J.Natn.Sci.Foundation Sri Lanka 2017 45 (3): 201-217 DOI: http://dx.doi.org/10.4038/jnsfsr.v45i3.8184
RESEARCH ARTICLE
Aerodynamics and right-left symmetry in wind dispersal of maple, dipterocarps, conifers and some genera of apocyanaceae and magnoliaceae
Kirthi Tennakone 6KHI¿HOG'ULYH3HDERG\0DVVDFKXVHWWV 86$
Revised: 30 September 2016; Accepted: 19 January 2017
Abstract: 7KHZLQGGLVSHUVLQJVHHGVKDYHHYROYHGUHPDUNDEOH INTRODUCTION aerodynamic optimisation to minimise the speed of descent on GHWDFKPHQWIURPWKHWUHHVRWKDWWKH\FRXOGEHFDUULHGDZD\ E\ WKH ZLQG DQG GHSRVLWHG D GLVWDQFH DZD\ IURP WKH SDUHQW ,Q FHUWDLQ SODQWV ZLQG LV DQ HIIHFWLYH PRGH RI VHHG $QHI¿FLHQWSK\VLFDOPHFKDQLVPWKDWHQDEOHVVORZLQJGRZQ GLVSHUVDO 3ULRU WR HYROXWLRQ RI ELUGV ZLQG GLVSHUVDO could have been the sole mode of long distance seed WKHGHVFHQWLVURWDWLRQZKLFKUHGXFHVWKHWUDQVODWLRQDONLQHWLF HQHUJ\ 5RWDWLRQ DQG WUDQVODWLRQ DOZD\V GH¿QH D ULJKW RU GLVWULEXWLRQRQODQG&RQVHTXHQWO\WUHHVDGRSWLQJZLQG OHIWKDQGHGQHVV7KHSUHVHQWZRUNH[DPLQHVWKHDHURG\QDPLFV GLVSHUVDOKDYHHYROYHGVHHGVZLWKLQWULFDWHDHURG\QDPLF of the dispersal of seeds of many species in relation to DSSHQGDJHVWRVORZGRZQWKHGHVFHQWHQDEOLQJWKHPWREH ULJKWOHIWDV\PPHWU\JLYLQJH[DPSOHVWRVKRZWKDWLQQDWXUH WUDQVSRUWHGE\ZLQG *UHHQH 4XHVDGD 3DQGRO¿ this symmetry is broken either spontaneously or intrinsically. ,]]R 6WHYHQVRQ et al ., 2015). Plumes attached In the former situation, seeds have no geometrical right-left to lighter seeds achieve this via increase of the air drag. asymmetry, but an initial instability chooses one sense of +RZHYHUKHDYLHUVHHGVVWRULQJIRRGIRUWKHLQLWLDOSKDVH rotation (right or left). Whereas in the latter, geometrical of germination cannot utilise this method and the viable asymmetry dictates the sense of rotation. Seeds of the maple SK\VLFDODOWHUQDWLYHZRXOGEHWRPDNHWKHPVSLQZKLOH IDPLOL\EHORQJVWRWKH¿UVWFDWHJRU\DQGDWKHRUWLFDOPRGHOLV falling. Here the gravitational potential energy of a falling presented to explain the motion. Seeds of dipterocarps, conifers seed dissipates largely via transformation into rotational and some genera of apocyanaceae and magnoliaceae are found kinetic energy, effectively inducing an aerodynamic lift. WRSRVVHVVDKDQGHGQHVVZKLFKGHWHUPLQHVWKHURWDWLRQGXULQJ seed fall. In dipterocarps, conifers and magnoliaceae, both 7ZR RI WKH PRVW GLYHUVH DQG ZLGHO\ GLVWULEXWHG right-handed and left-handed seeds are produced in the same angiosperm tree genera are maple (Aceraceae) and tree but correlated to the handedness of the sprial phyllotaxy dipterocarps (Dipterocarpaceae). Former is one of the of the branch that bears the fruit. Apocyanaceae is found to PRVWFRPPRQWUHHVSHFLHVLQWHPSHUDWH]RQHVZKLOHWKH be unique and seeds of all genera have the same handedness. ODWWHU GRPLQDWHV UDLQ IRUHVWV LQ WKH WURSLFV $VKWRQ 7KH GHWDLOV RI H[WHQVLYH ¿HOG REVHUYDWLRQV H[SHULPHQWV *XQDWLOOHNH &RUOHWW 3ULPDFN *XQDWLOOHNH and theoretical interpretations are presented to illustrate the et al ., 2006; Lamarque, 2013). Their success is due to relationship of right-left asymmetry to the aerodynamics of bearing of auto-gyrating seeds. Pines and other conifers VHHGZLQGGLVSHUVDOSRLQWLQJRXWWKHLPSOLFDWLRQVRIHYROXWLQDU\ &RQLIHUDH GLVWULEXWHG ZRUOGZLGH ZHUH WKH GRPLQDQW optimisation in practical aerodynamics. WUHHVSHFLHVLQWKH0HVR]RLFHUD7KHQWKHLUSUHGRPLQDQW PRGHRIGLVSHUVDOPD\KDYHEHHQWKHZLQG 6WHYHQVRQ Keywords: Apocyanaceae, dipterocarps, magnoliaceae, maple, et al ., 2015), and present day conifer seeds carry a trait plant handedness, seed aerodynamics. for auto-gyration.
[email protected] 202 Kirthi Tennakone
Spinning and translational motion of an object involves WKH ¿UVW YDULHWLHV KDYH FORVH VLPLODULWLHV FRPSDUHG D KDQGHGQHVV 6RPHWKLQJ IDOOLQJ GRZQ DQG URWDWLQJ to the 5 th and 6 th WKDWDUHOLJKWHUDQGGLIIHULQZLQJDQG FORFNZLVH ZKHQ YLHZHG IURP DERYH GLIIHUV IURP WKH VHHG JHRPHWU\ 7KHUHIRUH PRVW H[SHULPHQWV ZHUH VDPH URWDWLQJ FRXQWHUFORFNZLVH $ PDSOH VDPDUD FRQGXFWHGZLWK1RUZD\PDSOHDQG%R[(OGHUVDPDUDV GURSSHGIURPDKHLJKWURWDWHVHLWKHUFORFNZLVHRUFRXQWHU $YHUDJH ZHLJKWV ZHUH GHWHUPLQHG IURP VDPSOHV FORFNZLVH $OWKRXJK WKHUH DUH UHSRUWV LQ OLWHUDWXUH DQG VDPSOHV ZHUH XVHG IRU GHWHUPLQDWLRQ RI WKH discussing kinematics of maple seed motion (Lentink GLPHQVLRQV:LQJVWUXFWXUHZDVH[DPLQHGE\ORZSRZHU et al ., 2009; Varshney et al ., 2012; Sohn et al. , 2014), to PLFURVFRS\ EHIRUH DQG DIWHU FKHPLFDO WUHDWPHQW ZLWK WKHNQRZOHGJHRIWKHDXWKRUWKHDVSHFWRIKDQGHGQHVV acetone and hypochlorite. These treatments enhanced the of motion has not been investigated. Observations WUDQVSDUHQF\RIZLQJVWRH[DPLQHWKHYHLQV7KHSRVLWLRQ indicate that maple samaras or any of the macroscopic RIWKHFHQWUHRIJUDYLW\ZDVDVVHVVHGE\EDODQFLQJWKH organs of maple trees have no handedness (right-left VDPDUD RQ D YHUWLFDOO\ SODFHG KRUL]RQWDO HGJH ERWK distinguishing property) attributing maple seed rotation OHQJWKZLVHDQGEUHDGWKZLVH3K\VLFDODOWHUDWLRQVWRWKH to a spontaneous parity violation (right-left symmetry). VDPDUDVZHUHPDGHE\FXWWLQJWKHZLQJVDQGVHHGHGJHV In contrast, the dipterocarp fruit and other organs of ZLWK VFLVVRUV VDQGLQJ WKH VHHG VXUIDFHV V\PPHWULFDOO\ their trees possess handedness and the sense of rotation RU DV\PPHWULFDOO\ DQG FRDWLQJ ZLWK SDUDI¿Q ZD[ DQG of the fruit during falling is predetermined. Dipterocarp SROLVKLQJ 7HUPLQDO YHORFLWLHV ZHUH GHWHUPLQHG E\ trees have distinct right-handed (RH) and left-handed releasing the seeds from varying heights and timing /+ WZLJVÀRZHUVDQGIUXLWV6WUDQJHO\WKHH[WHQVLYH WKHPRWLRQIRUWKH¿QDOPHWUHV7R¿[WKHVHQVH OLWHUDWXUHRQWD[RQRP\RIGLSWHURFDUSDFHDH )R[ZRUWK\ RIURWDWLRQ UHODWLRQWRWKHGRZQZDUGYHUWLFDO DUHJLRQ 1911; Ashton, 1980; Smitinand et al ., 1980; Simmathri LQWKHVHHGVXUIDFHZDVOLJKWO\SDLQWHG$VOHDGLQJHGJH 7XUQEXOO GRHV QRW UHFRUG REVHUYLQJ RI WKLV LV WKH ULGJH RYHU WKH ZLQJ REVHUYLQJ WKH XSSHUPRVW property. Pines and magnolias are also found to have VLGHHQDEOHV¿[LQJWKHVHQVHRIURWDWLRQ(DFKVHHGZDV KDQGHGQHVV DWWULEXWHV VLPLODU WR GLSWHURFDUSV ZKHUHDV dropped 100 times from a height of ~ 3 m to assess the the entire family of apocyanaceae is ambidextrous. probability of the either sense of rotation. The effect of WKH SRVLWLRQLQJ RI WKH VHHG RQ UHOHDVLQJ ZDV FDUHIXOO\ The investigation reported here based on extensive REVHUYHG %HIRUH DQG DIWHU HDFK SK\VLFDO PRGL¿FDWLRQ observations on symmetries in plants theoretically WKH DYHUDJH GLVWDQFH RI IDOO DW ZKLFK VSLQQLQJ DQDO\VHVWKHDHURG\QDPLFVRIZLQGGLVSHUVDOLQUHODWLRQ starts, (2) terminal velocity, and (3) sense of rotation to right-left symmetry. Detailed observation of the ZHUHDVVHVVHG7KHHIIHFWRIGU\LQJWKHVHHGDVDZKROH PRUSKRJHQHWLFIHDWXUHVRIVHHGVDVZHOODVSDUHQWWUHHV RUH[SRVLQJRQHVXUIDFHWRWKHVXQZDVDOVRH[DPLQHG DQGVLPSOHH[SHULPHQWDWLRQLOOXVWUDWHKRZHYROXWLRQKDV 7ZLQVLQRQHIUXLWZHUHH[DPLQHGWRGHWHUPLQHZKHWKHU RSWLPLVHGWKHZLQGGLVSHUVDOVWUDWHJ\ there exist any correlations of sense of rotation.
