VOL. 20 NO. 4 (1997) 627-632
627
THE DIMENSION
OF
THE
BOUNDARY OF
THE
LiVY
DRAGON
R DUVALL
Department
ofMathematical SciencesUniversity ofNorth CarolinaatGreensboro Greensboro,
NC
27412,USA
and J. KEESLING
Department
ofMathematicsUniversity ofFlorida
P.O Box
118105,358 LittleHall Gainesville,FL
32611-8105,USA
(Received
April 7,1997)
ABSTRACT.
In
thispaper we describe the computations done by the authors in determining the dimensionoftheboundary of theLvy
Dragon. A
general theorywasdeveloped
for calculatingthe dimensionofa self-similar tileandthetheorywasappliedtothisparticularset. The computationswerechallenging
It
seemed thatamatrix which was 25 25would havetobe analyzed.It
waspossibletoreduce the analysistoa 752 x752 matrix.
At
lastitwasseen thatifA
wasthelargest
eigenvalue ofa certain734 734matrix,thendimH(K)
Perron-Frobeniustheory playedanimportantroleinanalyzingthis matrix.
KEY WORDS AND PHRASES:
Hausdorff dimension,iterated function systems, attractors, fractal geometry1991
AMS SUBJECT
CISSIFICATION CODES:
Primary28A20, 54E40, 57Nxx.1.
INTRODUCTION
Self-similarsetshave becomeanimportantareaof research. Theyare notonly interesting geometric figures, but
they
have alsoproved
tobeuseful inimage compression and storage],
[2],
and[3].
It
ispossible that thoseself-similar setsthatare tilesmight
play
animportantrole
inimage compressioninthefuture.
So,
thesesetsdeservetobestudied insomedetail. The worksummarized here is anattemptto extend whatisknownaboutthesesets.628 P DUVALL AND KEESLING
In
[4], P
L6vy studied thecomplexcurvewhich has cometobeknown as theLvy Dragon
He showedthat theDragon
isself-similar,andarguedthat theplanecanbe tiledby copies ofit L6vy’s paperis atourde
force
onthissubjectIt
is amazingthathewasabletodetermine somany propertms ofthis and other curves in[4]
withouttheuseofmodemcomputersAlthough L6vy found many properties ofthe
Levy Dragon,
he did not compute the Hausdorffdimensionofitsboundary.
In [5,
page236] G
Edgarasks whetherthedimensionoftheboundary oftheDragon
is greater than one Inapreviouspaper[6]
weanswered thatquestioninthe affirmative andin factdetermined aprecisevalue for this dimension.In
theprocess of doingthiswedevelopedageneral theory for computingtheHausdorffdimensionoftheboundary ofself-similarfractaltilesTheL6vy
Dragon
is theattractor for theIFS
F
{I’1, f2},
where1’1
andf2
are similitudes ofEuclidean
space R
given byfl(x,y)
(x-y
x+y)
2 2
f2(x,y)=
(x+y+l y-x+l)
2 2
Much that we discoveredabout the
Dragon
holdsin abroadersettingSo,
it isappropriatetohavefairly
general
notational conventions. TheHausdorffdimensionofa setE
isdenoted bydimn E,
andthe’lower
andupper box-countingdimensionsofE
aredenoted bydimsE
andd-sE.
Definitionsofthese dimensionfunctions can befoundin[7],
[8]
and[9].
Let
12denote the collectionof sequencesI
{il, i2,---}
withi
{
1,2,---,n}. 12
willdenote thesetof sequences of length kwith entriesfrom
{
1,2,.--,n}.
12 has the naturalmetricgiven byf
1,if’il
:/:
jld(I,
J)
c,
c.--c,,,
ifi
j for k<
mandim+l
For
I
12,
weletI
denotethe finitesequence{il,i2,...,i},
andwilloftenusethe shorthandfh
forthecomposition
f,a f,---f,
andch fortheproductc, c--.c,
Wewillalso needthemapsa andg"f2K
given bya(il,
i,...)
