Te oin-toug-te-ubbe tik: an elastially stabilized invagination Fanlong Meng Institute of Teoetial Pysis, Cinese Aademy of Sienes Pone: 86-15001345869 flm@itp.a.n Masao Doi Cente fo Soft Matte Pysis and its Appliations, Beiang Univesity Pone: 86-10831461 masao.doi @buaa.edu.n Zongan Ouyang Institute of Teoetial Pysis, Cinese Aademy of Sienes Pone: 86-6554467 oy@itp.a.n Xiaoyu Zeng Depatment. of Matematial Sienes, Kent State Univesity Pone: 1-330679089 zeng@mat.kent.edu Pete Palffy-Muoay Liquid Cystal Institute, Kent State Univesity Pone:1-33067604 mpalffy@kent.edu Abstat: A spetaula tik of lose-up magiians involves te appaent passing of a oin toug a ubbe seet. Te magi is based on te unusual elasti esponse of a tin ubbe seet: te fomation of an invagination, stabilized by fition and elastiity, wi olds te oin. By pessing on te oin, te invagination beomes unstable, and te oin is eleased. We desibe te defomation analytially using a simple Hookean desiption, and examine te stability of te invagination. We finally ompae te pedition of te Hookean analysis wit numeial solutions of te neo-hookean model, and povide a bief ommentay on te oigins of te tik. Keywods: magi, elastiity, invagination, ubbe dam, oop stess, apstan, stability. Matematis subjet lassifiation: 74B10, 74B0
1. Intodution One of te most impessive magi tiks wi an be pefomed witout any speial skills on te pat of te magiian is te oin-toug-te-ubbe tik (CTRT), wee an odinay oin appeas to pass toug a seet of odinay ubbe. In its simplest fom, te tik poeeds as follows. A tin ubbe seet, wit a oin on top, is plaed so as to ove te opening of an empty up. Te edges of te seet ae eld against te sides of te up eite by te magiian, o a ubbe band. A membe of te audiene is ten invited to pus down on te oin wit is/e finge. As e/se does so, te ubbe stetes a little, and ten, suddenly, te oin appeas to go toug te ubbe, and falls to te bottom of te up. Tis poess is illustated in Figs. 1 and. Fig. 1. Two pespetives of a latex ubbe dental dam wit a US quate on top of a glass beake. a. b.. Fig.. Wen pused down, te oin magially goes toug te ubbe, and falls to te bottom of te beake. Te tik is emakably effetive, te penomenon appeas to be eal magi. A bief istoy is given in te Appendix. Today, te CTRT is demonstated on youtube [1]; a vesion is used to make smat pone oves using balloons []. We ave also been able to pefom te CTRT wit a speial glass mable instead of a oin, as well as wit a vaiety of ylindial objets.
Te explanation of te magi is simple. Te ubbe is plaed on top of te oin, and is steted tigtly; so tigtly tat te ubbe beomes nealy tanspaent. Tis allows te illusion tat te oin is on top of, ate tan undeneat, te ubbe. Ten, te steted ubbe is allowed to elax unde te oin, ontating and foming an invagination wi olds te oin in plae below te ubbe on top. Tis is illustated in Fig. 3, sowing te oin patially enseated by te ubbe fom above, fom te bottom and fom te side. a. b.. Fig. 3 Images of te invaginated oin fom above, fom below and fom te side. Te eal magi is in te fomation of te poket - of te invagination tat olds te oin in plae. Te CTRT an be pefomed wit latex seets, su as dental dams, o ondoms o balloons. Invaginations in membanes ae ommon in biology. Tey an be ougly divided into two lasses. Te fist is wee ell membanes patially envelop and entap a vaiety of objets, anging fom pokayoti oganelles [3] to domain poteins [4]. In tese instanes, te invagination is diven by attative sufae foes [5], o uvatue of te membane [6-8]. Noneteless, elastiity makes impotant ontibutions wen etain poteins, su as latin, ae involved in endoytosis [9]. Failed latin mediated endoytosis wen, pio to sission, te ontents of te vesile say a vius is expelled, esembles somewat te expulsion of te oin fom te invagination in te ubbe seet. Te seond is wee invagination ous in goups of ells foming epitelial layes. Hee elastiity and ontatile stesses ae tougt to play impotant oles. Meanial models ave a long istoy [10]; a eent eview [11] igligts uent pysial models of mesodem invagination. Altoug we believe tat te pysis undelying te CTRT saes ommon featues wit invaginations in biology, establising a diet onnetion is beyond te sope of tis pape. Te penomenon of te ubbe patially enveloping te oin in CTRT is te fist instane of an invagination stabilized solely by ubbe elastiity and fition tat we ae awae of. Altoug tee ae some onnetions wit avitation[1], te penomenon in CTRT is fundamentally diffeent. Te aim of tis pape is to desibe, in simple tems, te pysis undelying elastiity stabilized invagination in elasti membanes.
