Helical Fractal

8/18/2019 Helical Fractal

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Division of Physics and Applied Physics,

School of Physical and Mathematical Sciences,

Nanyang Technological University.

April 9th, 29

roducing the Helical Fractal,

Discrete

Versions and

Sa! "ee#$iem, %r!an Ade P&tra.

per Fractals.


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The helicalised sine c&rve.

No!, ho! a(o&t

helicalising thehelicalised sinec&rve

The helicalised sine c&rveis a c&rve that !rapsaro&nd the sine c&rve.

3ertainly a moreinteresting '#dc&rve.

And helicalising thatonce more


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4elicalised helicalised helicalisedsine c&rve.

This is the c&rve !inds intoa heli, s&ch that theres&lting heli !raps aro&ndanother heli, !hich finallyc&rls aro&nd the sine c&rve.

%mpressive

A close#&p vie!of the innerstr&ct&re ofthe c&rve.

5No, not typo.6


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Some other helicalised c&rves.

para(ola

hyper(olictangent

ellipsestraightline

eponential c&rve


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%ntrod&cing7 The 4elicaliser8 .

• The helicaliser is a set of parametric form&lae s&ch that it /helicalises0 ac&rve.

• To say that a c&rve is /helicalised0 is to mean that the c&rve !o&ld (edescri(ed (y a ne! c&rve that !inds aro&nd the original c&rve, li1e a heli

(eing directed (y the original c&rve.

• ssentially, any c&rve can (e descri(ed (y its parametric e:&ation, i.e.

• 4ence, the helicaliser !o&ld yield the helicalised c&rve, ;&st (y s&(stit&tingthe f&nctions 5t6 , y5t6 , *5t6 ,


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The 4elical =ractal .

• An a(sol&tely self#similar fractal, called the Helical Fractal can (e&nderstood in t!o !ays7

# %t is li1e a c&rve that defines a heli, !hich defines a larger heli,!hich in t&rn defines yet some other (igger heli, ) ) )

# Another !ay to some!hat descri(e it is that the shape of the c&rveis defined (y the c&rve !inding aro&nd it, !here(y that shape of the!inding c&rve is act&ally defined (y the c&rve !inding a(o&t it, !hoseshape is ) ) )

• A formal mathematical definition of the helical fractal !ill (e givenlater, after its highly#la(orio&s derivation is finally accomplished.


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Some very (asic differentialgeometry.

tangent vector, v 5t6

normal, n 5t6

(inormal, b 5t6

c&rve in '#d space

The three &nit vectors v 5t6 , n 5t6 and b 5t6 areal!ays m&t&ally perpendic&lar to each other andthe directions depend on the point on the c&rve.


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Nat&ral parametrisation (y arclength.

• To even start thin1ing a(o&t helicalising a c&rve, it is important to&nderstand the nat&ral parametrisation of a c&rve. This is 1ey too(taining the normal and (inormal.

• To helicalise is to ma1e the c&rve !ind aro&nd in circles. This can (edone (y adding the oscillatory terms / cos >t 0 and / sin >t 0 to thenormal and (inormal.

• -ith these then, it is possi(le to !or1 o&t the helicaliser.

• A c&rve can (e parametrised (y any parameter t . %n partic&lar, thisparameter can (e chosen to (e s , !here s is the arc length. This is1no!n as the nat&ral parametrisation (y arc length.


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Tangent vector, normal and (inormalthro&gh nat&ral parametrisation.


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Messy differentions and an ellipticintegral.

• The tro&(le !ith the helical fractal is that its arc length eval&ation re:&iressolving the elliptic integral of the second 1ind.

# 3ertainly not something that is analytically solva(le, and also

ma1es it very cl&msy in the attempt to define a general form&la.

• Another iss&e is !ith the many differentiations involved.

# Tho&gh differentiations are al!ays possi(le, applying too many

derivatives invites comp&tational pro(lems (eca&se certain

soft!ares li1e MAT$A? are not smart eno&gh to carry o&t sym(ollic differentiations, or other!ise not efficient.

# Nevertheless, n&merical differentiation schemes are of co&rse o&t

there as a last resort aid. 5?UT let


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A simpler !ay approach.

• Act&ally, !hat are needed are not necessarily the normal and the(inormal.

• %n fact, any perpendic&lar t!o vectors sitting in the normal plane !ill dothe ;o(. &st remem(er to divide (y its length to ma1e it a &nit vector.

• Biven the parametric e:&ations of a c&rve, a vector tangent to thec&rve is simply ;&st the derivative.

• A trivial vector that is normal to the tangent vector can easily (e

o(tained (y inspection. The second normal !o&ld then (e the crossprod&ct.


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The !ay to helicalise.

• Then, the oscillatory terms cos5>t6 and sin5>t6 can each (e added tothe t!o normal vectors.

• The process is not diffic&lt in theory, (&t the possi(ility of act&allysolving the alge(ra is :&estiona(le, (eca&se it involves the inverti(ilityof a '' matri and very long#!inded epressions.

