1.1 Introduction Exercise

Transcription

1 Prt I The plne

2 1 Isometries 1.1 Introduction One prcticl im in Prt I is to equip the reder to build pttern-generting computer engine. The ptterns we hve in mind come from two min strems. Firstly the geometricl trdition, represented for exmple in the fine Moslem rt in the Alhmbr t Grnd in Spin, but found very widely. (See Figure 1.1.) Less bundnt but still noteworthy re the ptterns left by the ncient Romns (Field, 1988). The second type is tht for which the Dutch rtist M. C. Escher is fmous, exemplified in Figure 1.2, in which (stylised) motifs of living forms re dovetiled together in remrkble wys. Useful references re Coxeter (1987), McGillvry (1976), nd especilly Escher (1989). In Figure 1.2 we imitte clssic Escher-type pttern. The mgic is due prtly to the designers skill nd prtly to their discovery of certin rules nd techniques. We describe the underlying mthemticl theory nd how it my be pplied in prctice by someone climing no prticulr rtistic skills. The ptterns to which we refer re true plne ptterns, tht is, there re trnsltions in two non-prllel directions (opposite directions count s prllel) which move every submotif of the pttern onto copy of itself elsewhere in the pttern. A trnsltion is movement of everything, in the sme direction, by the sme mount. Thus in Figure 1.2 piece A cn be moved to piece B by the trnsltion represented by rrow, but no trnsltion will trnsform it to piece C. A reflection would hve to be incorported. Exercise The reder my like to verify tht, in Figure 1.1, two smllest such trnsltions re represented in their length nd direction by the rrows shown, nd determine corresponding rrows for Figure 1.2. These should be horizontl nd verticl. But there my be much more to it. More generlly, we ly bsis for understnding isometries those trnsformtions of the plne which preserve distnce nd look for the esiest wys to see how they combine or cn be decomposed. Exmples re trnsltions, rottions nd reflections. Our pproch is essentilly geometricl. An importnt tool is the ide of symmetry of plne figure; tht is, n isometry which sends every submotif of the pttern onto nother of the 3

3 4 Isometries Figure 1.1 Vrition on n Islmic theme. For the originl, see Critchlow (1976), pge 112. The rrows indicte symmetry in two independent directions, nd the pttern is considered to continue indefinitely, filling the plne. Figure 1.2 Plne pttern of interlocking birds, fter M. C. Escher. sme size nd shpe. (The trnsltions we cited for Figure 1.2 re thus symmetries, but we reiterte the ide here.) For exmple, the hed in Figure 1.3() is symmetricl bout the line AB nd, corresponding to this fct, the isometry obtined by reflecting the plne in line AB is clled symmetry of the hed. Of course we cll AB line of symmetry.in Figure 1.3(b) the isometry consisting of one third turn bout O is symmetry, nd O is clled 3-fold centre of symmetry. In generl, if the 1/n turn bout point A (n mximl) is symmetry of pttern we sy A is n n-fold centre of symmetry of the pttern.

4 1.1 Introduction 5 () Figure 1.3 (b) The key ide is tht the collection of ll symmetries, or symmetry opertions, of figure form group G (see Section 2.5). Here this mens simply tht the composition of ny two symmetries is nother, which is sometimes expressed by sying tht the set of symmetries is closed under composition. Thus, for Figure 1.3() the symmetry group G consists of the identity I (do nothing) nd reflection in line AB. For Figure 1.3(b), G consists of I,1/3 turn τ bout the centrl point, nd 2/3 turn which my be written τ 2 since it is the composition of two 1/3 turns τ. In fct, every plne pttern flls into one of 17 clsses determined by its symmetry group, s we shll see in Chpter 5. Tht is, provided one insists, s we do, tht the ptterns be discrete, in the sense tht no pttern cn be trnsformed onto itself by rbitrrily smll movements. This rules out for exmple pttern consisting of copies of n infinite br. Exercise Wht symmetries of the pttern represented in Figure 1.1 leve the centrl point unmoved? Section 6.3 on tilings or tesselltions of the plne is obviously relevnt to pttern genertion nd surfce filling. However, I m indebted to Aln Fournier for the comment tht it touches nother issue: how in future will we wish to divide up screen into pixels, nd wht should be their shpe? The nswer is not obvious, but we introduce some of the options. See Ulichney (1987), Chpter 2. A remrkble survey of tilings nd ptterns is given in Grünbum nd Shephrd (1987), in which lso the origins of mny fmilir nd not-so-fmilir ptterns re recorded. For study of isometries nd symmetry, including the non-discrete cse, see Lockwood nd Mcmilln (1978), nd for connection with mnifolds Montesinos (1987). Now, plne pttern hs smllest replicting unit known s fundmentl region F of its symmetry group: the copies of F obtined by pplying ech symmetry opertion of the group in turn form tiling of the plne. Tht is, they cover the plne without re overlp. In Figure 1.2 we my tke ny one of A, B, C s the fundmentl region. Usully severl copies of this region form together cell, or smllest replicting unit which cn be mde to tile the plne using trnsltions only. Referring gin to Figure 1.2, the combintion of A nd C is such cell.