. Modeling te invagination A wide vaiety of ontinuum models exist to desibe ubbe elastiity, wee te stain enegy density depends on invaiants of te left Cauy-Geen stet tenso [13]. One example is te geneal ypeelasti model poposed by Rivlin, wee, fo inompessible mateials, te enegy density as te fom [14] R p pq ( 1 3 3) ( 1 1 3 3 q 3), (1) pq, 0 W C wee te i 's ae pinipal stetes, and, fo inompessible mateials, 1 3 1. Te pinipal stetes ae obtained fom te eigenvalues of te stet tenso [13]. Te defomation an be desibed by R (), wee is te position of a point in te undefomed ubbe and R () is its position afte te defomation. In ou poblem, due to symmety and sine te seet is tin ompaed to its lateal dimensions, we assume tat R ( R ( ),0, R (, z)), wee, and z ae te usual ylindial oodinates. If we assume fute tat R / is small, ten te pinipal stetes ae and z R 1, () R, (3) R z 3. (4) z Rate tan stating wit an auate non-linea model to undestand ow te invagination is stabilized, tat is, to undestand ow te ubbe gips te oin, we onside a simple Hookean model of te seet. Te advantage of tis simple model is tat it allows an analyti desiption of te defomation, and so gives eady insigts into te undelying pysis. In spite of its simpliity, te simple Hookean model gives te qualitatively oet beavio, as an be seen fom ompaisons wit numeial solutions of te moe ealisti neo-hookean model in Set... z.1 Te Hookean model We begin ou analysis by assuming small defomations, wee i 1. Witing 3 in tems of 1 and and expanding te enegy density of Eq. (1) in te viinity of 1 1and 1, we obtain, to lowest ode, 10 01 1 1 W 4( C C )[( 1) ( 1) ( 1)( 1)], (5) R wee we take te i 's to be positive [15]. Sine te enegy is now quadati in te stains, tis is te Hookean model; we believe all ypeelasti models edue to tis fom in te limit of small stains. Beause of its tatability, we examine te peditions of tis Hookean model, even toug typial stains in te CTRT ae not small. In Set.., we ompae peditions of te Hookean model wit tose of te moe ealisti but less tatable neo-hookean model.
