A 2 ∗ , A 3 ∗ , O : [17 , 8] Given three co lline a r po in ts on a cubic, the asso ciated qua dr angles form a Desmic structure. Here are so me exa mples for our cubic A 1 A 2 A 3 ∞ e I 1 I 2 I 3 I 0 I 1 I 2 I 3 I 0 A 1 A 2 A 3 ∞ e E 1 E 2 E 3 [3 , 1 3] E ∗ 1 E ∗ 2 E ∗ 3 [19 , 13] A 1 A 2 A 3 ∞ e E ′ 1 E ′ 2 E ′ 3 [0 , 1 6] E ′∗ 1 E ′∗ 2 E ′∗ 3 [13 , 16] . W e see that ha ving r e c ognized an (a ffine) cubic curve as a Neuber g cubic of a tria ngle, we have lots of natura l and deep geometry that co nnects to the group mult iplica tion. A natur al questio n is: given a n elliptic curve can we find an affine view of it whic h is a Neub erg cubic? A nd if so , how can we classify such views, and use them to under stand elliptic curves? Such an approach oug h t to be esp ecially use ful in the conv enient lab ora tory provided by finite fields. 12 T angen t conics to an affine cubic Here is a quite different use of affine co ordinates in the study of an elliptic curve. F or mo re informatio n and exa mples inv olving tangent co nics, see [8]. Differen t metrical interpretations of tangent conics th us allows one to distinguish p o in ts on an affine cur v e from the nature of the tangent conic. Figure 4 gives a view of some tangent conics to [ x 0 , y 0 ] for the cur v e y 2 = x 3 − x ov er the decimal nu mber s. T angent co nics to p oin ts on the ‘egg’ are ellips e s , while others are either hyper bola s op e ning horizo n tally , a pair of lines, or h yp erb olas o pening vertically dep ending resp ectively on whether x 0 is less than, e q ual to, or g reater than q 2 3 √ 3 + 1 . Rather remark ably , these tang en t conics nowhere intersect. 5 2.5 0 -2.5 -5 5 2.5 0 -2.5 -5 x y x y Figure 4: T a ngen t co nics to y 2 = x 3 − x Theorem 5 O ver a field in which − 3 is not a squ ar e, any t wo tangent c onics of the affine curve y 2 = ax 3 + bx + c ar e either identic al or disjoint. Pro of. Recall that the ta ngen t conic to an affine curve is the part of the T aylor expansion of degree t wo or less (see [8, Chapter 19]). More s pecifically , to find the tangent conic to y 2 = ax 3 + bx + c at a p o in t A = [ x 0 , y 0 ] on it, first translate the curve by − A, yielding the equation ( y + y 0 ) 2 = a ( x + x 0 ) 3 + b ( x + x 0 ) + c or, after simplifica tion using the fact that A lies on the original curve, y 2 + 2 y y 0 − ax 3 − 3 ax 2 x 0 + x  − b − 3 ax 2 0  = 0 . Then take the q uadratic part: y 2 + 2 y y 0 − 3 ax 2 x 0 + x  − b − 3 ax 2 0  = 0 13 and translate this conic back by A, yielding ( y − y 0 ) 2 + 2 ( y − y 0 ) y 0 − 3 a ( x − x 0 ) 2 x 0 + ( x − x 0 )  − b − 3 ax 2 0  = 0 or after simplifica tion y 2 − 3 ax 2 x 0 + x  3 ax 2 0 − b  − ax 3 0 − c = 0 . T o find the in tersectio n b etw een tw o such ta ngen t conics y 2 − 3 ax 2 x 0 + x  3 ax 2 0 − b  − ax 3 0 − c = 0 y 2 − 3 ax 2 x 1 + x  3 ax 2 1 − b  − ax 3 1 − c = 0 take the differ e nce b etw een the t wo equa tions, which factor s as a ( x 1 − x 0 )  3 x 2 − 3 x ( x 0 + x 1 ) + x 2 0 + x 0 x 1 + x 2 1  . If x 0 = x 1 then the tangent conics coincide. Otherwise we get an intersection when the second qua dratic factor has a zero. But its disc r iminan t is 9 ( x 0 + x 1 ) 2 − 4 × 3  x 2 0 + x 0 x 1 + x 2 1  = ( − 3) ( x 1 − x 0 ) 2 and so if − 3 is not a square then there is no solution a nd so the tangent conics are disjoint. Note that the equation of the tang en t conic y 2 − 3 ax 2 x 0 + x  3 ax 2 0 − b  − ax 3 0 − c = 0 . can b e rewritten as y 2 −  ax 3 + bx + c  + a ( x − x 0 ) 3 = 0 . Figure 5 gives a v iew o f some tange nt conics for the curve y 2 = x 3 + x. The tangent conic to [0 , 0 ] is a par a bola , and other wise they are e ither hyperb olas op ening horizo ntally , a pair o f lines, or h y perb olas op ening vertically dep e nding resp ectively on whether x 0 is less than, e q ual to, or g reater than 1 3 √ 3 p 2 √ 3 − 3 . 5 2.5 0 -2.5 -5 5 2.5 -2.5 -5 x y Figure 5: T a ngen t co nics to y 2 = x 3 + x 14 Figure 6 shows the no dal cubic y 2 = x 3 + x 2 with tangent conic a t [ x 0 , y 0 ] given by y 2 + 3 xx 2 0 + x 2 ( − 3 x 0 − 1) − x 3 0 = 0 . W e get a par abo la when x 0 = − 1 / 3 , ellipses for x 0 less tha n that, hyp e r bola s op ening horiz o n tally till x 0 = 0 , when we get the pair of lines y = ± x, then hyperb olas op ening up wards. 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 x y Figure 6: T angent co nics to y 2 = x 3 + x 2 References [1] H. M. Cundy and C. F. Parry , Ge ometric al pr op erties of some Euler and cir cular cubics, Part 1 , J. Geom. 66 (1 999), 7 2-103. [2] H. M. Cundy and C. F. Parry , Ge ometric al pr op erties of some Euler and cir cular cubics, Part 2 , J. Geom. 68 (20 00), 58-75 . [3] H. M. Cundy a nd C. F. Parry , Some cubic curves asso ciate d with a triangle , J. Geom. 5 3 (199 5), 41-66. [4] J-P . Ehrmann and B. Gibe r t, S p e cial Iso cubics in t he T riangle Plane , av ail- able from http://pers o.w a nado o.fr/b ernard.gib ert/downloads.html . [5] C. Kimberling, T riangle Cent ers and Centr al T riangles , vol 1 29 of Congr es- sus Numerantium, Utilitas Mathematica Publishing, Inc, Winnipeg, Ma ni- toba, 199 8 . [6] C. Kimberling, Encyclop e dia of T riangle Centers , av a ilable at ht tp:// facult y .ev ansv ille.edu/ck6/encyclop e dia/ETC.html [7] W. Stothers , ‘Grassmann cubics and Desmic structures’, F o r um Geometri- corum 6 (2 006) 117- 138. 15 [8] N. J. Wildb erger, Divine Pr op ortions: Ratio n al T rigonometry to Universal Ge ometry , Wild Egg Bo o ks (http://wildegg.com), Sydney , 20 05. [9] N. J. Wildberger , ‘Affine and pro jective universal geometry’, arXiv:math.MG/06 12499 v 1 , 1 8 Dec., 2006. 16