On the number of inscribed squares of a simple closed curve in the plane

ON THE NUMBER OF INSCRIBED SQUARES IN A SIMPLE CLOSED CUR VE IN THE PLANE STRASHI MIR G. POPV ASSILEV Abstra ct. W e sho w that for every p ositive integer n there is a simple closed curve in the plane (which can b e tak en infinitely differen tiable and conv ex) whic h has exactly n inscrib ed s quares. Introduction It is an op en problem if for ev ery simple closed curv e in the plane there are four p oints from the cur v e that form th e v ertices of a squ are. Suc h a square is calle d inscrib ed in the curv e (though it is n ot requir ed that it is con tained in the region b ound ed b y the cur v e). The problem is simp ly stated, old, and has only p artial p ositiv e solutions. See [5] for a list of pap ers, and for comments. The present note a nsw ers in the negativ e wh at w e in terpret as a conjecture p osed b y Jaso n Can tarella on his w eb page [2]. The w eb site c ommen ts on his jo in t work with Elizab eth Denne and John McCleary on th is problem. Their resu lts ha ve b een announced in [3]. The author has recen tly b een informed by Elizab eth Denne and Jason Cant arella that the preprint presen ting the results announ ced in [3] is not yet ready to b e released. Our recen t discussion with Jason C an tarella and Eliza b eth Denne o n s ome of the idea s presen ted in [2] and [4] has b een helpf ul to the auth or, y et th e follo wing stat emen t made at the web site [2] h as not b een y et clarified: ‘Our results prov e that there are an o dd n umber of squares in any simp le c losed curv e w hic h is differen tiable or “not to o rough”.’ Apparent ly the exact statemen t of the ab ov e resu lt w ould app ear in the forth- coming p ap er by Jason Can tarella, Elizab eth Denne and John McCleary . The pur p ose of the p resen t n ote is the pro of of the follo wing. Theorem 1. F or every p ositive inte ger n ther e is a simple clos e d curve in the plane (which c an b e taken infinitely differ entiable and c onvex) which has exactly n inscrib e d sq u ar es. This seems to indicate (though w e provide some “evidence” only , and n o com- plete pro of ) that the follo w ing conjecture ab out th e n umb er of inscrib ed squares of an immers ed in the plane curve (self in tersections allo wed) made at the same w eb site, is not v alid, if only d ifferen tiabilit y is assumed: ‘W e migh t guess that the num b er of squares is equal to S t + ( J + − J − ) + 1 mo d 2.’ As indicated in [2], S t , J + and J − denote the in v ariants of the curve called strangeness, p ositiv e ju mp, and negativ e jump, int ro duced by Arnold. S ee [1]. Date : Octob er 26, 2008. Key wor ds and phr ases. simple closed curve, inscribed square, square peg problem. 1 2 STRASHIMIR G. POPV ASSILEV Figure 1. 1. How to con trol the numb er of inscr ibed squares First w e ske tc h the construction of an infi nitely different iable simple closed curve in the plane that has exactly t wo in scrib ed squares. Clearly t he unit circle has infi nitely man y inscrib ed s q u ares. On the other hand it is easy to mo d ify the unit circle to obtain a (non-differen tiable) simple closed curv e wh ic h has exac tly t wo in s crib ed squares. The construction is sho w n on Figure 1, left. The arc determined b y cen tral angles 5 π 4 and 7 π 4 is remo ved from the unit circle, and replaced by the semi-circle y = q 1 2 − x 2 − 1 √ 2 . The reader ma y verify that there are only t w o inscrib ed squares, as sh o wn on Figure 1, left. What looks lik e the equal sides (t hough they are not line segmen ts) of an isosceles triangle on that picture is the s et of p oin ts, that are endp oin ts of the base of a square s u c h that the top side of the square has endp oint s that are symmetric ab out the y -axis, b elong to the unit circle, and ha ve y -co ord inates ≥ 1 √ 2 . (Th e equation for the tw o equal sides of that triangle is y = √ 1 − x 2 − 2 | x | , − 1 √ 2 ≤ x ≤ 1 √ 2 .) T o get a differen tiable example we replace that arc with th e graph of y = − p 1 − x 2 + c exp  −  . 02 ( x + 1 √ 2 ) 2 + . 