We assume tat ou undefomed ubbe seet is a ylindial disk of adius m and tikness 0. Te total elasti enegy, fom Eq. (5), is R R F 4 G (( sr 1) ( 1) ( sr 1)( 1)) d, wee G ( C10 C01) [13] is te sea modulus. We ave dopped te subsipt on R, R' R /, and m 0 (6) 0 s sgn( R ) needed to maintain te positivity of. Te defomations of ou ubbe seet ae fully desibed by R (); te allowed defomations ae tose wi minimize te enegy F. In tis desiption, we assume tat te bending enegy is negligible, and te seet may be folded witout any enegy ost. Te Eule-Lagange (E-L) equation is R 3 R R (1 s ) 0, (7) wi is linea, ene solutions wi satisfy te bounday onditions ae unique. Te solution of te E-L equation is B 3 R A (1 s ) ln, (8) 4 and te onstants of integation A and B ae to be detemined fom bounday onditions. It is useful to onside te tensile foes pe lengt in te ubbe seet; tese ae in te adial, and 1 R T T0 s (sr 3) (9) R 1 1 R T T0 ( sr 3) (10) sr in te tangential dietions, and ae te adial and tangential stesses, T0 4G0, and 0 z 0 (11) sr R is te tikness of te defomed ubbe seet. In equilibium, te net foe on any pat of te ubbe seet must be zeo. If te adial stess vaies in te adial dietion, te ange must be balaned by te tangential stess, alled, in tis ase, te oop stess. On an element of te seet sown in Fig. 4, te net adial foe is d( T R) d, wile te net tangential foe ating on te element in te adial dietion is T drd. Fig. 4. Semati sowing foe balane via oop stess in an annula egion of te steted ubbe T at R T T T at R dr bottom view
Equating te net adial and net tangential foes in te adial dietion gives d( TR ) T, (1) dr wi is just te ondition tat te divegene of te stess in te bulk vanises. Substitution fo te tensile foes pe lengt gives Eq. (7); te E-L equation is te ondition fo foe balane..1.1 Solutions Desibing te Defomation It is onvenient to wok wit dimensionless quantities, we teefoe expess all lengts in units of te oin adius, and stesses in units of te modulus G. If te ubbe is steted so tat points on a ile of adius in te unsteted seet end up on te iumfeene of te oin, ten R ( ) 1. Te quantity, wose eipoal gives te stet atio of te ubbe on top of te oin, is a key paamete of te poblem. R Te defomation is ompletely desibed by R () of Eq. (8) wee te onstants of integation ave been osen to satisfy te bounday onditions. It is onvenient to divide te ubbe into tee egions, as sown in Fig 5. Region 1 is on te top of te oin, as sown in Fig. 3a., wee 0 ; ee te ubbe as been steted, R is positive, and s 1. Region is undeneat te oin, wee te ubbe is folded bak, and. We ave denoted te edge of te ole, seen in Fig. 3b., in te unsteted ubbe by ; ee te ubbe is steted, but R is negative, and s 1. Region 3 is te egion m extending fom te ole to te edge of te ubbe seet; ee te ubbe may be unsteted, as sown in Fig. 3., o steted, as sown in Fig.b.; in bot ases, s 1. Fig. 5. Semati of oin and ubbe sowing te tee diffeent egions.
