The Miimum Distace Eergy for Polygoal Ukots By:Johaa Tam Advisor: Rollad Trapp Abstract This paper ivestigates the eergy U MD of polygoal ukots It provides equatios for fidig the eergy for ay plaar regular -go ad for ay m-go, where the vertices lie o the vertices of a regular m -go ad o the midpoits of each edge I additio, we show that a regular 4-go miimizes the eergy for ay quadrilateral Fially, this paper icludes a proof showig that if we have a regular -go, R, iscribed i a circle ad ca move oly oe vertex of R, v, alog the circle betwee its two adjacet vertices, the the U MD is miimized whe v is a vertex o R 1 Itroductio The miimum distace eergy for polygoal kots, U MD, was itroduced by Joatho Simo i [] This eergy aalyzes the miimum distaces betwee the edges of a kot Furthermore, Rawdo ad Simo i [1] relates this miimum distace eergy to the Möbius eergy for smooth kots They show that U MD (P ) U MD (R ) approaches the Möbius eergy of the smooth kot K as P approaches K It is cojectured that the regular -go miimizes U MD for all kots with -sticks If this cojecture holds true, the U MD (P ) U MD (R ) will ever be egative, ad accordig to Joatho Simo [3], that limitig umber is the same as the miimum Möbius eergy for the kot type K My goal was to prove this cojecture This paper provides a itroductio to some of the observatios that ca be made about the eergy of the regular -go Now we will briefly cover some importat defiitios that are used throughout this paper To start with the basics, a kot is a cotiuous closed curve/loop i space It ca be thought of as a kotted strad of strig that is coected at the eds, where the strig has o thickess A polygoal kot is a kot equivalet to a polygo i R 3 It is depicted usig straight edges (sticks) ad vertices This is the type of kot that we will be focusig o The ukot is a kot that has a diagram with o crossigs ad therefore has a crossig umber of zero Oe type of ukot is the regular -go, which is a polygo that has edges of equal legth ad correspodig agles of equal degree which will be deoted as R throughout this paper 1
Fially, the mai topic of this paper is the miimum distace eergy Whe give two o-adjacet edges X, Y of a polygoal kot K, we defie the miimum distace eergy, U MD, by U MD (X, Y) = l(x) l(y) MD(X,Y) such that l(x), l(y ) are the legths of X ad Y cosecutively, ad MD(X, Y ) is the miimum distace betwee X ad Y Now we defie the miimum distace eergy of K by U MD (K) = U MD (X, Y) where the sum is take over all pairs of o-adjacet edges, X ad Y, of polygo K Note: The eergy U MD is scale ivariat This holds true because if we icrease the kot by a costat c, the the legths of all the edges ad the distaces betwee them would all icrease by c Therefore, we have U MD (X, Y ) = c l(x) c l(y ) (c MD(X,Y )) = c l(x) l(y ) c (MD(X,Y )) = l(x) l(y ) MD(X,Y ) Eergy Equatios for a Regular -go This sectio discusses two geeral equatios that ca be used to compute the U MD of a regular -go ad the methods we used to geerate them Oe equatio is for whe is odd ad the other is for whe is eve The U MD of ay regular -go ca be obtaied easily by just pluggig the equatio ito MAPLE ad chagig accordigly For the first step, we decided to view the regular -go as iscribed iside a uit circle (see Figure 1) sice U MD is scale ivariat Every vertex of the -go lies o the followig circle Thus vertex v j = (cos( π j Figure 1 Uit Circle ), si( π j )) For example, v 1 = (cos( π π ), si( )) We the foud the legth of a edge, l e, by fidig the distace betwee two cosecutive vertices of the polygo Sice the legth of each edge is equal by defiitio, we ca make it simple by just choosig to fid the distace betwee v 0 ad v 1, where v 0 = (1, 0) We have l e = (cos( π ) 1) (si( π )), which ca be simplified to l e = (1 cos( π ))