5 6 Isometries Section 6.4, the conclusion of Prt I, shows how the ide of fundmentl region of the symmetry group, plus smll number of bsic generting symmetries, gives on the one hnd much insight, nd on the other compct nd effective method of both nlysing nd utomting the production of ptterns. This forms the bsis of the downlodble progrm polynet described t the end of Chpter 6. This text contins commercil possibilities, not lest of which is the production of books of ptterns nd tech-yourself pttern construction. See for exmple Oliver (1979), Devney (1989), Schttschneider nd Wlker (1982), or inspect smple books of wllpper, linoleum, crpeting nd so on. We conclude by noting the ppliction of plne ptterns s test bed for techniques nd reserch in the re of texture mpping. See Heckbert (1989), Chpter Isometries nd their sense We strt by reviewing some bsic things needed which the reder my hve once known but not used for long time The plne nd vectors Coordintes Points in the plne will be denoted by cpitl letters A, B, C,...It is often convenient to specify the position of points by mens of Crtesin coordinte system. This consists of (i) fixed reference point normlly lbelled O nd clled the origin, (ii) pir of perpendiculr lines through O, clled the x-xis nd y-xis, nd (iii) chosen direction long ech xis in which movements re mesured s positive. Thus in Figure 1.4 the point A hs coordintes (3, 2), mening tht A is reched from O by movement of 3 units in the positive direction long the x-xis, then 2 units in the positive y direction. Compre B ( 2, 1), reched by movement of 2 units in the negtive (opposite to positive) x-direction nd 1 unit in the y-direction. Of course the two component movements could be mde in either order. Figure 1.4 Coordinte xes. The x-xis nd y-xis re lbelled by lower cse x, y nd often clled Ox, Oy. Positive directions re rrowed.

6 1.2 Isometries nd their sense 7 (i) (ii) (iii) Figure 1.5() Directed nd undirected line segments. Lines The stright line joining two points A, B is clled the line segment AB. Asinthe cse of coordintes, we need the technique of ssigning to AB one of the two possible directions, giving us the directed line segments AB or BA, ccording s the direction is towrds B or towrds A. This is illustrted in Figure 1.5(). Length AB denotes the length of the line segment AB, which equls of course the distnce between A nd B. Sometimes it is useful to hve formul for this in terms of the coordintes A ( 1, 2 ) nd B (b 1, b 2 ): AB = (b 1 1 ) 2 + (b 2 2 ) 2. (1.1) Exercise Prove Formul (1.1) by using the theorem of Pythgors. Vectors A vitl concept s soon s we come to trnsltion (Section 1.2.2()), vector is ny combintion of distnce, or mgnitude, nd direction in spce. (For now, the plne.) Thus every directed line segment represents some vector by its direction nd length, but the sme vector is represented by ny line segment with this length nd direction, s depicted in Figure 1.5(b). = AB = CD Figure 1.5(b) Directed line segments representing the sme vector. A letter representing vector will normlly be printed in bold lower cse thus:, nd lthough the directed line segment AB of Figure 1.5(b), for exmple, hs the dditionl property of n initil point A nd end point B we will sometimes llow ourselves to write for exmple = AB = CD = b, to men tht ll four hve the sme mgnitude nd direction. With the length (mgnitude) of vector x denoted by x, the sttement then includes = AB = CD = b. Also it is often convenient to drop the letters, in digrm, leving n rrow of the correct length nd direction thus:. The ngle between two vectors mens the ngle between representtive directed line segments AB, AC with the sme initl point. Components nd position vectors By contrst with the previous prgrph, we my stndrdise on the origin s initil point, representing vector by segment OA. Then