{i2,
i3,...}
andg(I)
liraf,t f,’" "f, (x)
Itisnotdifficulttok---*oo
seethat
g(I)
is independent ofthe choiceofxX.
Themapa is thewell-known shiftmap on thesequence
space2.
SUB-SELF-SIMILAR SETS
In [7],
Falconergave
an(abstract)
wayto determinethe dimension of sub-self-similarsets Given a setF
fl, f2,"’,
fn
}
of contractingsimilitudesonR
t,
theclosedsetE
issaid tobesub-self-similar
for
F
ifE
CI,.J,lf,(E).
It follows thatE
CK,
whereK
istheattractorforF.
Note
inparticularthat thetopological boundaryOK
ofK
iss.s.s.,since the interior intK
ofK
ismappedimo itselfbyeachoftheopen mappings
f,.
Thefirst crucial observationabouts.s.s,sets isPROPOSITION
2.1.Let
E
beaclosedset. ThenE
iss.s.s,forF
ifandonly
ifE
g(A)
forsomecompactset
A
C f2 such thata(A)
CA.
Furthermore, ifE
is s.s.s., such a setA
isgiven byA
{I
flg(a
(I))
E
for all k_> 0}.
Theset
A
inProposition2.1 isfundamentalin the calculationofdimnE.
For
k>
0,letAk
be thesetoffinite
sequences
obtainedby truncating elements ofA
after kterms.For
s>
0 definite629
U/,(u)
c
u,
ana
l,
(V)
l;(V)
,
if#
j.THEOREM
2.2.Let
E
be aclosedset which iss.s.s,with respecttothefamily{fl,
f2,
"-’,f,}
ofcontractingsimilitudes whichsatisfiesthe opensetcondition. Letsbethe numbersatisfying
-(s)
1 Then8dimnE
dirnBE
dimBE.
3.
BOUNDARIES OF ATTRACTORS
One
obvious difficulty in applying Theorem 2.2 is that one has to sum over finite sequencesdeterminedbyaninfinitecondition Itwasimportanttoshowthatmembershipin
A
canbe determined from finite dataPROPOSITION
3.1.Ak
{I
6_fklfI
(K)
NOK
:fi
0}
Once
wehad Proposition3.1 wehadastrategy forcomputingdimnOK. Suppose
forconveniencethat the c, all havethe samevalue c, and
suppose
thatwecan determine cr lira[Ak[
/’,
where denotesthe cardinality of thesetS.
ThenThus dimncgK is the solutionto ac 1, or
dimnOK
In(a)
In(c)
4.
THE
LIVY
DRAGON
We now revert to letting
K
denotetheIMvy
Dragon
with itsIFS{f],f2}
Lvy
[4]
showed,among
other things, thatK
tilestheplane
in the sensethattheplane
canbewrittenas the union of congruent copiesofK
thatmeetonly
intheirboundaries.Let
S
bethe unitsquare
with vertices{(0,
0),
(0,1), (1,1), (1,
0)}
and letTo
CS
be the trianglewith vertices
{
(0, 0), (1/2,1/2), (1,
0)}.
L6vy
viewedK
as liraF’(To).
k--oo
630 PDUVALL AND KEESLING
S
isthe union of fourtriangles inS
withhypotenuseanedge ofS
and thirdvertex at(1/2,
1/2)
Wedenote the triangulation of
R
consisting ofthesetrianglesand theirintegraltranslatesbyTo
To
and itsneighborhoodinTo
areshown inFigure2 IfT
is aright triangleintheplane,werefertothevertex opposite the hypotenuse as the top vertex ofT,
and the other vertices asleft
and rght so that we encounterthe vertices in theorderleft, top, andfightas we traversetheboundaryinclockwise order Theleft
(rght) edge ofT
is the edge determined by the left (fight) and top vertices ofT
Thetriangulation
To
has a subdivisionT
obtained by subdividing each triangleT
into the two trianglesdeterminedbythe lef (fight) edge of
T
andthemidpoint of thehypotenuse
ofT
Continuingin the obvious way gives a sequenceTo
-
T
-.--
-
Tk
--..
of subdivisions ofR
into fight isocelestriangles. Eachtrianglein
T
has diameterFor
eachT
T,
letT
r be thetrianglein/
whosehypotenuseistheleft edge ofT
and whichhasavertexoutsideofT.