thus, for the regular -go l(x) l(y ) = (1 cos( π )) where X ad Y are o-adjacet edges Next we ivestigated the miimum distaces betwee the o-adjacet edges Due to the symmetry of a regular -go, the U MD for e 1 ad its oadjacet edges is the same as the U MD for ay other edge ad its o-adjacet edges so we ca multiply 1 j=3 U MD(e 1, e j ) by However, ote that the U MD of each pair of o-adjacet edges have bee couted twice, ad hece we must divide by Therefore, the U MD (K) = j=3 U MD(e 1, e j ) where K is the ukot with edges The miimum distace betwee two edges is the distace betwee the head of oe edge ad the tail of the other where the head is farther couterclockwise tha the tail 1 The Miimum Distace Eergy of R Whe is Odd We will ow discuss how to fid the equatio for the U MD (R, = odd) Note that each edge of a -go where is odd, has a eve umber of o-adjacet edges We ca utilize this fact ad the symmetry of R to make fidig the equatio more feasible Throughout this subsectio we ca refer to the regular 5-go i figure as a example Figure We oticed that j=3 U MD(e 1, e j ) = j= 1 U MD(e 1, e j ) therefore we ca just calculate j=3 U MD(e 1, e j ) ad multiply by two So the first step is to fid a formula for the U MD (e 1, e j ) where e j is o-adjacet to e 1 We kow that l(e 1 ) l(e j ) = (1 cos( π )) from earlier Next we ee to fid the miimum distace betwee the head of e 1 ad the tail of e j Usig trigoometry ad geometry, we fid that the head of e 1 is at (cos( π π ), si( )) ad that the tail of e j is at (cos( π (j 1) ), si( π (j 1) )) Thus the MD(e 1, e j ) = (cos( π i π ) cos( )) (si( π i π ) si( )), where i = (j 1) Whe we put it all together, the U MD (R, = odd) = i= (cos( π i (1 cos( π )) ) cos( π )) (si( π i π ) si( )) The usig some trigoometric idetities, we ca simplify the equatio to the followig Propositio 11-The miimum distace eergy for R whe is odd is U MD (R ) = 3
i= (si( π )) (si( π (i 1) )) The Miimum Distace Eergy of R Whe is Eve Throughout this paper, fidig a formula for the miimum distace eergy is more complicated whe the kot has a eve umber of sticks tha whe there is a odd umber of sticks I this subsectio we ca refer to the followig 8-go as a example, where the bolded edge is e 1 ad the dotted lies represet the miimum distace betwee two edges (see Figure 3) Figure 3 We the ca follow the same procedures for whe is eve for R with oe additio Ulike whe is odd, a -go with a eve umber of edges will have a odd umber of o-adjacet edges for each edge as see i the 8-go above However, otice that with the exceptio of the edge opposite of e 1, it follows the same patter/symmetry as R whe is odd Due to this differece, the U MD of e 1 ad all its o-adjacet edges mius the farthest edge is the same as whe is odd with differet limits So to complete the geeral equatio for whe is eve, we eed to take the product of the legths of e 1 ad e ( 1) ad divide it by their miimum distace squared After that, we must multiply the resultig equatio by ad divide by to prevet double coutig Whe we put it all together the U MD (R, = eve) = 1 (1 cos( π )) )) i= π (1 cos( π ) si( )) (1cos( π However, (cos( π i ) cos( π )) (si( π i ) like the previous equatio, we ca simplify the equatio usig trigoometric idetities to the followig Propositio 1-The miimum distace eergy for R whe is eve is U MD ( go) = 1 (si( π )) i= (si( π (i 1) )) (ta( π )) 3 Eergy Equatios for a Flatteed -go I this sectio, we discuss how to fid the U MD for a specific irregular -go We let K be a plaar regular -go, ad let L be the plaar -go obtaied by takig K ad addig a vertex o the midpoit of each edge I other words, the flatteed -go is whe you take every other outer agle of a regular -go ad flatte it to 180 The equatios for U MD (L) were foud by followig the same steps used to fid the U MD for R Let e 1 s head be located o a 4