METHODOLOGY Dipterocarpus zeylanicus VDPSOHV ZHUH FROOHFWHG IURP *DPSDKD .XUXQHJDOD .DOXWUD 0DWDUD DQG 7KH DQDO\VLV DQG GLVFXVVLRQ LV EDVHG RQ D ¿HOG .HJDOOH'LVWULFWVRI6UL/DQND7HUPLQDOYHORFLWLHVZHUH observations and sample collection, and experimentation DVFHUWDLQHGE\GURSSLQJWKHIUXLWVIURPPKHLJKW ZLWK ZLQG GLVSHUVLQJ VHHGV E H[DPLQDWLRQV RI The angular speeds of falling fruits through height of WKH RUJDQV RI SODQWV ÀRZHUV OHDI DQG EUDQFKLQJ aPZHUHHVWLPDWHGE\VWURERVFRSLFPHDVXUHPHQWV DUUDQJHPHQW WKDWEHDUWKHVHVHHGVWRDVFHUWDLQZKHWKHU :LQJ VXUIDFH ZDV H[DPLQHG E\ PDJQL¿FDWLRQ DQG WKH right-left asymmetries exist; and (c) experimentation DQJOHV EHWZHHQ WKH ORQJLWXGLQDO YHLQV ZHUH PHDVXUHG ZLWKSDSHUWR\PRGHOVVRWKDWSDUDPHWHUVFDQEHUHDGLO\ after imprinting the vein structure on paper by pressing adjusted. WKHLQNHGZLQJV7KHQXWZDVFXWLQWRWUDQVYHUVHVHFWLRQV to determine the geometry of sepal arrangement. Wing D :LQGGLVSHUVHGPDSOHVDPDUDVZHUHFROOHFWHGIURP DQGVHHGPRGL¿FDWLRQZDVGRQHLQWKHVDPHZD\DVIRU Peabody area, Massachusetts, United States. Seeds of maple. Sugar maple ( Acer saccharum 3D[ 1RUZD\ PDSOH (A. platanoides L.), Sycamore ( A. pseudoplatanus L.), (b) Whenever an opportunity arose, observations Red maple ( A. rubrum ), Silver maple ( A. saccharinum ZHUH FRQGXFWHG GXULQJ WKH SDVW \HDUV WR DVFHUWDLQ Pax) and Box Elder ( A. negundo L.) often mixed the handedness of the seeds and sense of spinning to XS ZHUH VHSDUDWHG DQG VWRUHG DW DPELHQW FRQGLWLRQV GHWHUPLQHZKHWKHUVHHGDV\PPHWULHVWUDQVFHQGWRRWKHU (~15 oC, relative humidity ~ 60 %). The samaras of RUJDQV 7KH DV\PPHWULHV LQ ÀRZHUV IUXLWV OHDYHV DQG
September 2017 Journal of the National Science Foundation of Sri Lanka 45(3) Aerodynamics and right-left symmetry in seed wind dispersal 203
EUDQFKLQJ DUUDQJHPHQW ZHUH TXDOLWDWLYHO\ H[DPLQHG and dimensions of a maple samara varies from species 7KH UHODWLRQVKLS RI IUXLW KDQGHGQHVV WR WKDW RI ÀRZHU to species and perhaps climate conditions. Typically, a and leaf phyllotaxy in dipterocarps and related species 1RUZD\PDSOHVDPDUDKDVDOHQJWKaFPPD[LPXP ZDV H[WHQVLYHO\ LQYHVWLJDWHG$OPRVW DOO WKH JHQHUD RI EUHDGWKaFPZLWKDaPPVHHGDQGDPHGLDQZHLJKW DSRF\DQDFHDHLQ6UL/DQND ZLOGDQGRUQDPHQWDO ZHUH a PJ RI ZKLFK PRUH WKDQ a LV FRQWULEXWHG examined to identify traits of handedness in phyllotaxy, by the seed and the thick ridge of vein supporting the ÀRZHUVIUXLWVDQGVHHGV7ZR Sapu ( Michelia nilagrica ) ZLQJ )LJXUH 7KH%R[(OGHUVDPDUDVKDYHVPDOOHU YDULHWLHVJURZLQJLQWKH&HQWUDO3URYLQFHRI6UL/DQND dimensions; length ~ 3.8 cm, maximum breadth ~ 1 cm. DQG RUQDPHQWDO PDJQROLDV ZHUH XVHG WR DVFHUWDLQ 7KH\DUHDOVROLJKWHUZLWKDPHGLDQZHLJKWaPJZLWK WKH UHODWLRQVKLS RI OHDI SK\OORWD[\ WR ÀRZHU DQG IUXLW DQHORQJDWHGVHHGFDUU\LQJDERXWRIWKHZHLJKW7KH morphology. ZLQJV RI ERWK YDULHWLHV WDSHU LQ EUHDGWKZLVH GLUHFWLRQ WRZDUGV WKH OREH VXSSRUWHG E\ WKH KDUG DQG UHODWLYHO\ 2EVHUYDWLRQVDQGH[SHULPHQWVZHUHDOVRFRQGXFWHGWR thick ridge (Figure 1). A distinctive characteristic of the DVFHUWDLQZKHWKHUWKHKDQGHGQHVVRIWKHIUXLWRUVHQVHRI WZRYDULHWLHVLVWKHJHRPHWU\RIWKHVHHGDQGWKHULGJH URWDWLRQZDVDIIHFWHGE\WKHLQGLYLGXDOWUHHRULWVEUDQFK 7KH1RUZD\PDSOHVHHGKDVDQRYDOVKDSHZLWKDQDSH[ DQGÀRZHUJHRJUDSKLFORFDWLRQRUJHQHWLFYDULDWLRQVDQG SRLQWLQJWRZDUGVWKHULGJHZKHUHDVWKH%R[(OGHUVHHGLV ZKHWKHUWKHKDQGHGQHVVRIWKHQXWSDVVHVWRWKHVHHGOLQJ HORQJDWHGZLWKSRLQWHGDSH[HVRQERWKVLGHV )LJXUH Seeds of hibiscus varieties and papaya from right or left- ,IWKHWZRW\SHVDUHKHOGZLWKZLQJVSRLQWLQJYHUWLFDOO\ KDQGHG SODQWV ZHUH JHUPLQDWHG DQG KDQGHGQHVV RI WKH GRZQZDUGV WKH ULGJH DSSHDUV FRQFDYH LQ WKH FDVH RI SURJHQ\ZDVUHFRUGHG 1RUZD\ PDSOH DQG FRQYH[ LQ %R[ (OGHU )LJXUH 7KHFHQWUHRIJUDYLW\RI1RUZD\PDSOHDQG%R[(OGHU F 7R\PRGHOVRIV\VWHPVWKDWURWDWHGXULQJIDOOLQJZHUH VDPDUDVUHVLGHVaPPIURPWKHVHHGDSH[%RWK FRQVWUXFWHGRXWRISDSHUWRVLPXODWHVHHGZLQGGLVSHUVDO VLGHVRIWKHZLQJVXUIDFHVDUHFRYHUHGZLWKDSDWWHUQRI PHFKDQLVPV DQG LOOXVWUDWH KRZ DV\PPHWU\ LPSDUWV VWULDWLRQV DV VKRZQ LQ )LJXUHV E DQG F EXW PRUH URWDWLRQDOPRWLRQVORZLQJGRZQWKHUDWHRIIDOO densely populated than in the depiction. The texture and pattern of distribution on either side being identical as they originate from veins embedded in the middle of RESULTS AND DISCUSSION WKHZLQJWLVVXH7KHVXUIDFHDOVRSRVVHVVHVWKHSURSHUW\ RIQRWUHWDLQLQJZDWHU2WKHUZLVHWKHH[SHFWHGIXQFWLRQ Maple samara characteristics could not be performed effectively.
The maple fruit is a bilaterally symmetric structure A theoretical formulation of maple seed motion ZLWKWZRFRQMRLQHGZLQJHGVHHGV VDPDUDV ZKLFKDUH distinctly separated from each other and the stalk as the A maple samara falling under the gravity starts spinning, IUXLWULSHQVDQGGULHV )LJXUH *HQHUDOO\ZLQGGLVSHUVDO VORZLQJGRZQWKHGHVFHQWHQDEOLQJLWWREHFDUULHGDZD\ begins in late summer and continues into autumn, and the E\ZLQGWRIXUWKHUORFDWLRQV7KH\URWDWHDERXWDSRLQW SDWWHUQYDULHVIURPVSHFLHVWRVSHFLHV7KHPHGLDQZHLJKW located near the seed very close to the centre of gravity, ZKLOHWKHSODQHRIWKHZLQJVWD\VVOLJKWO\LQFOLQHGWRWKH KRUL]RQWDO7KHPRWLRQDQGPHFKDQLVPRIVORZLQJGRZQ during descent involve intricately complex aerodynamics RZLQJ WR QRQOLQHDULW\ RI DXWRURWDWLRQ /XJV $ VLPSOH H[SODQDWLRQ IROORZV IURP FRQVLGHUDWLRQ RI energy conservation. When a samara of mass m falls GRZQDYHUWLFDOGLVWDQFH h and acquire translational and angular velocities v and Ȧ UHVSHFWLYHO\ WKH FKDQJH LQ potential energy D V = mgh , QHHGVWREHDOZD\VJUHDWHU than the kinetic energy acquired D T = 1/2 ( mv 2 + I Z2), (b) (c) (a) ZKHUH I = moment of inertia of about a vertical axis through the centre of mass. This is because, a part of Fig.1 (a) Schematic diagrams of (a) a fruit of Norway maple showing two conjoined samaras, potential energy transforms into kinetic energy of air set Figure 1: D 6FKHPDWLF GLDJUDPV RI D D IUXLW RI 1RUZD\ PDSOH into motion, and a part dissipated as heat generated due VKRZLQJ WZR FRQMRLQHG VDPDUDV VHHGV DQG WKH VWDON WRWKHDLUUHVLVWDQFH7KXVZHREWDLQWKHLQHTXDOLW\ E ZLQJVKDSHDQGYHLQVLQ1RUZD\PDSOH F ZLQJVKDSHϲ 2 and veins in Box Elder v < 2[ gh − I /( m)ω ] ...(1)
Journal of the National Science Foundation of Sri Lanka 45(3) September 2017
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,Q HTXDWLRQ ¥ gh ) is the velocity acquired in the )RUDZKLOHLWPRYHVZLWKRXWURWDWLQJLQDQRULHQWDWLRQ absence of spinning action and air resistance. Rapid ZKHUHWKHKHDY\VHHGSRLQWVGRZQZDUGV6XEVHTXHQWO\ rotation, large moment of inertia and smaller mass WKH PRYHPHQW WXUQV HUUDWLF IROORZHG E\ DQ RUGHUO\ DOO IDYRXU ORZHULQJ RI WKH WLPH RI GHVFHQW 'HWDLOHG URWDWLRQLQHLWKHUFORFNZLVHRUFRXQWHUFORFNZLVHVHQVH analysis of the problem requires an understanding of )LJXUH 7UDQVLWLRQ RFFXUV ZKHQ WKH YHORFLW\ RI WKH
WKHDHURG\QDPLFOLIWDQGWRUTXHIRUFHVDULVLQJIURPÀXLG centre of the mass reaches a critical value ( vC). The motion and drag. These forces depend on the dimensions, transition process needs to conserve energy, and angular shape, and nature of the surface, mass and its distribution and linear momenta. If a distributed and massive body and elastic properties. takes up linear and angular momenta (in this case stream and vortices of air), the kinetic energy transferred to it The structural architecture makes a samara mirror ZRXOG EH UHODWLYHO\ VPDOO ,I WKH WUDQVLWLRQ LV DVVXPHG V\PPHWULFLQWKHVHQVHLILWLVODLGGRZQÀDWRQDWDEOHDQG instantaneous and resistive loss is minimal, conservation WXUQHGXSVLGHGRZQ WKHVKDSHDSSHDUVH[DFWO\WKHVDPH energy yields,
DVWKHPLUURULPDJH6LPLODUO\LQWKHPDSOHIUXLWWKHWZLQV 2 2 are mirror images of each other. This symmetry implies (1) mgh ≤ 2/1 mv C ≤ 2/1 IωC ...(2) that they do not possess handedness. Imagine a samara UHOHDVHG IURP WKH UHVW NHHSLQJ WKH ZLQJ LQ D YHUWLFDO ZKHUH I = moment of inertia of the about the centre of plane. One could argue that in this situation it cannot gravity (object rotates about vertical axis inclined at a auto-rotate during falling, because in the absence of a VPDOODQJOHWRWKHKRUL]RQWDOD[LV 2QUHOHDVLQJDVDPDUD ULJKWOHIWGLVWLQJXLVKLQJIHDWXUHWKHUHLVQRZD\WRGHFLGH from the rest, a sudden decrease in the translational
ZKLFK ZD\ WR WXUQ DQG URWDWH +RZHYHU DQ DOWHUQDWLYH YHORFLW\ EHORZ vC is noticeable at the point rotation ZD\ RI SUHVHUYLQJ ULJKWOHIW 5+ V\PPHWU\ H[LVWV ,I ensues. Since angular momentum needs to be conserved, WKHREMHFWURWDWHVFORFNZLVHDQGFRXQWHUFORFNZLVHZLWK LUURWDWLRQDO DLU ÀRZ LQYDULDEO\ WXUQV URWDWLRQDO 7KH HTXDO SUREDELOLWLHV WKH UHTXLUHG V\PPHWU\ ZRXOG EH system has all the characteristics of a spontaneous again preserved. Frequently, the symmetries in nature are EUHDNLQJ RI V\PPHWU\ 6%6 VRPHWLPHV VHHQ LQ ÀXLG respected in this manner; although individual symmetry PRWLRQ =KDQJ &KLOGUHVV %DJKHUL et al ., 2012). violations exist, they occur in equal probabilities. This Here an unstable initially symmetric state suddenly LV H[DFWO\ ZKDW ZH REVHUYHmb LQ GURSSLQJ D PDSOH VHHG GHYHORSV LQWR RQH RI WKH DV\PPHWULF VWDWHV ZKHQ D