Table 1 below summaizes te bounday onditions and te oesponding solutions. Region s BCs Defomation Comment R1 (0) 0 1 1 1 R A, B 0 R ( ) 1 1 1 1 1 R ( ) 1 B 3 detemined fom 1 ln T ( ) 0 R ( ) R ( ) R A 3 T ( ) 0 B detemined fom 3 1 T ( ) T R ( ) R ( ) 3 3 R3 A3 3 m ext 3 In ea egion, te onstants of integation in te solutions desibing te defomation an be detemined fom bounday onditions listed in Table 1. In egions and 3, te ole adius is not known a pioi; it is to be detemined fom te ontinuity ondition R ( ) R 3 ( ). Pusing down on te oin in te CTRT is equivalent to applying a tension pe lengt T ext at te edge m of te seet. So if T ext is given, an be detemined; onvesely, if is given, T ext an be detemined. If tee is no extenal tension applied at te edge, ten te ubbe in egion 3 is undefomed, R 3 () and letting R ( ) gives te elation between and. If tee is extenal adial tension T ext, te bounday ondition T Text at m gives te elation between and. Fo ou examples below, we ave osen 1. somewat abitaily. In egion 3, te adial tensions at two diffeent points ae elated by 1 1 T T ( 1) 1 ( ) m (13) Fig. 6 sows te elation of te ole adius R R( ) R3( ) as a funtion of fo vaious extenal tension T ext. As ineases, R ineases, i.e., as te top ubbe is less steted, te adius of te ole beomes lage. In tis gap, te uves end wen te ole adius te invagination beomes unstable and te oin is eleased fom te ubbe befoe onfiguation is disussed in Set..1.. R beomes equal to te oin adius, i.e., wen R 1. In patie, R eaes R ; te stability of te
In te example sown in Fig 3, is about 0.5, and R is about 0.77 aoding to te uve of T 0 in Fig. 6, in easonable ageement wit te ole size sown in Fig. 3. We note tat if 0, ten R 0, and if 1, ten R 1. ext Fig. 6. Plot of ole adius units of te oin adius R. R vs. fo diffeent values of extenal tension. Te lengts R and ae measued in Fig. 7 sows te solution R () in all tee egions fo diffeent applied tension at te edge. It is inteesting to note tat uves in te figue appea to be staigt line segments, altoug te defomations in egions and 3 ae nonlinea funtions of. In egion 1, R' 1/ 0, te ubbe on te top of te oin is steted unifomly, te slope of te line in egion 1 indiates te stet atio. In egion, R' 0, indiating tat te ubbe as folded bak, foming te poket wi olds te oin. In egion 3, R' 0, te ubbe is folded fowad. If tee is no extenal tension, Text 0, R() R () is a nonlinea funtion wi depends on T ext. is a staigt line, and te ubbe is undefomed. If tee is extenal tension,
Fig. 7. Defomation R () fo stet atios (a) 0.5 and (b) 0. and vaious extenal tensions Te oesponding tensions pe lengt, T and T, ae sown in Fig. 8. In egion 1, bot adial and tangential tensions ae unifom and equal. In egion, te adial tension T deeases wit and goes to zeo at, te bounday of egion. In egion 3, T T 0 if te extenal tension T ext is zeo. If Text 0, te adial tension T ineases to T ext, wile te tangential tension T deeases wit ineasing. In te limit tat m, T T. Fig. 8. Magnitudes of te adial and tangential tensions/lengt fo stet atios (a) 0.5 and (b) 0. and vaious extenal tensions. Te tikness of te seet, / 0 / ( sr R), is sown in Fig. 9. In egion 1, te tikness is unifom. If tee is no extenal tension, ten te tikness ineases in egion, and emains onstant 0 in egion 3. Wen an extenal tension T ext is applied, te tikness ineases wit T ext in egion, but deeases wit Text in egion