7 8 Isometries we write = ( 1, 2 ), B where 1, 2 ply double role s the coordintes of point A, nd b the components of vector. Further, since now defines uniquely the A position of the point A, we cll the position vector of A (with respect O to origin O). Similrly point B hs position vector b = (b 1, b 2 ), nd so on (Figure 1.6). Alterntively we my write r A for the position Figure 1.6 vector of A. Of course x, y will remin lterntive nottion for the coordintes, especilly if we consider vrible point, or n eqution in Crtesin coordintes, such s x = m for the line perpendiculr to the x-xis, crossing it t the point (m, 0). Sclr times vector In the context of vectors we often refer to numbers s sclrs, to emphsise tht they re not vectors. We recll tht the mgnitude or bsolute vlue of sclr λ is obtined by dropping its minus sign if there is one. Thus λ = λ if λ<0, otherwise λ =λ. If is vector nd λ sclr then we define λ s the vector whose mgnitude equls the product λ, nd whose direction is tht of if λ>0 nd opposite to if λ<0. If λ = 0 then we define the result to be the nomlous vector 0, with zero mgnitude nd direction undefined. As in the illustrtion below, we usully bbrevite ( 1) to,( 2) to 2, nd so on. Also (1/c) my be shortened to /c (c 0). Exmples (3/2) Adding vectors To dd two vectors we represent them by directed line segments plced nose to til s in Figure 1.7(). Subtrction is conveniently defined by the sclr times vector schem: b = + ( b), s in Figure 1.7(b). Digrms re esily drwn to confirm tht the order in which we dd the vectors does not mtter: + b = b + ( prllelogrm shows this), nd + (b + c) = ( + b) + c. C + b b b B b A () (b) Figure 1.7 Finding () the sum nd (b) the difference of two vectors by plcing them nose to til.

8 Rules Let, b be the position vectors of A, B. Then 1.2 Isometries nd their sense 9 + b = ( 1 + b 1, 2 + b 2 ), λ = (λ 1,λ 2 ), AB = b. (1.2) (1.2b) (1.2c) Proof For (1.2) we refer to Figure 1.7(), nd imgine coordinte xes with point A s origin, tking the x-direction s due Est. Then 1 + b 1 = (mount B is Est of A) + (mount C is Est of B) = mount C is Est of A = first component of C. The second components my be hndled similrly. Eqution (1.2b) is left to the reder. To estblish (1.2c), we note tht the journey from A to B in Figure 1.6 my be mde vi the origin: AB = AO + OB = ( ) + b. The section formul The point P on AB with AP : PB = m : n (illustrted below) hs position vector p given by p = 1 (mb + n). (1.3) m + n Often clled the section formul, this is extremely useful, nd hs the virtue of covering cses such s (i), (ii) shown below in which P does not lie between A nd B. (i) 2 1 This mens tht AP nd PB re in opposite directions nd so m, n hve opposite signs. Thus in A B P Cse (i) AP = 3PB nd we my write AP : PB = 3: 1 (or, eqully, 3 : 1), whilst Cse (ii) entils (ii) 3 2 AP = (3/5)PB, orap : PB = 3:5. P A B This sid, (1.3) is esily proved, for nap = mpb, so by (1.2) n(p ) = m(b p), which rerrnges s (m + n) p = mb + n. Exercise Drw the digrm for proving (1.2), mrking in the components of nd b. Appliction 1.1 This is hndy illustrtion of the use of vectors to prove well-known fct we will need in Chpter 6: the medins of tringle ABC ll pss through the point G (centre of grvity), whose position vector is g = 1 ( + b + c). 3 A To prove this, lbel the midpoints of the sides by F 2 E D, E, F s shown. By (1.3), D hs position vector d = (1/2)(b + c). So, gin by (1.3), the point tht divides medin AD in the rtio 2 : 1 hs position vector G 1 B (1/3)(2d + 1), which equls (1/3)( + b + c) on substituting for d. But this expression is symmetricl in D C, b, c, nd so lies on ll three medins, dividing ech in the rtio 2 : 1. Note The use of components gives n importnt wy to clculte with vectors, which will come into its own in Chpter 7. Before then, our rguments will be mostly geometricl,