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(a) ;ďͿ z z dt Fig.2 Schematic sketch illustrating (a) clockwise; (b) counter-clockwise spiraling of a falling maple Figure 2: 6FKHPDWLFVNHWFKLOOXVWUDWLQJ D FORFNZLVH E FRXQWHUFORFNZLVHVSLUDOLQJRIDIDOOLQJPDSOH ω VDPDUDRQFHLWUHDFKHVWKHFULWLFDOYHORFLW\ 9& 7KHSDWWHUQRIOLQHVGUDZQSHUSHQGLFXODUWR dt YHUWLFDOD[LV ]D[LV DUHWKHSURMHFWLRQVRIWKHDUURZPDUNHGLQWKHVDPDUDRQDSODQHSDVVLQJ WKURXJKWKH]D[LVDQGFLUFOHVLQGLFDWHWKHDSSHDUDQFHDQGWKHVHQVHZKHQYLHZHGIURPDERYH The darker shading of the samara corresponds to its thicker region. During motion thicker edge is − =DOZD\VWKHOHDGLQJHGJH(14) (15)
September 2017 Journal of the National Science Foundation of Sri Lanka 45(3) (16) (17) ϭϬ
Aerodynamics and right-left symmetry in seed wind dispersal 205
crucial parameter reaches a critical value; the inherent generating force and Ar = radial component, all acting V\PPHWU\EHLQJFRQVHUYHGLQUHDOLVLQJGLIIHUHQWDOORZHG DW & DQG ZHLJKW mg . For equilibrium of rotation, a 2 VWDWHVZLWKHTXDOSUREDELOLWLHV centrifugal force Fr = PȦ E6LQș , acting at the centre of JUDYLW\* 2& a2* b and l = length span) needs to Figure 2(a) and (b) give a schematic representation be provided aerodynamically. The centre of rotation O is RI WKH FORFNZLVH DQG FRXQWHUFORFNZLVH GRZQZDUG not exactly about centre of gravity. The torque necessary A . VSLUDOLQJRIDPDSOHVDPDUD$UURZVLQWKHVH¿JXUHVSRLQW IRU URWDWLRQ DURXQG WKH ]D[LV DJDLQVW SURYLGHG E\ ij DZD\IURPWKHVHHGWRZDUGVWKHRSSRVLWHHQGRIWKHZLQJ 7KHZLQJLVKHOGLQDQLQFOLQHGSRVLWLRQ DQDQJOH ș to the UHSUHVHQWLQJWKHSRVLWLRQRIWKHZLQJ,QERWKFORFNZLVH vertical axis) by these forces. DQG FRXQWHUFORFNZLVH PRWLRQV WKH WKLFNHU HGJH LV DOZD\VWKHOHDGLQJHGJH6KDGLQJLQGLFDWHVWKHWKLFNQHVV 7KH OLIW IRUFHV RSHUDWH QRUPDO ÀXLG ÀRZ DQG GUDJ YDULDWLRQ DFURVV WKH ZLQJ WKH GDUNHU HGJH EHLQJ WKH IRUFHVRSSRVLWHWRWKHGLUHFWLRQRIWKHÀRZ%RWKWKHVH WKLFNHVW PP DQGWKHZLQJWDSHUVGRZQ aLQD forces are proportional to the square of the radius of the ZLQJEUHDGWKaFP MXVWOLNHDQDHURIRLO$VWUDQVYHUVH blade and square of velocity, so that, FURVVVHFWLRQVRIWKHZLQJDUHQHDUO\V\PPHWULFDOOLWWOH 24 22 OLIWLVJHQHUDWHGDW]HURDQJOHRIDWWDFN Az = CzL l ω − zD vlC ...(3)
22 24 The i QYROYHPHQW RI WUDQVODWLRQ DV ZHOO DV URWDWLRQ Aϕ = ϕL vlC − CϕDl ω ...(4) and associated lift and drag forces complicates the 24 motion of falling maple seeds defying rigorous analysis Ar = CrL l ω ...(5) EDVHG RQ ¿UVW SULQFLSDOV (YHQ WKH XQLIRUP URWDWLRQ RI
D ULJLG SURSHOOHU SRLQWLQJ WR D ¿[HG GLUHFWLRQ RIIHUV ZKHUHA = FCL = and m CD ZLWK VXEVFULSWV / DQG ' GHQRWH WKH LQWULFDWH FRPSOH[LW\ DQG VROXWLRQV VXI¿FLHQWO\ SUHFLVH UHVSHFWLYH OLIW DQG GUDJ FRHI¿FLHQWV SURSRUWLRQDOLW\ RI for application has been arrived, incorporating empirical WKHVH IRUFHV WR ÀXLG GHQVLW\ KDV QRW EHHQ LQFRUSRUDWHG explicitly thus they have dimensions of a density). As LQSXWV7KHIROORZLQJVLPSOL¿HGWUHDWPHQWSUHVHQWHGLQ mb the next paragraph explains the salient features of this there is no mbtranslational motion in r direction, balancing intriguing system. mb of forces in the radial direction require, 2 Forces acting on an air foil can be related to force Ar = F r = m ω bSin θ (6) ...(6) dt DFWLQJDWDSRLQWDQGDFRXSOHZKLFKFKDQJHVZLWKWKH dt dt DQJOH RI DWWDFN +RZHYHU WKHUH H[LVWV RQH SDUWLFXODU The force Fr is generated aerodynamically and compensates centrifugal acceleration. From equations SRLQWWKHVRFDOOHGDHURG\QDPLFFHQWUH&DURXQGZKLFK the magnitude of the couple remains independent of DQG LWIROORZV mb the angle of attack (Carlton, 2007; Pope, 2010). Forces acting on a falling maple samara are: the aerodynamic 4 CrL l Sin θ = ...(7) force A WKH UHVXOWDQW RI GUDJ DQG OLIW IRUFHV ZLWK mb components A = lift force, A Z ij WKH D]LPXWKDO WRUTXH dt 7KXVZULWLQJ A in the form in equation (5) is in agreement r Az ZLWKWKHREVHUYDWLRQWKDWLQFOLQDWLRQRIWKHZLQJWRWKH Z dt ≠ vertical direction remains constant throughout motion. C Ar Taking moments of forces about O, the instantaneous Aφ Fr FHQWUHRIURWDWLRQZHREWDLQWKHIROORZLQJHTXDWLRQRI G motion, θ ' dt dt dt 2 O d θ mg (b) IO = ArbCos θ − AzaSin θ + mgbSin θ ...(8) (a) 2 dt ωω ω dt Fig.3 (a) Schematic representation of the forces acting on the samara, center of gravity, aerodynamic dt dt Thus the equilibrium occurs at ș = q0 Figure 3: (a) Schematic representation of the forces acting on the samara: centre of gravity, aerodynamic centre and centre Tan θ = A / A = 4blC ω 2[( lC ω 24 − vlC 22 )a − mgb ]−1 − = (14) 0 − −r =z = rL(15) zL zD RI URWDWLRQ DUH PDUNHG DV * & DQG 2 UHVSHFWLYHO\ dt ...(9) E SRVLWLRQLQJRIWKHSRLQWV*DQG&RQWKHVXUIDFH ω (16) dt (17) Journal of the National Science Foundation of Sri Lanka 45(3) September 2017 dt ≠ ≠ ω dt− = − dt = dt (16) ω ω dt dt − = − = (14) (14) (16) (16)
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Kirthi Tennakone $VWKHGUDJIRUFHVLQWKHSURSHOOHUDUHODUJHO\D]LPXWKDO C mg opposing rotation rather than translation, the vertical v 2 = [ φD ] ...(17) (16) T 2 motion experience little drag force and dissipation results CzL CϕL l 2 2 2 mainly from rotation. In this situation CzL l Ȧ > CzL Dv and equation (9) can be approximated as, )URPHTXDWLRQV DQG ZHREWDLQ 4 2 24 −1 ga Tan θ0 ≈ rL blC ω [( CzL al ω − mgb ] ...(10) Cos θ0 = 2 ...(18) ω bl T mgb Thus a necessary condition for≠ equilibrium of the object The parameters , , and v are readily measurable, LQWKHRULHQWDWLRQVKRZQLQ)LJXUH D LV ș0 Ȧȉ T ~ 68 o, ~ WKHW\SLFDOYDOXHVIRU1RUZD\PDSOHEHLQJ ș0 Ȧȉ -1 2 mgb v ~ 1 ms (b ≠ )0 +] T ([SHULPHQWVVKRZWKDWWKHFRQHDQJOH ω > 4 ...(11) alC ș is not highly sensitive to m as seen from equation zL 0 )URPWKHVDPHHTXDWLRQZHREWDLQ O ED ) ~ 1.6 cm, dt giving EDa 7KLVIDFWRUGH¿QHVWKHSRVLWLRQRIWKH If the cone angle ș is perturbed by a small amount aerodynamic centre+ relative to the centre of gravity. İ, inserting ș = q0 + İ LQ HTXDWLRQ ZH REWDLQ DQ Again from equation (17) for a typical value of l ~ 3.8 cm, equation of the form d2İGW 2 = - Nİ ( k > 0) indicating that ωdt ZHREWDLQWKHOLIWWRGUDJUDWLR C /C = 9.4. This ratio is equilibrium resist perturbations. Equation (11) explains ij/ ij' dt VLJQL¿FDQWO\KLJKHJFRPSDUHGWRWKDWRIDKHOLFRSWHU that auto-gyration could occur only if the angular velocity -1 ∂Theω average terminal velocity (~ 0.6 ms LVVORZHULQ exceedsω a critical value. This is experimentally observed; %R[(OGHUVDPDUDVFRPSDUHGWR1RUZD\PDSOH$PRUH samara starts spinning only after falling through some − dt = (14) striking difference is the time taken to attain the terminal distance h. From equation (2), the maximum angular YHORFLW\ ZKLFK LV OHVV WKDQ KDOI WKH WLPH WDNHQ E\ velocity attainable is ( 2mgh /I )1/2 , giving h > bI [2 C a4l]-1 . ]/ 1RUZD\PDSOH7KHHORQJDWHGVHHGRI%R[(OGHUUHGXFHV − = (14) ∂ωits moment of inertia. Box Elder is a shorter tree (~ 20 m) Motion in the vertical(16) direction is described by, FRPSDUHGWR1RUZD\PDSOH aP ,WLVKDUGHYHQWR 2 zd VSHFXODWHWKHVHFUHWRIWKHGLIIHUHQFHRIWKHLUZLQJVKDSHV m = A − mg ...(12) dt 2 z (16) (Figure 1). ∂ω :LQG WXQQHO H[SHULPHQWV KDYH VKRZQ WKDW LQ D Thus the equation of motion for rotation can be expressed ω spinning maple samara, vortices are developed at the as, leading edge, greatly increasing the lift (Hederstrom, dt 2002; Sane, 2003; Lentink et al 3RVVLEO\WKHZLQJ dω 2 2 24 Iφ = (a + b)Sin θ0 ( Cl φLv − CφDl ω ) ...(13) VWULDWLRQVSOD\DFUXFLDOUROHLQFRQ¿QLQJWKHYRUWH[WRWKH dt − = leading edge and facilitate streamlines to leave the trailing edge smoothly. Striations are smoothed out by coating I = I Cos 2 ZKHUH ij ȅ ș PRPHQW RI LQHUWLD RI LQFOLQHG SDUDI¿QZD[DQGSROLVKLQJDQG vT increases drastically. seed− about= the axis of (14) rotation. The falling seed soon ,IWKHWUHDWPHQWLVJLYHQWRRQHZLQJVXUIDFHWKHVHHG approaches the terminal translation and angular velocities J\UDWH ZLWK WKH XQWUHDWHG VXUIDFH XSZDUGV VXJJHVWLQJ v = v , Ȧ = Ȧ ZKLFKDUHWKHSDUDPHWHUVUHOHYDQWWR WKDW WKH VWULDWLRQV KDYH DQ LQÀXHQFH LQ HQKDQFLQJ OLIW T ȉ ZLQGGLVSHUVDO7KHWHUPLQDOYDOXHVFRUUHVSRQGWRVHWWLQJ IRUFHV3RVVLEO\WKH\DFWDVYRUWH[FRQ¿QHUVDVZHOODV (16) WKH WLPH GHULYDWLYHV LQ HTXDWLRQV HTXDO WR JHQHUDWRUV7KHSXUSRVHRIWKHYHLQVLVQRWRQO\WR¿[ ]HURLH WKHVWUXFWXUHRIWKHZLQJEXWDOVRWRVHUYHDQHYHQPRUH important aerodynamic requirement. giving, 6LPSOH H[SHULPHQWV FRQGXFWHG ZLWK 1RUZD\ A − mg = 0 ...(14) maple and Box Elder seeds support the validity of z the relationship in equation (17), if l is taken as the 2 22 PD[LPXP OHQJWK RI WKH ZLQJ 2Q UHGXFLQJ WKH ZLQJ − = (14) Cφ vTL − CφDl ωT = 0 ...(15) DUHDE\FXWWLQJWKHZLQJEUHDGWKZLVHWHUPLQDOYHORFLW\ UDSLGO\GHFUHDVHV,IWKHUHGXFWLRQLVOHQJWKZLVHXSWR 2 mg ω = ...(16) DSRLQWWKHWHUPLQDOYHORFLW\GHFUHDVHVPRUHVORZO\,W T C l 4 (16) zL September 2017 Journal of the National Science Foundation of Sri Lanka 45(3) Aerodynamics and right-left symmetry in seed wind dispersal 207