3. Te tikness appeas to be disontinuous at te edge of te oin, but ontinuous at te ole. Tis follows fom te ontinuity of sr, wi is ontinuous at te edge of te ole, but disontinuous at te edge of te oin. Fig. 9. Tikness / 0 of te ubbe vs. fo stet atio (a) 0.5 and (b) 0. and vaious extenal tension.1. Stability As an be seen in Fig. 8, tee ae lage diffeenes in bot adial and tangential tensions on te top, egion 1, and bottom, egion, of te oin at. Weeas in plana egions of ubbe, tangential oop stesses ompensate fo anges in te adial stess and povide foe balane, tis is not te ase at te edge of te oin, wee tangential tension is balaned by foe fom te edge of te oin. Te diffeene in te adial tension on top and bottom of te oin podues a net foe, wi must be balaned by fition of te ubbe against te edge of te oin if te invagination is to be stable and te ubbe is not to slip. A semati is sown in Fig. 10. Fig. 10. Semati of foes ating on te ubbe at te edge of te oin Te adial tension pe lengt, fom Eq. (9), on top of te oin, in egion 1 at, is
and below te oin, in egion, at, it is T T 1 3 (1 ), (14) 3 1 ln 3 To povide foe balane, te fition foe must equal te diffeene between te adial tensions on top and bottom at te edge of te oin. As te seet passes aound te edge of te oin fom top to bottom, te nomal foe deeases, as in te ase of a flexible ope passing aound a stationay apstan sown in Fig. 11. Hee te tension anges due. (15) Fig. 11. Semati of flexible ope passing aound a stationay apstan. to fition as te ope winds aound te apstan, te ange in tension is just dt Td wee is te oeffiient of stati fition. Te ontibution of fition to adial tension an be estimated fom Eule's equation, wi gives te atio of tensions in a ope o seet winding toug an angle aound a apstan, T T e (16) 1. (We ave negleted ee te ontibution of te tangential tension along te edge of te oin to te nomal foe, sine te oin is tin; uvatue of te edge assoiated wit tikness is mu geate tan uvatue assoiated wit te oin adius). In ou ase,, and we must ave, fo stability, tat T 1. (17) T e On substituting te expessions fo te tensions fom Eqs. (14) and (15), te itial fition oeffiient given by is (1 )(3 ) 1 ln( ). ln (18) Tis is ou geneal stability iteion. Fig. 1 sows te funtion (, ) given by Eq. (17). Note tat is a measue of te stengt of te tension T ext applied at te edge of te ubbe seet. Wen Text is ineased, deeases
and ineases. Te oin is eleased fom te ubbe wen beomes equal to te fition oeffiient of te ubbe. If 0.5, and no extenal foes ae applied at te edges, we ave 0.777, and 0.6508, i.e., te ubbe steted by te fato 1/ an old te oin if its fition oeffiient is geate tan 0.6508. As extenal tension is applied at te edges, T deeases, as an be seen in Fig. 8, ineases, and te itial fition oeffiient ineases. Fo example, if 0.5, and 0.6, ten 0.804. Pats of te uves sown in Fig. 1 ae dawn dased, indiating tat te uves in tis egion ae not pysially ealizable. In tis paamete ange, te E-L equation (7) does not ave a solution wi satisfies te ondition R 0 in egion. Te egion wee te Hookean epesentation beaks down is disussed fute below. Fig. 1. Te minimum fition oeffiient wi gives te onfiguation sown in Fig. 5 is plotted as a funtion of and fom Eq. (17). In te CTRT, te ange of stetes is 0. 0.5. Te oeffiient of fition fo latex on aluminum as been epoted [16] to be as ig as 1.1. Pusing down on te oin, as sown in Fig. b., is equivalent to applying and ineasing extenal tension at te edges of te ubbe seet. Wen te esulting itial fition oeffiient exeeds te atual one, te invagination loses stability. Hee is wee te magi ous: te seet moves aound te edge of te oin, egion disappeas, and te oin is ejeted fom te invagination towads te bottom of te ontaine.