midpoit of a edge i K The miimum distace is still from the head of oe edge to the tail of the other Oce agai we chose to focus o the U MD of oe edge ad its correspodig o-adjacet edges The we multiplied that quatity by The legth of a edge i L is equivalet to the legth of a edge i R divided by two We have l e = 1 (cos( π ) 1) (si( π )), which ca be simplified to l e = 1 (1 )) therefore, the product of the legths for two o-adjacet edges, X ad Y, for L is give by l(x) l(y ) = 1 (1 cos( 4 π )) As i the previous sectio, the U MD (L) requires two equatios, oe for whe = eve ad oe for whe = odd 31 U MD (L, = odd) This subsectio focuses o L whe is odd However, the -go still has a eve umber of edges ad thus L has a odd umber of o-adjacet edges for each edge Oce agai we ca rely o the symmetry of the -go to simplify the problem Figure 4 Before we compute the equatio, make ote that the head of e 1 is closest to the tails of e j whe 3 j ( 1) Also ote that the tail of e 1 is the closest to the heads of e j whe ( ) j ( 1) We ca ow fid U MD (L, / = odd) by breakig up the problem ito four sectios by lookig at the four possible cases of o-adjacet edge pairs First we fid the U MD of e 1 ad e j whe 3 j ( 1) where the tail of e j is located o a vertex of the regular -go, which equals (1 )) 4 (( )1 i )) ( si( 4 π ) si( 4 π i )) ) Secodly, we fid the U MD of e 1 ad e j whe 3 j ( 1) where the tail of e j is located o a midpoit of a edge i K, which equals 4 (( )1 ( i (1 )) )cos( ) )) ( Next we fid the U MD of e 1 ad e j whe ( si( 4 π ) ( si( 4 π i )si( ) )) ) ) j ( 1) where the 5
head of e j is located o a vertex from the regular -go, which equals 4 1 (( (1 )) ) cos( )) (si( 4 π ) si( )) ) Lastly, we fid the U MD of e 1 ad e j whe ( ) j ( 1) where the head of e j is located o a midpoit of a edge i K, which equals 4 (( (1 )) i )cos( ) ) ( )) (si( 4 π si( 4 π i ) ( )si( ) )) ) Now after all these computatios, we oly eed to add up these summatios, simplify ad multiply the result by Therefore, we have U MD (L, = odd) = ( 4 4 (( 4 1 4 (( )1 ( (( (( )1 i i ) cos( i ) ( (si( π )) )cos( ) si( 4 π ) si( 4 π i )) ( (si( π )) )) ( (si( π )) )) (si( 4 π )cos( ) (si( π )) )) ) si( 4 π ) si( 4 π i ( )si( ) ) si( )) (si( 4 π )) ) si( 4 π i ) ( )) ) )si( ) )) ) ) 3 U MD (L, = eve) I this subsectio we will show how we fid a formula for the U MD (L, = eve) We follow the same procedures as whe fidig the U MD (R ), where the equatio for the case eve builds from the equatio used for the whe odd case Use the 16-go below as a example for this subsectio (see Figure 5) Oce agai the dotted lies represet the miimum distace betwee two o-adjacet edges Figure 5 For fidig the equatio for U MD (L, = eve), we ca use the same method used above whe is odd Actually we ca use the exact same summads with differet bouds to represet the same cases of o-adjacet edge pairs of the ukot However, whe is eve, we have a odd umber of oadjacet edges for each edge Similar to computig the U MD (R, = eve), the U MD (e 1, e ( 1) ) requires a extra term added o to the formula for whe is odd The product of these two edges is the same as the rest of the oadjacet pairs The miimum distace is foud by computig the miimum distace betwee the head of e 1 ad the head of e ( 1), which is equal to (( (si( π )) )1) (si( 4 π )) Therefore if we add this quatity to the summatios 6
foud for U MD (L, = odd) ad chage the limits appropriately, we ed up with U MD (L, = eve) = ( 4 1 4 1 (( (( )1 )1 i ( i (1 )) )) ( )cos( ) si( 4 π ) si( 4 π i (si( π )) )) ( )) ) si( 4 π ) si( 4 π i ( )si( ) )) ) 4 1 4 1 (( (( ) cos( i ) ( (si( π )) )) (si( 4 π )cos( ) ) si( (si( π )) )) (si( 4 π )) ) si( 4 π i ) ( )si( ) )) ) (si( π )) ( )1) (si( 4 π )) ) 4 Regular 4-go Miimizes U MD for all Quadrilaterals This sectio provides a proof that the regular 4-go miimizes the U MD for all quadrilaterals, which icludes irregular, ot covex, ad o-plaar 4-gos Before we begi the proof we must calculate the actual value of U MD (R 4 ) Let the edge legths of R 4 equal a,b,c,d where the edges with legths a ad b are o-adjacet WLOG, U MD (R 4 ) is give by a b d c d a Sice we kow that for a regular 4-go, a=b=c=d,the above quatity ca be writte as a a a a a a = a a a a = 1 1 = Theorem 1: The regular 4-go miimizes the miimal distace eergy U MD for all kots with four edges Figure 6 Proof Let F be ay ukot with 4 edges, e a, e b, e c, e d, with edge legths a, b, c, d respectively, where e a ad e b are o-adjacet ad a < b ad d < c 7