DSSHDUVWKDWWKHVROHFULWHULRQRIZLQJGHVLJQLVQRWRQO\ ZRXOGEHUHVSHFWHGLIERWKVHQVHVRIURWDWLRQKDSSHQZLWK minimisation of terminal velocity but also propelling HTXDO SUREDELOLWLHV ,Q GURSSLQJ D VDPDUD IURP D ¿[HG DORQJ WKH ZLQG GLUHFWLRQ ,I PDSOH VHHGV DUH GURSSHG height, a large number of times, a higher probability of IURP VRPH KHLJKW DW WKH WLPH RI D EORZLQJ ZLQG WKH\ RQHVHQVHRIURWDWLRQLVIUHTXHQWO\REVHUYHG,IVL]HDEOH VRPHWLPHVPRYHXSRUGRZQLQZLGHKHOL[HVDVHGGLHV VDPSOHVDUHH[DPLQHGWKLVZD\WKHKLVWRJUDPREWDLQHG HQHUJLVH PRWLRQ 3RVVLEO\ WKH\ DUH ZRQGHULQJ LQ YRQ corresponds to a normal distribution, concluding that there .DUPDQVWUHHWV 1DUDVLPKDQ DVZLQGEORZSDVWDQ H[LVWVQRLQWULQVLFELDVLQJWRZDUGVFORFNZLVHRUFRXQWHU obstacle. FORFNZLVHDQGWKHREVHUYHGLQGLYLGXDOYLRODWLRQVKDSSHQ DV HIIHFWV RI DFFLGHQWDO LQÀXHQFHV RSHUDWLQJ ZLWKRXW D A question also arises regarding the stability of the universal bias. As only R-L distinguishing characters can ZLQJDJDLQVWURWDWLRQVDERXWWKHOHQJWKZLVHD[LV*&RI LQWURGXFHDELDVZHSHUIRUPHGDQXPEHURIH[SHULPHQWV WKHZLQJ,QDV\PPHWULFDOZLQJ WUDQVYHUVHFURVVVHFWLRQ to identify the factors that could introduce such biases. of Maple samara is symmetrical), the point of action of the position of the aerodynamic centre remains unaltered (i) In maple samaras the embedded seed symmetrically irrespective of changes in the attack angle. Therefore SURWUXGHV DERYH WKH ZLQJ VXUIDFH RQ ERWK VLGHV :H WKHPRPHQWIRUFHVDERXWWKHD[LV*&LV]HURLPSO\LQJ gradually sanded the protrusion on one side and observed that the move about axis is in neutral equilibrium. Thus, WKHVHQVHRIVSLQQLQJ:KHQWKLVLVGRQHWZRULJKWOHIW the system could adjust to maintain the angle of attack GLVWLQJXLVKDEOHREMHFWVDUHREWDLQHG )LJXUH 1RZWKH optimally. object (b) cannot be superposed on (a) by translations and rotations. If they are dropped from a height, during The importance of right-left asymmetry in wind descent the sanded surface tends to remain uppermost. dispersal of seeds $VOHDGLQJHGJHLVWKHRQHZLWKWKHULGJH D SUHIHUVWR URWDWH FORFNZLVH DQG E FRXQWHUFORFNZLVH EHFDXVH RI :LQG GLVSHUVDO QHFHVVLWDWHV VORZLQJ GRZQ RI WKH UDWH WKH FHQWUH RI JUDYLW\ ELDV 7KH ELDV LQFUHDVHV ZLWK WKH RI IDOO RI WKH VHHG VR WKDW ZLQG FRXOG FDUU\ WKH VHHG degree of sanding. When the seed protrusion is fully IXUWKHUDZD\IURPWKHSDUHQWWUHH7KHSK\VLFDOSULQFLSOH levelled, the probability for this trend is about 60 % for that enables optimising this strategy is generation of 1RUZD\PDSOHDQGIRUVXJDUPDSOH rotation so that transitional velocity of the centre of mass is continuously reduced. The additional advantage RIURWDWLRQLVDHURG\QDPLFOLIWRSSRVLQJJUDYLW\ZKLFK further reduces the rate of fall. Rotation and the direction RIWUDQVLWLRQDOPRWLRQGH¿QHDULJKWRUOHIWKDQGHGQHVV Such a handedness can be achieved either spontaneously (a) (b) or intrinsically. In the former situation, the object (seed) possesses no geometrical right-left asymmetry, but one sense of rotation is chosen as a consequence of the aerodynamics resulting from the design of the seed. In Figure 4: 7ZRPDSOHVDPDUDVZKHUHWKHSURWUXGLQJVHHGRQRQHVLGH the latter case the seed itself possesses a geometrical has been sanded off (a) sanded seed on the upper surface; right-left asymmetry giving preference to one sense of (b) sanded seed backside PRWLRQ,WLVDPD]LQJWRREVHUYHWKDWERWKWKHVHSULQFLSOHV DUH DGRSWHG LQ QDWXUH 7KH GLVFXVVLRQ EHORZ SUHVHQWV H[DPSOHVRIWZRFDVHV compensate for any weight gain, the trend continued
Right-left symmetry of falling maple seeds ϭϴ Most symmetries in living and non-living systems ;ĂͿ ;ďͿ including elementary constituents of matter, are not perfectly realised giving rise to frequent instances of biasing. The same feature is seen in the sense of rotation of maple samaras. As discussed earlier maple seeds possessFig.5 Two Maple samaras where one wing surface has been polished by rubbing paraffin wax (a) upper QRFKLUDOLW\WRGHFLGHDVHQVHRIURWDWLRQZKHQIDOOLQJ Figure 5: 7ZR PDSOH VDPDUDV ZKHUH RQH ZLQJ VXUIDFH KDV EHHQ nevertheless they auto-rotate as a result of spontaneous SROLVKHGE\UXEELQJSDUDI¿QZD[ D XSSHUVXUIDFHSROLVKHG breaking of the R-H symmetry. The required symmetry (b) backside polished
Journal of the National Science Foundation of Sri Lanka 45(3) September 2017
ϭϵ 208 Kirthi Tennakone
LL ,QDQRWKHUH[SHULPHQWRQHZLQJVXUIDFHZDVOLJKWO\ genetic accident of no value to the plant. There are FRDWHGZLWKSDUDI¿QZD[DQGSROLVKHGWROHYHOVWULDWLRQV also species having individuals of either handedness, )LJXUHLQGLFDWHVWZRVDPDUDVODLGRQDWDEOH D XSSHU FODVVL¿DEOHDV5+DQG/+PHPEHUV+DYLQJH[DPLQHG surface is polished and (b) the backside is polished. KXQGUHGVRISODQWVIDOOLQJLQWRWKLVFDWHJRU\ZHIRXQGWKDW When they are dropped, (a) preferentially spins counter- if they propagate from seeds, the RH and LH individuals FORFNZLVHDQG E FORFNZLVH$VWKHOHDGLQJHGJHLVWKH occur in equal (50 %) probabilities. Here the cause ridge, the result implies that they prefer to spin, facing seems to be a result of the dynamics of morphogenesis. WKHXQSROLVKHGVXUIDFHXSZDUGV:HFRQ¿UPHGWKDWWKH During cell proliferation, instability sets in and the plant HIIHFWZDVQRWFDXVHGE\WKHLQFUHDVHRIZHLJKW(YHQLI KDVWRVSLUDOHLWKHU5RU/ZLWKUHVSHFWWRWKHGLUHFWLRQ DSRUWLRQRIWKHVHHGVXUIDFHRQWKHRSSRVLWHZDVVFUDSHG RI JURZWK D FDVH RI VSRQWDQHRXV EUHDNLQJ RI 5+ WRFRPSHQVDWHIRUDQ\ZHLJKWJDLQWKHWUHQGFRQWLQXHG symmetry. We have observed that the phyllotaxy of leaf arrangement in un-branched papaya ( Carica papaya L.) 7KH REVHUYDWLRQ VXJJHVWV WKDW LQWHUIHULQJ ZLWK LVHLWKHU5RU/KDQGHG/DWHUDOEUDQFKHVLIWKH\JURZ striations affect the lift, and they seem to play an favour the handedness of the stem but reversals also important role in vortex formation and attachment. RFFXU:KHQSDSD\DVHHGVJHUPLQDWHWKH¿UVWIHZOHDYHV VKRZ QR VSLUDOLW\ EXW DV WKH JURZWK SURJUHVVHV WKH\ LLL $ELDVLQLQLWLDOFRQGLWLRQVZLOOJLYHDSUHIHUHQFHWR IROORZDSDWWHUQRIHLWKHU5RU/KHOLFLW\7KHUHDUHDOVR WKHVHQVHRIVSLQQLQJ,IWKHVDPDUDLVKHOGZLWKWKHSODQH VSHFLHV DQG HQWLUH IDPLOLHV ZKHUH PDQ\ RI WKH RUJDQV RIZLQJKRUL]RQWDODQGJLYHQDWRUTXHWRWXUQWKHOHDGLQJ SRVVHVV D KDQGHGQHVV +RZHYHU LQ D PDWXUHG SODQW RI HGJHIRUZDUGWKHREVHUYHGWUHQGZLOOEHWRFRQWLQXHVSLQ this class, RH and LH organs bear in equal probabilities. in the sense induced. The plants of malvaceae generally have branches or H[WHQGHGUHJLRQVLQDEUDQFKZKHUHWKHOHDIDUUDQJHPHQW A system governed by dynamics of SBS displays VKRZV D KDQGHGQHVV DQG WKH VHQVH RI LPEULFDWLRQ RI very high sensitivity even to minutest biases near the SHWDOV LQ ÀRZHUV FRUUHODWLQJ ZLWK WKH KDQGHGQHVV RI point of criticality. The observed biases in individual the parent branch (Tennakone et al ., 1982). The feature VHHGV RULJLQDWH IURP DFFLGHQWDO LQÀXHQFHV 7KH PDSOH displays more conspicuously in herbs belonging to the tree has no signs of R-L asymmetries. At this scale of family or young plants. Here the plant itself possesses a SKHQRPHQDWKH&RUROLVIRUFHZRXOGQRWKDYHDQHIIHFW KDQGHGQHVVEHFDXVHLWEHDUVPRUHEUDQFKHVDQGÀRZHUV of one kind (R or L) than the other. As the plant matures, LY 7KH VHQVHV RI URWDWLRQ RI WKH WZLQV RI WKH WZLQ it gradually sheds the handedness in bearing organs of VDPDUDV VKRZ D VOLJKW RSSRVLWH FRUUHODWLRQ 7KLV FDQ both types in equal numbers. The R-L distinction to be understood as drying and action of light could be XVKXPDQVLVRISDUDPRXQWLPSRUWDQFH:HZRXOGQRW different on the faces of the fruit. EH DEOH WR HQJDJH HYHQ LQ URXWLQH PDWWHUV ZLWKRXW WKLV ambidexterity. Although macroscopic R-L disparities Right-left symmetries in plants and the fascinating originate in plants as genetic accidents or consequences Dipterocarpus of morphogenesis, it is hard to see any advantages for survival. The exception seems to be the use of this trait Not only the fruit and the samaras, maple trees have no DVDQDGYDQWDJHIRUZLQGGLVSHUVDO macroscopic organs carrying an attribute of handedness. A majority of plant species falls into this group. Here Plants of dipterocapaceae and families in the order the leaves and their phyllotaxy, the pattern of branching PDOYDOHV WR ZKLFK WKH\ DOOLHV 'D\DQDQGDQ RU ÀRZHUV DQG IUXLWV GLVSOD\ QR 5/ GLVWLQJXLVKDEOH HQGRZ WKH GLIIHUHQWLDWHG RUJDQV ZLWK WKHVH SHFXOLDU characters. After extensive observations of symmetries characteristics. The R-L symmetry seen in dipterocarps is in plants, the author noticed that plants more primitive rarely seen in other families. They have spiral phyllotaxy in the phylogenetic ladder and the ones that mature fast DQGVKRRWVZLWK5DQG/OHDIVSLUDOVGLVWULEXWHGLQHTXDO possess R-L symmetries to a higher degree. There seems probabilities, and induce this symmetry to other organs to be a universal trend in all systems that, as complexity GHYHORSHGLQWKHWZLJ,QWKHGLVFXVVLRQWKHFRQYHQWLRQ LQFUHDVHDQGZKHQWKH\DGDSWWRSHUIRUPPRUHIXQFWLRQV XVHGWRGH¿QH5DQG/ZLOOEHDVIROORZV LIDVKRRWLV the symmetries disappear. Some plants demonstrate a KHOGYHUWLFDOZLWKWKHDSH[XSZDUGVDQGRQHOHDIDSH[ unique handedness in one or more of their organs as a SRLQWLQJWKHREVHUYHUWKHVSLUDOLQJLVGH¿QHGDV5 / genetically inherited quality common to the species. LI WKH QH[W OHDI MXVW DERYH SODFHV LWVHOI WRZDUGV 5 / 7KHDXWKRUQRWLFHGWKDWLQDSRF\DQDFHDHÀRZHUVLQDOO 7KH VDPH ZLOO EH DGRSWHG WR DVVLJQ WKH KDQGHGQHVV genera and species imbricate in the same sense- making WR WKH DUUDQJHPHQW RI SHWDOV DQG VHSDOV LQ D ÀRZHU WKHZKROHIDPLO\OHIWKDQGHG7KLVPD\EHDQLUUHYRFDEOH ,Q PRVW GLSWHURFDUSV ÀRZHUV EHDU DV D[LDO UDFHPHV