As te itial tension is eaed, te ubbe stats to slip aound te edge of te oin. As te ubbe slips, ineases. As sown in Fig. 17, as ineases, even geate fition oeffiient is equied to stabilize te seet tan peviously. Futemoe, as te ubbe stats to slip, stati fition between te ubbe and te oin is eplaed by (smalle) kineti fition, wi fute ontibutes to te loss of stability. Te loss of stability is tus subitial, one te ubbe stats to slip, it beomes ineasingly unstable; te oin is ejeted, and te ubbe elaxes to te unfolded onfiguation..1.3 Failue of te Hookean Desiption Altoug te Hookean model geneally desibes te defomations of te ubbe seet supisingly well, it does beak down in etain egions of paamete spae. Conside te adial tension in egion in te viinity of of lage extenal tension, wen. Te adial tension, fom Eq. (9), is 1 R T To (sr 3). R in te ase (19) Nea, R / 1/. Sine te adial tension as to vanis at, we must ave 1 sr 3. Clealy, tis is not possible if 1/ 3. We see teefoe tat te Hookean desiption beaks down in te ase of lage stain ( 1/ 3 ) and lage extenal tension, wee wi is outside of te egion of validity of te linea appoximation of te nonlinea enegy density..1.4 Disussion Te simple Hookean model above desibes well te elastially stabilized invagination foming te poket wi olds te oin and makes possible te magi. A key aspet of te onfiguation is te folding of te ubbe seet; it is inteesting to onside ow it omes about. Fist, it is not neessay fo te seet to fold at te edge of te oin at ; in te egion m, te seet ould emain unfolded wit R' 0, satisfying te E-L equation wit te defomation m m m m (1 ) 1 R ( ) ( ), (0) wi satisfies te bounday onditions R ( ) 1 and T( m) 0. Fo tis solution, R 1fo, indiating ompession ate tan tension, and te fee enegy is geate tan tat of te solution oesponding to te folded invagination. Te situation is simila unde te oin. If te seet folds at, it is not neessay to fold again at te ole. It ould emain unfolded, but again te defomation fo enegy tan te solution wit te fold at would be a ompession wit ige. Te seet tus folds wee neessay to avoid ompession yet satisfy
te neessay bounday onditions.. Te Neo-Hookean Model A moe auate desiption of wat appens is given by te neo-hookean model of elastiity, wi is bette suited to lage defomations tan te Hookean model pediated on small stains. It is a speial ase of Rivlin s geneal 1 model in Eq. (1) wee C10 Gand all te ote oeffiients ae zeo. Assuming again te defomation to be of te fom R ( R ( ),0, R (, z)) and tat R / is small, te enegy is given by z z 1 m R 0 [( ) ( ) ( ) 3]. 0 (1) F G R d RR Te E-L equation 1 R 1 3 3 3 3 RR 3 4 RR R ( R ) () is now nonlinea. Te adial and tangential tensions pe unit lengt ae and 1 0 T T ( R ), 4 RR RR T 1 R T0 ( ). 4 RR R R (3) (4) Unlike in te linea Hookean model, ee in egions and 3, te E-L equation annot be solved analytially; te solution must be obtained numeially. Te bounday onditions ae te same as in te Hookean ase. Te numeial metod we use ee fo te solution is te sooting metod. Results of te numeial alulations ae sown below. Figs 13 and 14 allow ompaison of te defomation R () fom te Hookean and te neo-hookean models fo diffeent values of. Te two models gives qualitatively simila esults: in Fig. 13 (a), tey ae emakably lose to ea ote. Fig 15 allows ompaison of te pedited ole adius simila esults; te diffeene is less tan 0%. R as funtion of. Again, te two models give
Fig. 13. Peditions of Hookean and Neo-Hookean models of te defomation R () wit 0.5 (a) T ext =0, (b) Text 0.10T. 0 Fig. 14. Peditions of Hookean and Neo-Hookean models of te defomation R () wit 0. (a) T ext =0, (b) Text 0.10T. 0
Fig. 15 Peditions of Hookean and Neo-Hookean models fo te ole adius R as funtion of. As in te ase of te Hookean model, te invagination emains stable so long as 1 T ( ) ln( ). T ( ) (5) Fig. 16. Peditions of Hookean and neo-hookean models fo te itial fition oeffiient as funtion of.