DefineT
similarly.One
can easily see thatF(T0)
T0
U
T0
.
In
fact,thefollowing propositionis immediatePROPOSITION
4.1.For
each>
O,
F(To)
is a union of 2 triangles inT.
F+(To)
is obtainedfromF (To)
by replacingeachT
inF
(To)
by
T
andT
Weused theformula at the end ofSection 3 to compute the Hausdorffdimension of
OK
The difficulty thatwe encountered was thatK
andOKare very complicated objects, andwecan onlyseefiniteapproximationstothem,so it wasnotclearhowtocompute
A
What we were abletowork with wasBk
(Ik
e ak If,
(To)
i)Fk(To)
=fi
0
}.
Thenwewereabletodeterminetherelationshipbetween
Ak
andBk
tocompletethe calculation\
/
Figure3 TheNdghborhood Ordering
One
ofthe difficulties indealingwithK
is that it is noteasytolocate pointsinthe interiorofK
by lookingatF
k(T0).
One
needsto findobjects of positivearea whichpersist fromiteration to the next, butassoon as atriangle
T
appearsatonelevel,its interior is discardedatthe nextleveland isreplaced by twotrianglesexteriortoT. It
is not untilthe iterationshave become sufficiently intertwinedto create covered triangles thatthe discarded mass is filledback in by adjacent triangles Through computer experimentationwediscoveredthatcoveredtrianglesdo not exist until the 14thiteration, andonly8ofthe
214
trianglesthatcompriseFI4(To)
arecovered!However,
iratriangleT
ETk
iscovered,thenT
istheunion oftwotriangles in
Tk+l
which are also covered.It
follows that intTCFk+"’(To)
for all6 31
Figure4.
Ns
andFSTo
PROPOSITION
4.2.Bk
CItfollows that
[Bk[ <_
[Ak[
We
alsousedthefact thatateach stageF
(I)
isa tilingofR
So,
inusing
B
twoways, wewereabletodetermine lower andupper bounds fordimHoaK
whichwerethe same number thusgivingus acompletedetermination of this dimensionLet/3
liraIBm[
1/TIIEOREM
4.3.n(/)
5.
COMPUTATIONAL ISSUES
In
thelightof Theorem4.3,ourgoalwastounderstandtheasymptoticbehavior of the number[Bk[
oftriangles in
Fk(To)
which are notcovered.To
this end, we needed to study the neighborhoodstructuresintroduced in Section 4.
In
ordertokeeptrackof thenumberof neighborhood typesateach stage inF
k(To)
weneededamatrix which was215
x215
However,
by experimentingwith this matrix, wefound thatwecould reduceit to a smaller 734 x 734 matrix withnonnegative integerentries which was irreducible Perron-Frobeniustheory
was used to determine that there was a unique largest eigenvalueA
tothis matrix. Thiseigenvaluedeterminedthecardinalityof[Bk[
tobeA
kasymptoticallyThis allowed usto compute the dimension oftheboundary of the Lrwj
Dragon
We
were abletodetermineprecise bounds for
A
usingintegerarithmetic and also determine as estimate forA
tohigh precision usingthepower
method.By
Section4,In(A)
dimn,
OK
ln(v/)
Floating point computations using the power method give the estimate
A
1.934007183 Therigorous estimatethat we obtainedusing integerarithmetic was 1.824190
<
A
<
1.974189. Thislastestimate
proved
thatA
wasgreater thanV/,
and that theHausdorff
dimension ofoakisgreater than1Thus, the
boundary of
theDragon
hasHausdorff
dimension greater than 1, andEdgar’s
questionwas632 P DUVALLAND KEESLING
ACKNOWLEDGEMENT.
Thisresearch was done while the first authorwasaVisiting Professoratthe University of Florida He gratefully acknowledges UF’s hospitality, as well as research support from the University of NorthCarolina, Greensboro
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