Note that for each arbitrary 4-go above the largest possible MD(e a, e b ) equals d, ad the largest possible MD(e c, e d ) equals a Sice this maximizes the deomiator, it miimizes the U MD (F ) The we have U MD (F ) = a b c d (MD(e a,e b )) (MD(e c,e d )) a b d c d a > a d d a = a4 d 4 a d We kow that x y x y Let x = a ad y = d Thus a 4 d 4 a d It follows that a 4 d 4 a d Therefore the U MD (4 go) is miimized by a regular 4-go sice the U MD (R 4 ) = 5 Movemet of Oe Vertex Alog the Circumferece of a Circle If we have a regular -go, R, iscribed i a circle, all of its vertices will lie o the circumferece of the circle Now if we are allowed to move oly oe of these vertices, v, alog the circle betwee its two adjacet vertices, how does this affect the U MD of the kot? I this sectio we will show that if v moves from its origial positio o R, it will oly icrease the miimum distace eergy of the kot To start the ivestigatio, let this altered R be deoted as A where oe of its vertices have the coordiates (1, 0), ad let v be adjacet ad located at (cos(θ), si(θ)), where 0 < θ < 4 π Before computig the miimum distace eergy, otice that oly two of the edges, e 1 ad e, from R will be altered Therefore, U MD of those two edges ad their correspodig o-adjacet edges will chage, while the cotributios to U MD of the remaiig edge pairs will remai the same Let e 1 have the edpoits (1, 0) ad (cos(θ), si(θ)), ad e have the edpoits (cos(θ), si(θ)) ad ( 4 π ), si( )) Therefore we have l(e 1) = si( θ ) ad l(e ) = si( 4 π θ ) 51 U MD (A, = odd) To fid the U MD (e 1 ) we eed to fid U MD (e 1, e i ), where e i is o-adjacet to e 1, there are two cases to cosider: whe (1, 0) is the poit closest to the oadjacet edge ad whe (cos(θ), si(θ)) is the closest The eergy for whe (1, 0) is the poit earest to e i equals i= si( π ) si( θ ) The eergy for whe v is si( π (i 1) ) the closest to e i is equal to i= si( π ) si( θ ) The sum used for computig si( θ π i ) the U MD (e ) is similar to the U MD (e 1 ), but istead of si( π ) si( θ ), the 8
umerator chages to si( π 4 π ) si( θ ) Therefore, the U MD (e ) equals 4 π θ 4 π θ to i= (si( π )) si( ) si( π (i 1) )) i= (si( π )) si( ) The remaiig si( θ π i )) edge pairs have the same U MD as if the -go were regular, ad therefore equal i= ( 4) (si( π )) We ca ow add up these summatios to get the (si( π (i 1) )) miimal distace eergy of the kot Whe simplified, U MD (A, = odd) = i= ( si( π ) (si( θ )si( (si( π (i 1) 5 U MD (A, = eve) 4 π θ )) )) si( π (si( π )) (si( π (i 1) )) ) Figure 7 ) (si( θ 4 π )si( θ )) (si( θ π i )) ( 4) Oce agai whe fidig the equatio for U MD ( go, = eve), we ca employ the same method used above whe is odd, which will give us the same summads, but with differet limits ad some extra terms The summads from U MD (A, = odd) do ot iclude the U MD (e i, e (i ) ) where e i is ay edge i the -go Whe e i ad e (i ) are ot e 1 or e, the we kow from the U MD (R ) that the U MD (e i, e (i ) ) = ( ) (ta( π )) We eed to fid the U MD (e 1, e ( 1) ) ad the U MD (e, e ( ) ) The MD(e 1, e ( 1) ) is the distace betwee (cos(θ), si(θ)) ad ( 1, 0) The MD(e, e ( ) ) is the distace betwee ( 4 π π () ), si( )) ad (cos( ), si( π () )) Therefore the 4 π θ U MD (e 1, e ( 1) ) = (si( π )) si( θ ) ad the U (cos( θ MD (e, e )) ) = (si( π )) si( ) (si( π ( ) )) We ca ow add up these equatios ad simplify to get the followig equatio U MD (A, = eve) = 1 4) i= ( (si( π )) (si( θ )si( (si( π )) (si( π (i 1) (si( π (i 1) 4 π θ )) )) (si( π ) ( )) ) (ta( π )) (si( π )) si( θ ) (cos( θ )) Figure 8 53 U MD (A ) is Miimized Whe θ = π )) (si( θ 4 π )si( θ )) (si( θ π i )) ( (si( π )) si( 4 π θ ) (si( π ( ) )) To aalyze the U MD (A ) further, we will look at the rate of chage of θ The followig proof will use calculus to show that U MD (A ) is miimized whe θ = π However, before the proof we eed to fid the derivative of U MD(A ) 9