Fig. 16. sows te itial fition oeffiient as funtion of fo bot Hookean and neo-hookean desiptions; it may be ompaed wit Fig. 1. We note tat te neo-hookean model an pedit te itial fition oeffiient fo abitay paametes and witout diffiulty, unlike te Hookean model. Fig 17 sows te ole adius, te ole adius R as funtion of fo diffeent values of extenal tension T ext. Fo given value of R ineases wit T ext, but annot inease beyond te oin adius R. In patie, te maximum tension is detemined by fition at te edge of te oin. Te stability limit detemined by Eq. (5) is indiated by te dased uves in Fig. 17. Fo given values of te fition oeffiient and given extenal tension, te oin is eleased fom te ubbe wen te solid line and dased uve inteset. Fig. 17 Te ole adius model. Te solid uves indiate R as funtion of fo diffeent values of extenal tension Text in te neo-hookean R vs ; te dased uves indiate te stability limit. Summay In tis pape we pesented a simple analysis of te elasti defomation wi undelies te CTRT wee a oin, appaently on top of a ubbe seet oveing a up, appeas to go toug te ubbe and fall into te up. Te oin, in fat always unde te ubbe seet, is eld in an invagination wi is stabilized by elastiity and by fition; pusing
down suffiiently ad on te oin (o equivalently, applying suffiient tension at te edges of te ubbe) auses te invagination to beome unstable, te oin is eleased, and falls to te bottom of te up. A simple volume onseving Hookean model fo te ylindially symmeti system povides two distint linea Eule-Lagange equations fo diffeent potions of te ubbe wi may be solved analytially. Te solution indiates an invagination wi olds te oin; stability of te invagination is povided by elastiity and fition between te ubbe and te edge of te oin; te domain of stability an be detemined via Eule s apstan equation. To test te validity of te Hookean model, we ave aied out a neo-hookean analysis. Tis povides a non-linea Eule-Lagange equation, wi needs to be solved numeially. Peditions of te Hookean and neo-hookean models wee found to be in good ageement in a lage egion of paamete spae; te failue and egion of stability of te Hookean model wee identified. Altoug we ae unawae of simila invaginations, stabilized by elastiity and fition, ouing in natue, we suspet tat tey may exist in biology. Aknowledgements We aknowledge useful disussions wit David Andelman, Jon Ball and Epifanio Viga. M. D. aknowledges te finanial suppot of te Cinese Cental Govenment in te pogam of Tousand talents and te NSFC gant (No. 1434001). F. M. aknowledges te finanial suppot of NSFC unde gant numbes 3118004, 9107045, 10834014, 1117550, National Basi Resea Pogam of Cina (973 pogam, No. 013CB93800). P.P-M and X. Z. aknowledge suppot by te NSF unde EFRI-13371, IIP-111433 and DMS-11046 Appendix: A Bief Histoy of te CTRT. Te CTRT, o oin-toug-te-ubbe tik was a favoite of Matin Gadne. One of us (P.P-M.) leaned tis tik fom William G. Unu, wo leaned it fom Si Roge Penose, wo leaned it fom Matin Gadne. Rudy Ruke mentions Gadne sowing im te tik [17] and indiating tat te tik was well known among lose-up magiians. Ruke eoded Gadne s explanation wile pefoming te tik [18]: I want to move tis dime toug te membane into te glass. How? I'll use te fout dimension. If someting tavels toug te fout dimension it an go "aound" a 3D obstale. Tink of a squae in Flatland. If I daw a up wit a lid in te plane, it's a single unboken line. (Matin dew tis sape on a piee of pape, like a U wit a line aoss te top wit a squae beside it. And a squae an't pus someting inside tat line witout ossing te edge. But if someting ises out of te squae's D spae into te 3d dimension it an move ove te line and settle down inside it.