September 2017 Journal of the National Science Foundation of Sri Lanka 45(3) Aerodynamics and right-left symmetry in seed wind dispersal 209
popping out at the leaf brackets (Figure 6). As a result their population in Sri Lanka. Possibly the same applies the axles of racemes contain the same spirally. Again to other members of this genera, irrespective of the VSLUDOLW\ DSSHDUV LQ WKH VHWWLQJ RI ÀRZHU EXGV RQ WKH geographical location. raceme axle (Figure 6). The handedness of this spiral FKDQJHV DOWHUQDWLYHO\ LI WKH SDWWHUQ RI ÀRZHU EXGV LQ winged dipterocarps fruits of have very similar morphology A RQHUDFHPHLVULJKWWKHRQHVDERYHDQGEHORZDUHOHIW O Progression of spirals continues to other organs. The CR CL sepals and petals contort and imbricate in the same sense, F F H GH¿QLQJDKDQGHGQHVV+HUHDJDLQWKHVHQVHRIWKHVSLUDO GL GR A DOWHUQDWHV IURP RQH ÀRZHU EXG RU ÀRZHU WR WKH QH[W G In a majority of dipterocarps all sepals do not develop mg θ θ mg HTXDOO\DQGDIHZHORQJDWHDVZLQJVDWWDFKHGWRWKHQXW )LJXUH *HQHUDOO\WZRVHSDOVVWDQGLQJDWDQJOHV o N RIDSHQWDJRQHORQJDWHFRQFRPLWDQWO\DVWKHIUXLWJURZV JUHDWO\VKXQWLQJWKHJURZWKRIWKHRWKHUV1HYHUWKHOHVV WKHRULJLQDOKDQGHGQHVVRIWKHVHSDOVUHWDLQVHQGRZLQJ a right or left handedness to the fruits (Figure 7). Thus Mg ZLQJV VKDSHG DV 5+ RU /+ SURSHOOHUV IDOO URWDWLQJFig.8. LQ Forces acting on Dipterocarpus fruit and the points of their action: A - aerodynamic forces acting RSSRVLWH VHQVHV $ SURSHOOHU ZLWKRXW D KDQGHGQHVV Figure 8: Forces acting on Dipterocarpus fruit and the points of their action: A - aerodynamic forces acting at centres C and C ; cannot rotate and propel. We observed that there is no L R detectable disparity in the abundance of RH and LH )FHQWULIXJDOIRUFHVDFWLQJDW* LDQG* R PJZHLJKWRIWKH ZLQJVDFWLQJDW* DQG* 0JZHLJKWRIWKHQXWDFWLQJDW1 nuts in an individual tree of Dipterocarpus zeylanicus or L R
7KHSURSHOOHUVRIDQDLUSODQHRUDVKLSJHQHUDWHDIRUZDUG WKUXVWZKHQURWDWHGLQRQHVHQVHDQGEDFNZDUGWKUXVWLQ the opposite sense of rotation. The trailing edge of the Ϯϯ GLSWHURFDUSZLQJLVWKHVOLJKWO\EXOJHGOREH7KHUHIRUH WKH 5+ IUXLW )LJXUH D ZLOO URWDWH FRXQWHUFORFNZLVH ZKHQ YLHZHG IURP DERYH DV LW IDOOV DQG WKH /+ IUXLW )LJXUHE ZLOOURWDWHFORFNZLVH
Forces acting on a falling fruit are the aerodynamic forces A ( A , A , A ) Z r ij DFWLQJRQWKHZLQJVDWWKHFHQWUHV CL and C R; centrifugal forces F acting at the centres of mg (a) (b) JUDYLWLHV* LDQG* RRIWKHZLQJVZHLJKW RIWKHZLQJV DFWLQJDWWKHLUFHQWUHVRIJUDYLW\* LDQG* RDQGZHLJKW Mg of the nut acting at N (Figure 8). When the fruit is VSLQQLQJDQJOHEHWZHHQWKHZLQJV ș) open up a little Figure 6: Right-handed (a) and left handed (b) axle racemes of a ELWXQWLOLWLVHTXLOLEUDWHGE\WKHHODVWLFIRUFHVLQWKHZLQJ and centripetal force. If the fruit is perturbed by tilting typical dipterocarp ϮϮ inclination from ș to ( ș + İ ), the resulting motion is DERXW*GHVFULEHGE\
d 2ε I −= dA ε G 2 z ...(19) dt ZKHUH d 2* s +* GDQG l = *1:KHQWKHWHUPLQDO (a)(a) (b)(b) velocity has reached A = (M+2m)g and the equilibrium Z Fig.7 Right-handed (a) and (b) left handed fruits of Dipterocarpus zeylanicus . Other two is stable, provided d > 0. Thus there exist limitations to Figure 7: Right-handed (a) and left handed (b) fruits of Dipterocarpus PDVVDQGDUHDGLVWULEXWLRQRIWKHZLQJV6LPLODUO\LIWKH CR CzeylanicusL 2WKHUWZRZLQJHGIUXLWVRIGLSWHURFDUSVKDYHD centre of gravity is shifted to either right or left of the line very similarGL morphologyGR 21VWDELOLW\ZRXOGEHDIIHFWHG G + − ∆τ N Journal of the National Science Foundation of Sri Lanka 45(3) September 2017 ∂ω
Ϯϯ ∂ω
∂ω 210 Kirthi Tennakone
The dynamics governing rotational and translational result of inherent handedness in the sepal arrangement in motion of the stabilised system parallel that of the WKHFDO\[IUXLWYDULDWLRQVZLWKORQJHUVHSDOVZRXOGVSLQ maple seed and equations (14) – (17) illustrate this. IDVWHUVORZLQJGRZQWKHGHVFHQWDVVLVWLQJZLQGGLVSHUVDO During the decelerating phase of motion, stability DQGWKLVDGYDQWDJHRXVWUDLWZRXOGKDYHDPSOL¿HGOHDGLQJ JHWVFRPSOLFDWHGGXHWRWKHÀH[LELOLW\RIWKHZLQJ2Q WRVSHFLHVZLWK¿YHHORQJDWHGVHSDOV$OWKRXJK¿YHZLQJV ORDGLQJWKHQXWDUWL¿FLDOO\DSRLQWLVUHDFKHGZKHQWKH ensure stability, there are number of disadvantages. It ZLQJV GR QRW RSHQ DQG WKH URWDWLRQDO HIIHFW FHDVHV WR LQFUHDVHVWKHZHLJKWDQGDOVRWKHPRPHQWRILQHUWLDDERXW IXQFWLRQSRVVLEO\EHFDXVHRIWKHLQVXI¿FLHQF\RIWKHOLIW the rotational axis, making it harder to rotate in gaining forces or their diversion. In nature this does not happen, the lift, and both factors diminish the terminal velocity unless evolution has turned retrograde due to changing and increase the time elapsed to reach the terminal value. HQYLURQPHQW:HLJKWRIWKHZLQJVLQUHODWLRQWRWKHQXW )XUWKHUPRUHORQJHUZLQJV\LHOGPRUHOLIWEHFDXVHWKH DQG ZLQJ GLPHQVLRQV EHLQJ ZHOO EDODQFHG WKH\ UDUHO\ counting factor (velocity) 2 YDULHV DV ZLQJ OHQJWK 2. FROODSVHZKHQIDOOLQJ$XWRJ\UDWLQJVHHGVKDYHHYROYHG ,QFUHDVLQJWKHOHQJWKRIWKH¿YHZLQJVZRXOGEHWRRPXFK to optimise the lift keeping drag to a minimum and also RIDEXUGHQWRWKHRUJDQLVPDQGEURDGHU¿YHZLQJVZLOO to acquire the terminal value as soon as possible. Just DIIHFWDLUÀRZ7KXVWKHRQO\RSWLRQDYDLODEOHZRXOGEH as in maple, the strategy has been to dissipate potential WRFXWGRZQWKHQXPEHURIZLQJV,IDYDULDWLRQSURGXFHV energy via rotation. According to equation (17), larger DYDULHW\ZLWKIRXUZLQJVWKDWVWUDLQZLOOEHLQIHULRU OLIWDQGORZHUGUDJFRHI¿FLHQWVIDYRXUVPDOOHUWHUPLQDO because of the lesser degree of stability in falling of the YHORFLWLHV 'LSWHURFDUSV ZLWK KHDYLHU QXWV KDYH ORQJHU fruit, as the centre of gravity shifts furthest from the EXWQRWVREURDGHQHGZLQJV7RDJRRGDSSUR[LPDWLRQ FHQWUHZKHQWKHQXPEHURIZLQJVLVIRXU7KHQH[WRSWLRQ the terminal motion of a falling dipterocarpus nut can be ZLOOEHWRKDYHWKUHHZLQJVLQSRVLWLRQVDQGRIWKH described by equation (17). It has been suggested that the SHQWDJRQZKHQWKHFHQWUHRIJUDYLW\VKLIWVRQO\[ WHUPLQDOYHORFLW\DFTXLUHGE\ZLQGGLVSHUVLQJVHHGVYDU\ IURPWKHFHQWUH$VH[SHFWHGLQWKUHHZLQJHGVSHFLHVWKH ZLWKWKHVTXDUHURRWRIWKHPDVVWRDUHDUDWLR $XJVSXUJHU ZLQJVDUHSRVLWLRQHGDWYHUWLFHVRIWKHSHQWDJRQ )UDVRQ *UHHQH 4XHVDGD (TXDWLRQ $PD]LQJO\DOPRVWDOOH[WDQWDQGH[WLQFW0LRFHQHIRVVLO LQFRUSRUDWHV WKLV WUHQG EXW PRUH VSHFL¿FDOO\ WKH GLSWHURFDUSV 6KL /L )HQJ et al ZLWK parameter ‘ l’ in equation (13) much more closely the WKUHHZLQJVKDYHRQHFRQVSLFXRXVO\VKRUWHULHLIWKH OHQJWKRIWKHZLQJ2XUH[SHULPHQWZLWK Dipterocarpus ZLQJVDUHDI¿[HGDWWKHYHUWLFHVWKHVKRUWHUZLQJ zeylanicus VKRZHGDYDULDWLRQYHU\FORVHWRWKHVTXDUH is either 1 or 2. Clearly, this variation adjusts the centre URRWRIWKHPDVVWRZLQJOHQJWK of gravity closer to the centre of the pentagon. Possibly dipterocarps proliferated and became invasive after Evolution of dipterocarp wings WKHRULJLQRIWKUHHZLQJHGVSHFLHVDQGWKHLUVXFFHVVIXO GLVSHUVLRQE\ZLQG1RZWKHGLSWHURFDUSVKDYHDFTXLUHG 'LSWHURFDUSVKDYHVSHFLHVZLWKIUXLWVKDYLQJWZRWKUHH DJHQHWLFIDFLOLW\WRFRQWUROWKHOHQJWKRIRQHZLQJLH RU¿YHVHSDOVPRUHHORQJDWHGWRIRUPZLQJV7ZRZLQJHG D ZLQJ LQ SRVLWLRQ RU LQ WKH DUUDQJHPHQW ones seem to be more abundant compared to three and &HUWDLQO\VSHFLHVZLWKDYHU\VKRUWDQGDOPRVWVKXQWHG ¿YHZLQJHGVSHFLHVDQG¿YHZLQJHGYDULHWLHVDUHOHDVW WKLUGZLQJ RUSRVLWLRQV KDYHHPHUJHGDQGDWUDLWWR DEXQGDQW7KHDXWKRUFRXOGQRW¿QGDQ\HYLGHQFHLQWKH OLWHUDWXUHIRUH[LVWHQFHRIVSHFLHVZLWKIRXUZLQJV:LQJV 1 KDYHHYROYHGIURP¿YHVHSDOVFHQWUHGRQWKHYHUWLFHVRI DSHQWDJRQ7KHIROORZLQJFRQVLGHUDWLRQVFRXOGEHDFOXH DVWRKRZWKHZLQJVFRXOGKDYHHYROYHG,IHTXDOZHLJKWV 2 DUHSODFHGDWWKHYHUWLFHVRIDUHJXODUSHQWDJRQZLWKVLGH 5 RIOHQJWK[WKHFHQWUHRIJUDYLW\ZLOOEH REYLRXVO\ at the centre of the inscribing circle if all the vertices are loaded; (2) ~ 1.4 x from the centre if the vertices 1, 2, 3,4 or any of the equivalent vertices are loaded; (3) ~ 1.3 x from the centre if the vertices 1, 2, 3 are loaded; 4 3 (4) ~ 0.88 x from the centre if the vertices 1, 3,Fig. 4 9 are Sepals of deterocarps are placed at the vertices of pentagon. The vertices are marked 1 loaded; (5) ~ 1.5 x from the centre if the vertices 1, 2 are Figure 9: Sepals of dipterocarps are placed at the vertices of a ORDGHG a [LIWKHYHUWLFHVDUHORDGHG7ZR SHQWDJRQ7KHYHUWLFHVDUHPDUNHGIRUWKHSXUSRVHRI ZLQJHGRQHVVHHPWREHPRUHDEXQGDQWFRPSDUHGWRWKUHH LGHQWL¿FDWLRQRIWKHUHODWLYHSRVLWLRQVRIHORQJDWHGVHSDOV DQG¿YHZLQJHGDQG¿YHEHLQJWKHOHDVWDEXQGDQW$VD WKDWIRUPZLQJV