9 10 Isometries with components s tool in some exercises. However, we give both geometricl nd coordinte proof of (1.14) little further on, which the reder my find interesting for comprison purposes t tht point. Exercise Use position vectors nd (1.3), which pplies eqully in 3-spce (indeed, in ny dimension), to prove the following fcts bout ny tetrhedron ABCD. (i) The four lines joining vertex to the centroid of its opposite fce re concurrent t point G which divides ech such line in the rtio 3 : 1, (ii) the three lines joining midpoints of pirs of opposite edges ll meet in G Isometries Definition 1.2 A trnsformtion g of the plne is rule which ssigns to ech point P unique point P g,orp, clled the imge of P under g. (Note tht P g does not men P to the power of g.) We think of g s moving points round in the plne. We lso cll g mp or mpping of the plne onto itself, nd sy g mps P to P.Anisometry of the plne is trnsformtion g of the plne which preserves distnces. Tht is, for ny two points P, Q: P Q = PQ. (1.4) The reder is dvised not to think first of the formul (1.4) but to strt from the ide of isometries preserving distnce. Of course the sme definition is pplicble to 3-spce or even higher dimensions, nd we pursue this in Prt II (Chpters 7 8). An importnt first consequence of the definition is s follows. An isometry g trnsforms stright lines into stright lines, nd preserves the (unsigned) size of ngles. (1.5) Proof of (1.5). We refer to Figure 1.8. It suffices to show tht if points A, B, C lie on stright line, then so do their imges A, B, C. Suppose B lies between A nd C. Then elementry geometry tells us tht AC = AB + BC, B A D C E F C' Figure 1.8 Points A, B, C on stright line, tringle DEF, nd their imges under n isometry. The mgnitude of ngle φ is preserved. A' B' D' E' F'

10 1.2 Isometries nd their sense 11 Figure 1.9 The trnsltion T is n isometry s P Q = PQ lwys. nd therefore, from condition (1.4) of n isometry, the sme holds with A, B, C replced by A, B, C. Consequently, A, B, C lso lie on stright line, nd the first ssertion of (1.5) is estblished: stright lines re trnsformed to stright lines. Now, given this, let us view the ngle φ between two lines s the vertex ngle of some tringle DEF, trnsformed by g into nother tringle, D E F which must be congruent to DEF becuse the lengths of the sides re unchnged by g. Thus the vertex ngle is unchnged, lying side considertions of sign. This completes the proof. Nottion 1.3 The following re convenient t different times for referring to the imge of P under trnsformtion g: (i) P, (ii) P g, (iii) g(p). We shll explin in Section the significnce of our choosing (ii) rther thn (iii). In ech cse the nottion llows us to replce P by ny figure or subset F in the plne. Thus figure F g consists of the imges of the points of F, orf g ={x g : x F}. For exmple, if F is the lower plm tree in Figure 1.9, with g = T (see previous pge) then F g is the upper. The heds of Figure 1.11 provide further exmple. Three types of isometry At this juncture it is pproprite to discuss the three most fmilir types of isometry in the plne. The remining type is introduced in Section () Trnsltion For ny vector the trnsltion T of the plne is the trnsformtion in which every point is moved in the direction of, through distnce equl to its mgnitude. Thus PP = (in mgnitude nd direction). To show tht T is n isometry, suppose it sends nother point Q to Q. Then = QQ, so tht PP nd QQ re prllel nd equl, mking prllelogrm PP Q Q. Hence, by n elementry theorem in geometry, P Q = PQ, nd T hs indeed preserved distnces. Nottion 1.4 P is lso clled the trnslte of P (by T ). Notice tht T sends x to x +, when we identify point X with its position vector x. More geometriclly, if = PQ we my write unmbiguously T = T PQ, the trnsltion which tkes P to Q. (b) Rottion As illustrted in Figure 1.10, let the trnsformtion R A (φ) be R A (φ) = rottion bout the point A through the ngle φ.