d dθ (U MD(A, = odd)(θ)) = (si( π )) ( 1 cos( θ ) 1 π cos( θ )) (si( θ π i )) (si( π d dθ (U MD(A, = eve)(θ) = 1 π cos( θ )) (si( π (i 1) )) )) (si( θ π ) si( θ )) cos( θ π i ) ) (si( θ π i ))3 i= ( (si( π )) ( 1 cos( θ ) 1 π cos( θ )) (si( π (i 1) )) π ) si( θ )) cos( θ π i ) ) (si( θ π i ))3 π ) cos( θ ) (si( π ( ) )) i= ( (si( π )) ( 1 cos( θ ) 1 (si( π )) ( 1 cos( θ ) 1 π cos( θ )) (si( π (si( θ π i )) (si( θ )) si( π ) si( π cos( θ ) (si( θ )) si( π ) (cos( θ ))3 Figure 9 Theorem : The U MD (A ) is miimized whe A = R Proof Give U MD (A ) ad d dθ (U MD(A )), we eed to show that the fuctio U MD (A )(θ) icreases if θ starts at π ad decreases towards 0 or icreases towards 4 π Due to symmetry (see Figure 9), we ca aalyze the oe-sided derivative of U MD (A ) where θ π We wat to show that d dθ (U MD(A ))( π ) > 0 Usig trigoometric idetities we are able to simplify d dθ (U MD(A ))( π ) where d dθ (U MD(A, = odd)( π )) = 4 (si( π )) cot( (i 1) π ) i= (si( (i 1) π )) d dθ (U MD(A, = eve)( π )) = 1 i= 4 (si( π )) cot( (i 1) π ) (si( (i 1) π )) ) (ta( π ))3 Sice we proved that the regular 4-go miimizes the eergy U MD for all 4- gos, we ca let 5 Whe we plug i the limits, ad, we fid that π (i 1) π < ( ) π Thus, 0 < (i 1) π < π Hece, 4 (si( π )) cot( (i 1) π ) > (si( (i 1) π )) 0 ad (ta( π ))3 > 0 sice we kow that cot(θ) > 0 ad ta(θ) > 0 whe 0 < θ < π So it follows that U MD(A ) is icreasig as θ is icreasig from π Therefore, U MD(A ) is miimized whe θ = π Remark: Sice d dθ (U MD(A ))( π ) 0, we kow that U MD(A ) is ot differetiable at θ = π 6 Coclusio I coclusio, we have foud some equatios for computig the miimum distace eergy for the regular -go ad other types of polygoal ukots I additio, we proved that the regular 4-go miimizes U MD for all ukots with 4 sticks We also proved that if we have a regular -go, R, iscribed i a circle ad ca move oly oe vertex of R, v, alog the circle betwee its two adjacet vertices, the U MD of that kot is miimized whe the kot is a regular -go Overall, we employed differet methods to aalyze the U MD of the 10
ukot We focused o the miimum distace betwee edges i sectio ad sectio 3 The i sectio 4 we focused o edge legths Fially, for sectio 5 we focused o the agles betwee the edpoits The miimum distace eergy of a regular -go has yet be prove to miimize the U MD for all ukots with sticks There are defiitely differet ways to ivestigate this cojecture Oe directio to go is to fid equatios for the regular -go ad other polygoal ukots that ca be compared effectively This ca maybe be doe by ivestigatig whether the ukot is easier to aalyze whe we are focusig o the edge legths, miimum distaces, agles, etc Sice this paper discusses maily plaar, o-covex -gos, maybe we ca aalyze covex or o-plaar -gos If the regular -go is prove to miimize the U MD for all -gos, the maybe U MD of other kot types with higher crossig umbers ca be miimized by equalizig the edges ad expadig the agles betwee the edges evely Overall, it will give us a better uderstadig of the miimum distace eergy U MD ad its properties 7 Ackowledgemets I wat to thak Dr Rollad Trapp for advisig me throughout this project ad for his time ad patiece Also, I wat to thak Dr Joatho Simo for his advice ad ideas for for my project Fially, I wat to thak the CSUSB REU 006 group ad Dr Joseph Chavez for their support All this was made possible by the Natioal Sciece Foudatio REU Grat DMS-0453605 It was truly a hoor to be give this opportuity Thak you 8 Refereces [1] Rawdo, Eric J; Simo, Joatha K Polygoal approximatio ad eergy of smooth kots J Kot Theory Ramificatios 15 (006), o 4, 49 451 58E10 (57M5) [] Simo, Joatha K Eergy fuctios for polygoal kots Radom kottig ad likig (Vacouver, BC, 1993), 67 88, Ser Kots Everythig, 7, World Sci Publishig, River Edge, NJ, 1994 57M5 [3] Simo, Joatha K Persoal Commuicatio 11