September 2017 Journal of the National Science Foundation of Sri Lanka 45(3)
Ϯϳ Aerodynamics and right-left symmetry in seed wind dispersal 211
EHQGZLQJVVRDVWRDGMXVWWKHFHQWUHRIJUDYLW\HYROYHG $VFKHPDWLFUHSUHVHQWDWLRQ )LJXUH VKRZVKRZWKH VXEVHTXHQWO\'HYHORSLQJWZRZLQJVUHOLHYHWKHWUHHRI ZLQJVRIDW\SLFDOGLSWHURFDUSDUHDWWDFKHGWRWKHQXWWKH WKHEXUGHQRIV\QWKHVLVLQJPRUHELRPDVVDQGWZRZLQJV RULHQWDWLRQDQGVWUXFWXUHRIWKHZLQJVXUIDFH Longitudinal SHUIRUP EHWWHU LQ VORZLQJ WKH IDOOLQJ VSHHG *HQHUDOO\ veins covering the surface are more or less straight on evolution proceeds by gradual trial and error corrective RQHVLGHRIWKHZLQJDQGVOLJKWO\FXUYHGGLYHUJLQJRQWKH VWHSV DQG QRW E\ TXDQWXP MXPSV ,W ZRXOG EH KLJKO\ RSSRVLWHVLGHRIWKHVDPHZLQJDQGWKHSDWWHUQLVUHYHUVHG LPSUREDEOHIRUWKHWZRZLQJHGVSHFLHVWRKDYHHYROYHG LQWKHWZRZLQJV'XULQJIDOOLQJWKHQXWURWDWHVDURXQG LQRQHVWHSE\FXWWLQJGRZQWKUHHZLQJVIURPWKH¿YH the vertical axis (extension of the straight line segment at 7KH K\SRWKHVLV ZH KDYH SURSRVHG FDQ EH WHVWHG E\ WKHERWWRPRI)LJXUH FORFNZLVHDVHGJHVRIWKHZLQJV examining the phylogeny of species of Dipterocarpaceae PDUNHGZLWKDUURZVDUHOHDGLQJHGJHV9HLQVGLYHUJHDQG ZLWKGLIIHUHQWQXPEHURIZLQJV FXUYH WRZDUGV WKH WUDLOLQJ HGJH DV LI WKH YHLQV LPSULQW WKH ÀRZ OLQHV7KH ZLQJV DUH DOVR FRYHUHG ZLWK D ¿QH The conclusions of some experiments conducted PRVDLF RI YHLQV PRUH FRQFHQWUDWHG WRZDUGV WKH XSSHU WRPHDVXUHGLVWDQFHRIGHÀHFWLRQ D) of dipterocarpus HQGRIWKHZLQJDQGWKHOHDGLQJHGJH:HEHOLHYHWKDW VHHGVVXJJHVWWKDWWKH\GRQRWPRYHEH\RQGDIHZWHQV WKHVHPRVDLFVVHUYHDVYHUWH[FRQ¿QHUVWRHQKDQFHWKH of meters from the parent tree, and under conditions of OLIW:LQGWXQQHOH[SHULPHQWVKDYHVKRZQWKDWDWWDFKHG QRPLQDOZLQGVSHHGVDQGRFFDVLRQDOKLJKVSHHGZLQGV OHDGLQJHGJHYRUWLFHVIRUPZKHQDPDSOHVHHGVSLQVDQG may carry them a longer distance (Smith et al ., 2015; WKHUHVXOWLQJORZSUHVVXUHFUHDWHVWKHOLIWDQGWRUTXHDQG Song, 2015). The crucial deciding parameter should be also provides centrifugal force. It appears that the same mechanism operates in the case of dipterocarps. The the terminal velocity vT and mere measurement of D does not give the information necessary for determining angular momentum of the spinning seed is compensated WKH GLVSHUVDO SRWHQWLDO )XUWKHUPRUH ZLQGV KDYH ERWK by the vorticity. u and u , KRUL]RQWDODQGYHUWLFDOO\XSZDUGFRPSRQHQWVangular momentumH of thev spinning seed is compensated by the vorticity respectively. As aerodynamic forces are proportional to the square of air speeds relative to a moving object, the terminal velocity in the presence of an updraft of speed u is given by, mg C ϕD 2/1 ...(20) vT = vT 0 − uV = [ 2 ] − uV a CφL CzL ZKHUH v is the terminal velocity in the absence of the T0 XSGUDIWFRPSRQHQWRIWKHZLQG7KHXSGUDIWJHQHUDWHVDQ+ − ∆τ additional lift and the possibility of v becoming negative T LPSOLHV WKDW WKH QXW FRXOG HYHQ SURSHO D VLJQL¿FDQW v GLVWDQFH XSZDUGV EHIRUH T DSSURDFKLQJ ]HUR DQG VWDUW GHVFHQGLQJ 6XSSRVH D ZLQG KDV FRQVWDQW KRUL]RQWDO speed∂ω u and vertical component u greater than v H Fig.10.9'IJ Schematic diagramT0 illustrating the orientation of wings of a typical dipterocarp, relative to lasting for time then remaining constant at u v . 'IJ V T0 Figure 10: 6FKHPDWLFGLDJUDPLOOXVWUDWLQJWKHRULHQWDWLRQRIZLQJVRI If the canopy height = hWKHGHÀHFWLRQGLVWDQFHZLOOEH D W\SLFDO GLSWHURFDUS UHODWLYH WR WKH QXW 2QO\ WZR YHLQV GRWWHG OLQHV GUDZQ WR LQGLFDWH KRZ WKH\ GLYHUJH DQG h + (u − v )∆τ ∂ωD = u ∆τ +[ V∆τ T 0 ]u OHDGLQJ HGJHV DUH PDUNHG ZLWK DUURZV:LQJV DOVR KDYH H H ...(21) ¿QHPRVDLFRIYHLQVWKURXJKRXWEXWFRQFHQWUDWHGQHDUWKH (vT 0 − uV ) VKDGHGVSRWVVXJJHVWLQJWKDWYRUWLFHVDUHFRQ¿QHGWRWKDW region. We∂ω found that the average velocity for Dipterocarpus -1 zeylanicus fruits is vT0 ~ 3 ms *LYLQJ WKH IROORZLQJ∂ω SODXVLEOH YDOXHV IRU RWKHU SDUDPHWHUV LH Entertaining and intriguing motion of paper h = 40 km, u = 2 ms -1 , u = 5 ms -1 , u = 1 ms -1 H 9'IJ V ZH helicopters simulating dipterocarps obtain D = 2.4 km. Dipterocarps have intricately GHVLJQHGZLQJVDWPXFKFRVWWRWKHLQGLYLGXDODVVXFK∂ω 7RVLPXODWHGLSWHURFDUSZLQGGLVSHUVDOZHDOVRH[DPLQHG WKHLU SXUSRVH FRXOG QRW EH MXVW IRU D VWRQH¶V WKURZ SDSHUKHOLFRSWHUVZKLFKFKLOGUHQPDNHDQGVRPHWLPHV deposition from the parent. GHPRQVWUDWHG LQ D ZLQG WXQQHO LQ VFLHQFH PXVHXPV
Journal of the National Science Foundation of Sri Lanka 45(3) September 2017 ∂ω
Ϯϵ 212 Kirthi Tennakone
Despite the simplicity of construction, they display the FORFNZLVHEUHDNLQJWKH5/V\PPHWU\VSRQWDQHRXVO\,I necessity of a handedness. They can be crafted giving a DQ\DV\PPHWU\LVLQWURGXFHGWRWKHZLQJVRIWKHPRGHO KDQGHGQHVVWR¿[WKHVHQVHRIURWDWLRQRUWREHV\PPHWULF the performance greatly improves. In models (d) and and break the R-L symmetry spontaneously. H GDUNHQHG UHFWDQJXODU SRUWLRQV LQ WKH ZLQJV KDYH been cut-off, and the performance remarkably improves 7KHWR\V D DQG E VKRZQLQ)LJXUHDUHIDEULFDWHG ZLWK G URWDWLQJ FRXQWHUFORFNZLVH DQG H FORFNZLVH by cutting a single sheet of paper and folding; the dark $VLPLODUDV\PPHWU\FDQEHLQWURGXFHGE\WZLVWLQJWKH UHFWDQJOHGHQRWHVWKHSDSHUFOLSXVHGDVDZHLJKW7KH\ ÀDSV ,I D ZLOIXO DV\PPHWU\ LV QRW LQWURGXFHG WR WKH are mirror images of each other and distinct, because PRGHO F DW KHDYLHU ORDGLQJV WKH ZLQJV WZLVW RQ LWV WKHUH H[LVWV QR DXWRPRUSKLVP EHWZHHQ WKHP :KHQ RZQ EUHDNLQJ WKH V\PPHWU\ 2XU ¿QGLQJV DJUHH ZLWK GURSSHG DQG YLHZHG IURP DERYH D URWDWHV FORFNZLVH WKHUHVXOWVRIVLPLODUH[SHULPHQWVFDUULHGRXWZLWKSODQH DQG E FRXQWHUFORFNZLVH&RSWHU F LVPDGHE\SDVWLQJ VLQJOHDQGGRXEOHZLQJVPDGHRISDSHU 6WHYHQVRQ et al. , WZRKDOIJOXHGUHFWDQJXODUVWULSVDQGIROGLQJ0RGHO F LV 5HVXOWVVKRZWKDWWZRZLQJVV\VWHPVURWDWHVWDEO\ 5/V\PPHWULFQHYHUWKHOHVVZLOODOVRURWDWHLQRQHVHQVH DQGGULYHDVLJQL¿FDQWOLIWSURYLGHGWKHUHLVDQLQKHUHQW other. Careful examination reveals a clear difference of asymmetry. the behaviour of (c) compared to asymmetric models D DQG E ,IFDUHLVWDNHQWRNHHSWKHZLQJVKRUL]RQWDO Right-left symmetry and wind dispersing conifers ZLWKRXWWZLVWLQJWKHSHUIRUPDQFHRIPRGHO F LVIRXQG WR EH SRRU DQG XQVWDEOH XQOHVV ZHLJKW RI WKH SDSHU ,WZDVLQWHUHVWLQJWRREVHUYHWKDWFRQLIHUVDOVRSRVVHVV FOLSVLVLQFUHDVHG2QFHVXI¿FLHQWO\ORDGHGLWZLOOIDOOD right-left symmetries though not so conspicuous, but ORQJHUGLVWDQFHDQGURWDWHHLWKHUFORFNZLVHRUFRXQWHU akin to the pattern seen in dipterocarps. Examination
(a) (b)
(c)
(d (e)
Figure 11: Toy paper helicopters (a) and (b) are mirror images of each other and (c) is symmetric and has no handedness. (d) and (e) are asymmetric models made by UHPRYLQJWKHVKDGHGUHFWDQJXODUSRUWLRQIURPHDFKZLQJ
September 2017 Journal of the National Science Foundation of Sri Lanka 45(3)
ϯϬ Aerodynamics and right-left symmetry in seed wind dispersal 213
RISLQHVUHYHDOHGWKDWWZLJVKDYHKDQGHGQHVV7KHOHDI Right-left symmetry in magnoliaceae and auto- IDVFLFOHVLQSLQHDUHDUUDQJHGDVWZRVSLUDOVZLQGLQJLQ gyrating samaras of the tulip tree RSSRVLWH GLUHFWLRQV +RZHYHU WKH QXPEHU RI ZLQGLQJ turns the spiral takes to reach the same angular position The R-L symmetry in magnoliaceae resembles that RIWKHIDVFLFOHLPPHGLDWHO\DERYHRUEHORZDUHGLIIHUHQW of pines and dipterocarps. Having examined Walsapu LQWKHWZRVSLUDOVJLYLQJDQHWKDQGHGQHVVWRWKHWZLJ -XVW OLNH LQ GLSWHURFDUSV WKH WZLJV RI ERWK NLQGV DUH equally distributed. Cones have the same feature, there DJDLQWKHVDPDUDVLQWKHIUXLWDUHDUUDQJHGIROORZLQJWKH VDPHSDWWHUQDVWKHIDVFLFOHVVRWKDWWKHUHDUHWZRW\SHV of cones, RH or LH born in equal abundance. Samaras are bilaterally asymmetric, but all of them, similar in one kind of cone (R or L) but the mirror image in the RWKHUW\SH /RU5 7KHYHQWUDOYLHZRIWKHVDPDUDVRI5 and L cones of Eastern White Pine ( Pinnus strobes ) are VKRZQLQ)LJXUH$VERWKOREHVDUHIROGHGXSZDUGV the objects are geometrically different and cannot be superposed on each other. Wind dispersing species of pine and other conifers generally have more asymmetric͘ϭϮ͘ Figure 12: 7KHYHQWUDOYLHZRIWKHVDPDUDVRIULJKWDQGOHIW seeds. handed cones of Eastern White Pine.