02 ( x − 1 √ 2 ) 2   ( ∗ ) Notice that this graph for p ositiv e and not t o o big v alues of c intersects the unit circle only at the end-p oin ts of the arc that w as remo v ed. Among these v alues of c , for larger c th e graph in tersects eac h of the t w o equal sides of that isosceles triangle in t wo p oin ts (w e do not coun t the endp oin ts of the arc that w as remov ed). F or sm aller v alues of c th e graph d o es n ot int ersect the equal sides of the triangle (except at th e endp oin ts of the remov ed arc). Therefore for a certain v alue of c (appro xim ately 1 . 18264) on eac h side of the triangle there is a unique p oin t that b elongs to the graph (apart fr om the end p oint) . W e sketc h the pro of that the t w o inscrib ed squares sh o wn on Figure 1, right, are the on ly ones. The ab ov e considerations sho w that these t w o squ ares are the only inscrib ed squares that ha ve a horizon tal sid e. Assume S is an inscrib ed square with no horizon tal side. If S h as three vertice s on the 3 4 -circle (i.e. on the un ion of arcs g AB , g B C , g C D , see Fig.2 ) then it follo ws that the fou r th v ertex would b e on the unit circle, on the arc that w as r emo ved from our cu rv e, a con tradiction. Let g D A denote th e grap h of ( ∗ ). Let the vertices of S b e E , F , G, H (in this order) and consider the case when E , F b elong to the 3 4 -circle, and G, H b elong to g D A . W e only consider tw o t yp ical cases. ON THE NUMBER OF INSCRIBED SQUARES I N A SIMPLE CLOSED CUR VE ... 3 Figure 2. Case 1. E b elongs to g B C , and H b elongs to the part of g D A that is below the line segmen t D A (Fig.2, ( a )). Let H ′ b e the in tersection of the line through E , H with the arc remo v ed from the unit circle. Then H ′ and F are diametrically opp osite (since the angle at E is righ t). He nce F b elongs to g B C , an d if d d enotes the distance fu nction, then d ( E , F ) < √ 2 < d ( E , H ), a con tr adiction. Case 2. Assume that S is just a rectangle, not necessarily a square, and that F E is a line segmen t with p ositiv e slop e and endp oin ts on the 3 4 -circle, and G, H are o n g D A with H ab o ve D A (Fig.2, ( b )). Let l be th e perp endicular bisector of E and F . Then l go es thr ough the origin O and th rough the midp oin t M of G and H . Let k b e the ray starting at M and going though G , and let G ′ b e the “first” p oint on k that b elongs to g D A . T he reader ma y v erify that d ( M , G ′ ) > d ( M , H ) and h ence G do es not b elong to g D A , a contradictio n . 2. How to obt ain exactl y n sq uares The next set of examples is also based on th e idea that w e ma y r eplace a c ertain arc of the unit circle. It will ev entually lead to a d ifferen tiable conv ex curv e with a num b er of ins crib ed squ ares sp ecified in adv ance. The idea is v ery simple, and the pr o ofs are ea sy (though might be tec hnical) so w e omit some of the details. Start with the unit square and this time remo ve the arc [ − π 4 , π 4 ]. F or con ve nience w e iden tify any real num b er P with the corresp onding p oin t on the unit circle, if w e treat P as an angle. Pic k an y P ∈ ( − π 4 , π 4 ) a nd c onnect P to A = − π 4 , and P to B = π 4 , with a circular arc of radius close to 1 bu t less than 1. Clearly the resu lting simple closed cur v e has only tw o inscrib ed squares, as sho wn on Fig.3, ( a ). Of cour s e if w e add circular arcs of smaller than 1 radius then we do not get a differen tiable curve, b ut we m a y instead add an arc of the form (p olar co ordin ates): r ( θ ) = 1 + c exp  −  . 02 ( θ − U ) 2 + . 02 ( V − θ ) 2   in order to connect an y giv en pair of points U and V on the unit circle, w here c > 0. F or example, on Fig.3, ( b ), th e p oints P and B are conn ected with an arc of the ab o ve t yp e, with c = 0 . 