If we wall off a little volume of ou spae wit a glass and wit a ubbe membane, as I ave done, we an't get inside tat volume witout beaking te glass o puntuing te membane. In patiula, I an't put tis oin inside te glass. But if I an lift te oin up into te fout dimension, I an move "ove" te membane and land inside te glass. I'll let you elp. It would be dangeous fo you to pus te oin toug te fout dimension wit you bae finge. So I'll est a quate on top of te dime. And you pus down on te quate. And we'll fous on te fout dimension wile you do it. And te dime will slide toug ypespae and end up in te glass. Ready? Pus. (latte of te dime in te glass). You see? Te dime tavelled toug fou dimensional spae to get aound te ubbe membane. Te CTRT, also known as te dental dam tik is edited [19] as a eation of Lubo Fiedle. Fiedle (1933-014) was bon in Bno, Czeoslovakia. He moved to Vienna, wee was selling magi tiks to te publi on te steets and at te entanes to supemakets in ode to make ends meet. Te fist desiption of te CTRT was publised by im [0] in 1958. Te tik was subsequently maketed by Gene Godon witout pemission o edit as Dam Deeption [1] in 1963. Te dental dam, used in CTRT, was invented by D. Sandfod Cistie Banum, in Montiello, New Yok in 1864 []. Rate tan patenting it, Banum pesented it as a fee gift to te dental pofession. In Euope, te dental dam beame known as te Koffedam. 1. ttps://www.youtube.om/wat?v=64gdjmwa-m. ttps://www.youtube.om/wat?v=3ib7wtkvgmw 3. Muat, D., Quinlan, A., Vali, H., Komeili, A.: Compeensive geneti dissetion of te magnetosome gene island eveals te step-wise assembly of a pokayoti oganelle. P Natl Aad Si USA 107(1), 5593-5598 (010). 4. Ito, T., Edmann, K.S., Roux, A., Habemann, B., Wene, H., De Camilli, P.: Dynamin and te atin ytoskeleton oopeatively egulate plasma membane invagination by BAR and F-BAR poteins. Dev Cell 9(6), 791-804 (005). 5. Sens, P., Tune, M.S.: Teoetial model fo te fomation of aveolae and simila membane invaginations. Biopysial jounal 86(4), 049-057 (004). 6. Ben-Dov, N., Koenstein, R.: Enanement of ell membane invaginations, vesiulation and uptake of maomoleules by potonation of te ell sufae. PloS one 7(4), e3504 (01). 7. Wolff, J., Komua, S., Andelman, D.: Budding of domains in mixed bilaye membanes. Pys Rev E 91(1) 01708-9 (015). 8. Lipowsky, R.: Remodeling of membane ompatments: some onsequenes of membane fluidity. Biol Cem 395(3), 53-74 (014). 9. MMaon, H.T., Bouot, E.: Moleula meanism and pysiologial funtions of latinmediated endoytosis. Nat Rev Mol Cell Bio 1(8), 517-533 (011). 10. Lewis, W.H.: Meanis of invagination. Te Anatomial eod 97(), 139-156 (1947). 11. Rauzi, M., Hoeva Bezavsek, A., Ziel, P., Leptin, M.: Pysial models of mesodem invagination in Dosopila embyo. Biopysial jounal 105(1), 3-10 (013). 1. Ball, J.M.: Disontinuous Equilibium Solutions and Cavitation in Non-Linea Elastiity. Pilos T R So A 306(1496), 557-611 (198).
13. Bowe, A.F.: Applied meanis of solids. CRC Pess, Boa Raton (010) 14. Boye, M.C., Auda, E.M.: Constitutive models of ubbe elastiity: A eview. Rubbe Cem Tenol 73(3), 504-53 (000). 15. Audoly, B., Pomeau, Y.: Elastiity and geomety : fom ai uls to te non-linea esponse of sells. Oxfod Univesity Pess, Oxfod ; New Yok (010) 16. Hu, P., Motawa, B., Seo, N.J.: Hand beakaway stengt model-effets of glove use and andle sapes on a peson's and stengt to old onto andles to pevent fall fom elevation. J Biome 45(6), 958-964 (01). 17. ttp://matin-gadne.og/testimonials.tml 18. Rudy Ruke (piv. omm.) 19. ttp://geniimagazine.om/magipedia/lubo_fiedle 0. Fiedle, L.: Eine fast unmöglie Münzenwandeung. Zaubekunst 4(10), 19 (1958). 1. ttp://www.geniimagazine.om/magipedia/gene_godon. Ko, C.R.E., Tope, B.L.: Histoy of dental sugey, vol.. Te National At Publising Company, Ciago, Ill., (1909)