(a) (b)
Figure 13: D 7KHGRUVDOYLHZRIWKHVDPDUDVRIULJKWDQGOHIWKDQGHGSRGVRIWKHWXOLSWUHH E UHFWDQJXODU VKHHWVRISDSHUSDVWHGZLWKDOHWWHU/FXWIURPSDSHU anticlockwise when viewed from above (Figure 14a). (Michelia nilagrica DQG D IHZ UHODWHG VSHFLHV LQ Sri Lanka, the tulip tree ( Liriodendron tulifera L.) DQG RUQDPHQWDO PDJQROLDV LQ WKH 8QLWHG 6WDWHV ZH REVHUYHG WKDW DOO WKHVH JHQHUD KDYH WZLJV RI VSLUDO phyllotaxy and R-L helicities occur in equal proportion. Sepals and petals are almost indistinguishable but the KDQGHGQHVV RI WKH WZLJ SDVVHV RQ WR WKH ÀRZHU DQG WKH IUXLW ,Q ÀRZHU EXGV WZR RSSRVLWH VSLUDOV DUH VHHQ (normally 5/8 Fibonnacci) correlated to the sense of leaf phyllotaxy. Just as in dipterocarp and pine there are R (a) (b) and L fruits. The R (L) fruit of the tulip tree has R(L) samaras readily distinguishable from the direction of WZLVWLQWKHULGJHSODFHGPLGZD\DQGLQWKHOHQJWKZLVH direction (Figure 13a). Samaras of R and L pods are Figure 14: (a) Spinning of a falling Pumeria ÀRZHU ORDGHG ZLWK mirror images of ϯϰeach other and no automorphism exists a match stick; (b) a parachuting seed of Wrighta because of ventral-dorsal difference. During descent ϯϮ Wrighta antidyscentericaantidyscentericaas viewedDVYLHZHGIURPDERYH from above D DQG E J\UDWHLQRSSRVLWHVHQVHVZLWKPLGGOHULGJH
Journal of the National Science Foundation of Sri Lanka 45(3) September 2017 ϯϯ 214 Kirthi Tennakone as the axis. R- L asymmetry provides torque for the DGRSWHG $Q HQWHUWDLQLQJ ZD\ WR VHH WKLV ZRXOG EH WR rotational motion, creating a lift. The moment of inertia ORDGDQDSRF\DQDFHDHÀRZHU HJ Pumeria / VWDONZLWK is minimum about the mid axis and motion remains DPDWFKVWLFNDQGOHWLWIDOO$OZD\VWKH\URWDWHFRXQWHU stable. It has been suggested that this stability greatly FORFNZLVHZKHQYLHZHGIURPDERYH )LJXUHD helps dispersal, although the terminal velocities attained are smaller compared to maple (Mccutchen, 1977). A majority of apocyanaceae has no edible fruits, LQVWHDG WKH\ FRQWDLQ SRLVRQRXV DONDORLGV )ORZHUV DUH We have also conducted simple paper toy experiments IUDJUDQWO\ VFHQWHG DQG KHDYLO\ ORDGHG ZLWK QHFWDU to simulate the auto-gyration of tulip tree samaras. suggesting that their ancestry has been in the insect era Figure 13(b) depicts four rectangles (~ 2.5 x 10 cm 2) ZHOOEHIRUHWKHELUGV0DQ\VSHFLHVRIDSRF\DQDFHDHKDYH FXWRXWRIRUGLQDU\WKLQZULWLQJSDSHU,QRQHRIWKHP SOXPHGVHHGVGLVSHUVHGE\ZLQG7KHOHIWKDQGHGQHVVRI a thin rectangular strip of paper is pasted symmetrically the family transcends to fruits and seeds. As a result of DQGWKHUHPDLQLQJWZRZLWKSDVWHGOHWWHU/FXWIURPWKH WKH FKLUDOLW\ SDUDFKXWLQJ VHHGV VORZO\ URWDWH +HUH WKH VDPH SDSHU DV VKRZQ %HFDXVH RI WKH WKLFNQHVV RI WKH advantage of rotation seems to be not the generation of a OHWWHU/SDVWHGWKHWZRFRQ¿JXUDWLRQVLQ E ZLWKWKH lift, but an enhancement of the drag. The centripetal force letter L are distinct objects and mirror images of each RI URWDWLRQ VSUHDGV WKH SOXPH LQ WKH KRUL]RQWDO SODQH RWKHUDQGWKH¿UVWWZRKDYHKDQGHGQHVV:KHQGURSSHG increasing the drag. The appearance of the long hairs IURPDKHLJKWDOOIRXUJOLGHDQGURWDWHZLWKOHQJWKZLVH in a parachuting Wrighta ( Wrighta antidyscenterica ) is VLGHUHPDLQLQJQHDUO\KRUL]RQWDOEXWZLWKWKHIROORZLQJ VKRZQLQ)LJXUH E differences: the symmetric ones rotate in either sense ZKHQ YLHZHG IURP D ¿[HG GLUHFWLRQ UHODWLYH WR WKH On origin of handedness in plants UHFWDQJOH KRZHYHU WKH RQH ZLWK WKH PLGGOH ULGJH LV PXFKPRUHVWDEOH WKHRQHVSDVWHGZLWKOHWWHUV/URWDWH Recent experiments suggest that auxins trigger spiral in opposite directions. A simple experiment sometimes positioning of leaves (Reinhhardt et al ., 2003; Niklas JHWV FRPSOLFDWHG RZLQJ WR WKH WZLVW GHYHORSHG ZKHQ et al ., .UDPHU &KLWZRRG +RZHYHU the strips are pasted. Many experiments have been ZKDW GHWHUPLQHV WKH VLJQ RI KDQGHGQHVV UHPDLQV performed to study the aerodynamics of falling paper, XQDQVZHUHG 7KH OHDYHV RI WRPDWR KDYH EHHQ PDGH WR cards and discs and the general conclusion has been break the bilateral symmetry by application of auxins WKDWWKH\DUHWXPEOLQJDQGXQVWDEOH 3HVDYHQWR :DQJ to the leaf primordium. A rare example of a herb &KDQJTLX ;X 9DUVKQH\ et al ., 2013). conspicuously displaying breaking of bilateral symmetry We have found that the mid ridge greatly stabilises the is Agalonema pseudo-bracteatum 7HQQDNRQH ,TEDO PRWLRQ$QRWKHULPSRUWDQWGLIIHUHQFHZHREVHUYHGLQRXU ZKHUHULJKWDQGOHIWKDQGHGOREHVJUHDWO\GLIIHULQ H[SHULPHQWVZLWKIDOOLQJWKLQQHUÀH[LEOHSDSHUUHFWDQJOHV DUHDDQGGLVSDULW\LVKLJKHUDWVHDVRQVRIIDVWHUJURZWK is the stability and absence of tumbling. The scaling of 7KHVH DUH FOHDUO\ DPSOL¿FDWLRQ HIIHFWV RI DQ DOUHDG\ these systems depend not only on the aspect ratios and the existing spiral phyllotaxy. 5H\QROGV QXPEHU ZKHQ HODVWLF GHIRUPDWLRQ HQWHU LQWR the picture. If the length of the bare rectangle is increased Leonardo da Vinci noted that the cross-sectional WRFPWKHSDSHUURWDWHVYHU\VWDEO\LQKRUL]RQWDO area of a tree trunk approximates to the sum of areas of RULHQWDWLRQDQGERWKVHQVHVRIURWDWLRQLVREVHUYHGZLWK EUDQFKHVDVLIJURZWKLVOLNHWKHÀRZRIDQLQFRPSUHVVLEOH near equal probability. Here, the paper rectangle during ÀXLG 5LFKWHU %DVHG RQ WKH LGHD ZH JLYH DQ motion spontaneously acquires handedness because of H[SODQDWLRQIRUDX[LQVHIIHFWDVIROORZV the elastic deformation and spin stably. The effect could have important implications in biological systems. The Navier Stoke equation in vorticity form reads,
The case of apocyanaceae ∂ω 1 2 ×∇= (u ×ω) + ∇ ω ...(22) dt Re One of most unusual chirality features are seen in
DSRF\DQDFHDH ,Q DOPRVW DOO JHQHUD ZH H[DPLQHG WKDW ZKHUH u ÀXLGYHORFLW\ Ȧ ¨î u and Re = Reynolds signs of a handedness is not seen in phyllotaxy or number∂ω EUDQFKLQJ +RZHYHU ÀRZHUV RI WKH HQWLUH IDPLO\ ZH have not seen exceptions) are imbricated in the same The vorticity equation (22) can also be cast into the form VHQVH OHIWKDQGHGDFFRUGLQJWRWKHFRQYHQWLRQZHKDYH (uîȦ
∂ω
September 2017 Journal of the National Science Foundation of Sri Lanka 45(3)
Aerodynamics and right-left symmetry in seed wind dispersal 215 ∂ω 1 2 CONCLUSION = ω.∇u − u.∇ω + ∇ ω ...(23) dt Re ∂ω 1 We make the reasonable assumption that ∇u2 = G and 0DSOH DQG GLSWHURFDUSV DUH WZR WUHH VSHFLHV PRVW R VXFFHVVIXOO\SUROLIHUDWHGRZLQJWRWKHHYROXWLRQRIVHHGV Ȧ DUHLQSDUDOOHOGLUHFWLRQV JURZWKGLUHFWLRQ DQG e u and ∂ω 1 2 ZLWKZLQJDSSHQGDJHV:LQJVHIIHFWLYHO\VORZGRZQWKH ∇∂ȦDUHRUWKRJRQDO7KXVHTXDWLRQ DSSUR[LPDWHVWRω gravitational descent via generation of a rotational lift. Re a reaction diffusion equation, $ WKHRUHWLFDO IRUPDOLVP ZDV GHYHORSHG WR H[SODLQ WKH ∂ω observed motion, account for the stability and calculate ∂∂ωω 1 2 = ωωG G + ∇ ω ...(24) terminal velocities - the crucial parameter determining dt R ∂ω e WKH GHÀHFWLRQ RI VHHGV GXULQJ ZLQG GLVSHUVDO 5HVXOWV DQGDX[LQVWKDWFUHDWHODUJHJURZWKJUDGLHQWVHQKDQFHWKH indicate that under favourable conditions, these seeds spirality generated spontaneously or as a result of bias. can disperse kilometre distances.