05. Clearly this approac h results in an infinitely differen tiable curve. If w e select th e constant c small enough then the signed curv ature w ould b e p ositiv e for all θ ∈ [ U, V ], and therefore the (region boun ded by the) simple closed curv e obtained in this mann er w ould b e con vex. W e can pick an y finite n umb er of p oint s b etw een A an d B on the unit circle and replace the consecutiv e unit circle arcs that connect these 4 STRASHIMIR G. POPV ASSILEV Figure 3. p oints with arcs of the t yp e describ ed ab o v e, and sin ce exactly one inscrib ed squ are w ould corresp ond to eac h of these p oints we ma y o btain an infinitely differen tiable, con vex s imple closed cur v e with exactly n inscrib ed squares, for any p ositive integ er n give n in ad v ance, as stated in Theorem 1. See Fig.2, ( b ). 3. On the role of S t , J + and J − In this section w e ind icate a p ossible pro of of the follo win g conjecture. Conje ctur e 2 . Give n an y imm ersed curve T in the plane, there is a p ositiv e in teger m s u c h that for eve ry n ≥ m th er e is an immersed curve T n whic h has the same v alues of S t , J + and J − as T , and such that T n has exactly n inscrib ed squares. Moreo v er there is k (indep enden t of n ) such that all bu t k man y of the inscrib ed squares of T n ha ve the p rop erty that their v ertices app ear in the same order in whic h they app ear on T n . The idea is the follo w in g. Start w ith an immersed curv e T (e.g. the one sho w n in Fig .3, ( c ), in the middle of the circle ). Pull one of the lo ops of T and wrap it around the unit circle, and at the s ame time make the rest of T muc h smaller, so that we ha v e a v ery small cop y of T , v ery clo se to B , as shown in Fig.3, ( c ), except for the lo op that is wrapp ed around the u nit circle. Call the resulting cur v e T ′ . More p recisely we assume that a p oint P ∈ ( − π 4 , π 4 ) has b een fi x ed , the un it circle arcs from A to P , and from P to B h a ve b een replaced by arcs of the t yp e describ ed ab o ve, and then T ′ has b een formed by wrapping one of its loops aroun d, so that, except for this lo op, a ve ry sm all (top ologica l) copy of T remains v ery close to B , and “b et we en” P and B . W e also assum e that T ′ is differen tiable. Clearly T and T ′ ha ve the same v alues for S t , J + and J − . W e will giv e a pro of of our conject ure based on the fol lo wing genericit y assump tion (GA), whic h we le a ve without p r o of. (W e d o not know ho w to prov e it, but we b eliev e it is correct.) Genericity Assumpt ion 3 . T he ab o v e transformation of T to T ′ can b e done in suc h a wa y that T ′ has only fin itely many inscrib ed squares. No w in ord er to pro v e our co njecture based on our GA, let m b e the finite n um b er of inscrib ed squares of T ′ . Let k be the num b er of them for whic h the v ertices app ear in order d ifferen t from the order in w hic h they app ear on T ′ . W e can pic k p oin ts Q on the un it circle, b et w een A and P , one at a time, replacing an arc of the t yp e describ ed a b ov e with t wo smal ler arcs, so that e v ery time a new inscrib ed squ are w ith one v ertex at the new p oin t Q w ould b e in tr o duced, and no ON THE NUMBER OF INSCRIBED SQUARES I N A SIMPLE CLOSED CUR VE ... 5 other inscrib ed squares would b e introdu ced. No tice that the new squares hav e v ertices whic h ap p ear in th e same order as in T ′ (see Fig.3, ( c )). This completes the pro of. Referen ces 1. V.I. Arnold, Plane curves , their invariants, p er estr oikas, and classific ations , Singularities and Bifurcations, (V.I. Arnold ed.), Adv . Sov. Math. v ol.21, (1994), pp.39-91. 2. Jason Cantarella, Squar e Pe g pr oblem , w eb si te http://w ww.jas oncanta rella.com/w ebpage/index.php ?title=Square P eg problem 3. Jason Cantarel la ∗ , Elizabeth D enne, and John McCleary , New R esults on the “Squar e Pe g” pr oblem , Abstract 1020-54-119, 2006 F all C entral Section Meeting Cincinnati, OH, Octob er 21-22, 2006 (Saturda y - Sunday) Meeting #10 20. 4. Elizab eth Denne, Inscrib e d squar es: Denne sp e aks , 31Aug07, web site http://quomodo cumque.wordpress.co m/2007/0 8/31/inscribed-sq u ares-denne-sp eaks/ 5. Igor P ak, The discr ete squar e p e g pr oblem , a rXiv:0804.06 57 v1 [math.MG] 4 Apr 2008. The City College of New Y ork, 160 Convent A venue, New York, NY 10031 E-mail addr ess : strash.po p@gmail.c om