3UDFWLFDO VLJQL¿FDQFH ± ELRPLPHWLF DHURG\QDPLF Despite the similarity of the basic aerodynamics, a innovations GLVWLQFWGLIIHUHQFHH[LVWVEHWZHHQPDSOHDQGGLSWHURFDUSV Maple samaras possess no handedness; consequently they $HURG\QDPLF GHVLJQV DUH EDVHG RQ SRZHUIXO EXW VSLQHLWKHUFORFNZLVHRU counter- FORFNZLVHLQGHVFHQGLQJ yet imperfect theory and experimentation. Although as a result of spontaneous breaking of the symmetry. evolutionary corrections have been incorporated, +RZHYHU V\VWHPV H[KLELWLQJ WKLV EHKDYLRXU DUH KLJKO\ optimisation via testing varying models is prohibitively susceptible even to the smallest biasing perturbation. time consuming and expensive. Wind dispersing seeds 7KH IDFWRUV LQGXFLQJ VXFK ELDVHV ZHUH LGHQWL¿HG DQG are the aerodynamic systems most extensively optimised experiments indicate that although individual seeds via trial and error based selection. Consequently, they frequently violate the symmetry, globally it is respected. are expected to possess favourable attributes blind to It is not possible to quantify the variations in maple modern engineering. An important implication of the seed attributes and correlate them to a bias in the sense DERYHZRUNLVWKDWDXWRJ\UDWLQJVHHGVKDYHVXFFHVVIXOO\ RIURWDWLRQKRZHYHUIUHTXHQF\DQDO\VLVUHYHDOVQRUPDO achieved a design perfection to divert drag forces to distribution indicative of no preference. In contrast generate rotation, increasing the lift. Wing shape and dipterocarpus nuts possess a natural handedness and a striations of veins seem to play an important role. Veins JHRPHWULFDOGLIIHUHQFHFDQEHLGHQWL¿HGWRDFRQ¿GHQFH level more than 99 %. Here, both right and left handed SOD\ D GXDO UROH DV WKH VWUXFWXUH VXSSRUWLQJ WKH ZLQJ fruits are produced in equal abundance, and during falling DQG RSWLPLVHG DLU ÀRZ 7KHVH PD\ EH LQFRUSRUDWHG WR rotate in opposite directions. The investigation disclosed EXLOG OLJKWZHLJKW EXW VWUXFWXUDOO\ VWURQJ DLUIRLOV DQG WKDW WKH PDSOH VSHFLHV KDYH QR RUJDQV ZLWK ULJKWOHIW URWRU EODGHV 5HPRWHO\ FRQWUROOHG OLJKW ZHLJKW VPDOO GLVWLQJXLVKLQJ DWWULEXWHV ZKHUHDV LQ GLSWHURFDUSV WKH airplanes and helicopters consuming minimal quantities WZLJV UDFHPHV ÀRZHUV DQG VHSDOV KDYH KDQGHGQHVV RI IXHO ¿QG SHDFHIXO DSSOLFDWLRQV LQ DJULFXOWXUH DQG 7KH VHSDOV DUH SODFHG LQ D WZLVWHG DUUDQJHPHQW DW WKH DHULDOSKRWRJUDSK\6LPLODUO\VRODUSRZHUHGDLUSODQHV vertices of a pentagon. Sepals elongate to form either QHHG QHZ GHVLJQV WR EHDU WKH ZHLJKW RI WKH FHOOV DQG ¿YHWKUHHRUWZRZLQJVH[FOXGLQJIRXUDQGPDMRULW\RI compromise light harvesting and lift. WKHVSHFLHVKDYHRQO\WZRZLQJV7KLVLVH[SODLQHGDVDQ evolutionary reaction to the centre of gravity instabilities, (YHQDVLPSOHREVHUYDWLRQUHODWHGWRZLQGGLVSHUVDO aerodynamic effectiveness and the natural requirement of seeds could have profound implications on practical RI UHGXFLQJ EXUGHQ ELRPDVV SURGXFWLRQ /LNHZLVH WKH aerodynamics. As pointed out earlier Box Elder samaras R-L asymmetry of pine cones and samaras associate KDYHDGLVWLQFWO\GLIIHUHQWZLQJVKDSHFRPSDUHGWRWKH LWVHOIZLWKWKHKDQGHGQHVVRIWKHOHDIVSLUDOSK\OORWD[\ FORVHO\ UHODWHG RWKHU PDSOH VSHFLHV ZKLFK DUH WDOOHU Conifers and some genera of magnoliaceae also possess Experiments indicated that Box Elder samaras acquire R-L asymmetry characters similar to dipterocarps. In the terminal velocity in a comparatively shorter duration. apocyanaceae, unlike dipterocarps and conifers, the 7KHRUHWLFDO XQGHUVWDQGLQJ RI WKH GLVWLQFWLRQV LQ ZLQJ SK\OORWD[\GRHVQRWGLVSOD\KDQGHGQHVVEXWWKHÀRZHUV VWUXFWXUHVRIPDSOHVSHFLHVPLJKWOHDGWRQHZDLUIRLODQG DUHDOZD\VOHIWKDQGHGDQGWKLVDV\PPHWU\WUDQVFHQGVWR URWRUGHVLJQV:HOOSUHVHUYHGIRVVLOZLQJHGGLSWHURFDUS the fruit and seeds. Several genera of apocyanaceae have VHHGVKDYHEHHQGLVFRYHUHG 6KL /L DQGFDUHIXO SOXPHGVHHGVGLVSHUVHGE\ZLQG+HUHWKHDGYDQWDJHRI study of their structure in comparison to extant species handedness seems to be spreading of hairs of parachuting could yield clues to variations resulting from climate seeds by centrifugal acceleration, thereby enhancing the change. air drag.
Journal of the National Science Foundation of Sri Lanka 45(3) September 2017 Kirthi Tennakone
:LQG GLVSHUVLQJ VHHGV LQ SODQWV RI ZLGHO\ VHSDUDWHG DOI: https://doi.org/10.1103/PhysRevLett.109.154502 families adopt auto-gyration as an effective mechanism of 5. Carlton J. (2007). Marine Propellers and Propulsion . VORZLQJGRZQWKHGHVFHQW$XWRJ\UDWLRQIXQGDPHQWDOO\ %XWWHUZRUWK+HLQHPDQQ(OVHYLHU/WG/RQGRQ8. necessitates a breaking of the R-L symmetry. Maple 6. &KDQJTLX - ;X . 1XPHULFDO VWXG\ RI has designed the seeds to achieve this as a dynamically the unsteady aerodynamics of freely falling plates. Communication in Computational Physics 3 generated SBS. In dipterocarps, magnolias and 7. &KLWZRRG '+ +HDGODQG /5 5DQMDQ $ 0DUWLQH] SLQHV D PRUSKRORJLFDOO\ LQGXFHG 6%6 HQGRZV VHHGV %UD\EURRN6$.RHQLJ'3.XKOHPHLHU&6PLWK56 ZLWK D KDQGHGQHVV HLWKHU ULJKW RU OHIW :KHUHDV LQ 6LQKD15 /HDIDV\PPHWU\DVDGHYHORSPHQWDO apocyanaceae, presumably a persisting accidently constraint imposed by auxin-dependent phyllotactic introduced genetic R-L disparity induce unique chirality patterning. The Plant Cell 24 : RIWKHVHHGV$QRWKHUSRVVLELOLW\ZRXOGEHWRJHQHUDWH DOI: https://doi.org/10.1105/tpc.112.098798 an asymmetry to the seed appendage via a spontaneous 8. &RUOHWW5 3ULPDFN5 'LSWHURFDUSV WUHHVWKDW asymmetric deformation, as a consequence of motion. dominate the Asian rain forest. Arnoldia 63 It is not impossible that this method is also realised Available at KWWS SHRSOHEXHGXSULPDFN'LSWHURFDUSV VRPHZKHUHLQZLQGGLVSHUVDO$SDUWIURPDUHTXLUHPHQW pdf, Accessed 22 November 2015. of morphogenesis, it is hard to conceive the advantage 9. 'D\DQDQGDQ 6 $VKWRQ 36 :LOOLDPV 60 3ULPDFN R.B. (1999). The phylogeny of the tropical tree family of a handedness to a plant in any of its functions other dipterocarpaceae based nucleotide sequence of the WKDQ ZLQG GLVSHUVLQJ 3ODQWV HYHQ LQ IDPLOLHV ZLGHO\ chloroplast RBCLgene. American Journal of Botany 86 : separated, possessing R-L distinctive characteristics have exploited it. Wind dispersal have utilised the available DOI: https://doi.org/10.2307/2656982 traits and optimised in the course of evolution (Wrights 10. )HQJ ;7DQJ % .RGUXO70 -LQ - :LQJHG et al. , 2008); handedness is one such trait. fruits and associated leaves of Shorea (Dipterocarpaceae) from Late Eocene of South China and their phytographic Acknowledgement and paleoclimatic implication. American Journal of Botany 100 DOI: https://doi.org/10.3732/ajb.1200397 $ VLJQL¿FDQW SRUWLRQ RI WKLV LQYHVWLJDWLRQ LV EDVHG RQ 11. )R[ZRUWK\ ): Philippine Dipterocarpaceae . WKH NQRZOHGJH JDLQHG ZKLOH FRQGXFWLQJ WKH UHVHDUFK Manila Bureau of Printing, Manila, The Philippines. SURMHFW ދ5LJKW/HIW 6\PPHWU\ LQ 3ODQWV 5* ތ 12. *UHHQH ') 4XHVDGD 0 6HHG VL]H GLVSHUVDO supported by the Natural Resources, Energy and Science DQG DHURG\QDPLF FRQVWUDLQWV ZLWKLQ WKH %RPEDFDFHD Authority of Sri Lanka (NARESA). The author also American Journal of Botany 92 JUDWHIXOO\ DFNQRZOHGJH WKH DVVLVWDQFH JUDQWHG E\ WKH DOI: https://doi.org/10.3732/ajb.92.6.998 staff, Department of Physics and Department of Biology, 13. *XQDWLOOHNH &96 *XQDWLOOHNH ,$81 (VXIDOL 6 8QLYHUVLW\RI6UL-D\HZDUGHQHSXUD DQGWKH +DUPV .($VKWRQ 306 %XUVOHP ')53 $VKWRQ 'HSDUWPHQW RI 3K\VLFV 8QLYHUVLW\ RI 5XKXQD P.S. (2006). Species-habitat associations in a Sri Lankan 1988). dipterocarp forest. Journal of Tropical Ecology 22 384. DOI: https://doi.org/10.1017/S0266467406003282 REFERENCES 14. Hederstrom A. (2002). Aerodynamics and evolution of DYLDQÀLJKW Trends in Ecology and Evolution 13 1. Ashton P.S. (1980). Dipterocarpace. A Revised Handbook 422. to the Flora of Ceylon HGV 0' 'DVVDQD\DNH )5 15. .UDPHU( &RPSXWHUPRGHOVRIDX[LQWUDQVSRUW D )RVEHUJ YROXPH SS $PHULQG 3XEOLVKLQJ UHYLHZDQGFRPPHQWDU\ Journal of Experimental Botany &RPSDQ\1HZ'HOKL,QGLD 59 2. $VKWRQ 36 *XQDWLOOHNH &96 1HZ OLJKW RQ DOI: https://doi.org/10.1093/jxb/erm060 plant geography of Ceylon. Journal of Biogeography 14 : 16. Lamarque L.J. (2013). Ecology and evolution of invasive maple tree species. PhD thesis , York University, Toronto, DOI: https://doi.org/10.2307/2844895 Canada. 3. $XJVSXUJHU &. )UDVRQ 6( :LQG GLVSHUVDO 17. /HQWLQN''LFNVRQ:YDQ/HHXZHQ- 'LFNLQVRQ0+ RIDUWL¿FLDOIUXLWVYDU\LQJLQPDVVDUHDDQGPRUSKRORJ\ (2009). Leading-edge vortices elevate lift of autorotating Ecology 68 plant seeds. Science 324 DOI: https://doi.org/10.2307/1938802 DOI: https://doi.org/10.1126/science.1174196 4. %DJKHUL 6 0D]]LQR $ %RWWDURO $ 7KH 18. Lugs H.J. (1983). Annual Review of Fluid Mechanics 15 : VSRQWDQHRXV V\PPHWU\ EUHDNLQJ RI D KLQJHG ÀDSSLQJ ¿ODPHQW JHQHUDWHV OLIW Physical Review Letters 109: DOI: KWWSV GRLRUJDQQXUHYÀ 19. McCutchen C.W. (1977). The spinning rotation of ash and
September 2017 Journal of the National Science Foundation of Sri Lanka 45(3) Aerodynamics and right-left symmetry in seed wind dispersal 217
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