• ?&t the pro(lem is finally solved, tedio&sly . Th&s, there lies thedefinition of the helical fractal, as the limit of repeated iteration of

the helicaliser form&lae.


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The f&ll derivation of the helicaliser.


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Tangent vectors and t!o normals.


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Celationship (et!een the ne! (asis!ith the original ones.


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Celationship (et!een the ne! (asis!ith the original ones.


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The c&rve in terms of the tangentvector and the normals.


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Addition of oscillatory terms.


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Accomplishment.


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=ormal mathematical definition ofthe helical fractal.

• The helicaliser ;&st derived is sort of a r&le that helicalises a c&rve!ith the inp&t of the parametric f&nction and their derivatives.

• ?y ta1ing the *ero#th level of the helical fractal as the straight line,

applying the helicaliser once gives a heli .

• Second application helicalises the heli to prod&ce !hat is called theslin1y .

• The reperc&ssion of many repeated applications !ill then res&lt in thehelical fractal.


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=ormal mathematical definition ofthe helical fractal.

So here is the long#a!aited definition7


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Presenting7 The 4elical =ractal .

$evel *ero7Straight line.

$evel one74eli. $evel t!o7

Slin1y.


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4elical =ractal7 Up close.

%t is a reminiscent ofitself at all scales definition of a fractal.


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lliptic integral of the second 1ind.

• A thoro&gh analysis on this fractal 5li1e its length and dimension6, especially!ith its mathematical definition in place sho&ld (e carried o&t to give a morein#depth &nderstanding of this fractal.

• 4o!ever as mentioned (efore, the arc length involves an elliptic integral of thesecond 1ind and th&s cannot (e analytically solved.

• This seems to ma1e the eval&ation of its dimension a (it complicated.

• %t has to (e noted that this is an a(sol&tely self#similar fractal.

•?&t !hat is its self#similar dimension, especially that its no! not a simple c&(eor !hatever, (&t a (east of coiling monster that is mind#(oggling

• Please do not offer the idea of eval&ating the 4a&sdoff dimension. Thee:&ation of the c&rve is already complicated eno&gh to !rite o&t on paper@@


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%ntrod&cing7The 3irc&lar =ractal .

• No!, there eists a different fractal, !hich some!hat possesses thesame manifestation as the helical fractal.

• %nstead of defining a c&rve (y coiling aro&nd it, the c&rve can (e

descri(ed (y circles that are normal to it .

• Then starting at level one !ith a circle, the net level !o&ld (e to removethis circle, and replace it !ith > smaller circles that defines the circle.

•

4o! it defines the original circle is (y laying normal to it.

• Pict&res spea1 lo&der than !ords, tho&gh !itho&t emitting any so&nd.4ere goes.

A discrete version,


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Presenting7 The 3irc&lar =ractal .

=irst level7one circle.

Second level7>EF circles, AEF .

Third level7>EFG circles, AEFG .

The 3irc&lar =ractal is the limit !here > more circles smaller (y a factor of A

defines the circles of the previo&s level.

Hf co&rse, the infinite n&m(er of circlesget so small s&ch that its infinitesimal si*e

ma1es it almost invisi(le.

Note that its self#similar dimension is one.5> copies scaled (y a factor A, !here >EAEF .6


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3om(ining together all levels,

• -ell, the Circular Fractal is a fractal. %t has a self#similar dimensionof d E ln>I ln A .

• ?&t ;&st going ;&st slightly more advent&ro&s as to piece &p A$$ levelsof the circ&lar fractal, the res&lt is the Super Circular Fractal .

%ntrod&cing7 The S&per 3irc&lar =ractal .


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4elical =ractal , A&iliary =ractal ,Discrete 4elical =ractal .

• %n this case, the circ&lar fractal is called an auxiliary fractal , since it is &sedto (&ild &p the s&per circ&lar fractal.

• No!, the circ&lar fractal is essential a discrete version of the helical fractal.4ence it is d&((ed as the

• So here, everything (ecomes o(vio&s. All these three fractals have a self#similar dimension of d E ln>I ln A .

• %n other !ords, the dimension of the fractal highly depends on relationship(et!een the n&m(er of ne! copies as !ell as the scale factor.

Discrete 4elical =ractal.Discrete 4elical =ractal.


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S&per 4elical =ractal

• ?&t !ait. %t seems really nat&ral that the s&per circ&lar fractal eists.There is no do&(t that each smaller circle is an eact copy of thelarger preceding level.

• 4ave another good loo1 at it.


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%ntrod&cing7 The S&per 4elical =ractal .

• Since the circ&lar fractal isthe discrete helical fractal,the helical fractal sho&lditself also (e an a&iliary

fractal.

• Piecing &p all its individ&allevels event&ally forms theS&per 4elical =ractal .

• 4ave a good loo1 at it.

%ndeed@


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Analysis7 Arc length.


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Ma1ing arc length finite.


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The :&antity associated !ith arclength.

Note that from the originalform&la for arc length, it is here!here the po!er n occ&rs s&chthat the nat&ral logarithms areta1en.


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3omparison (et!een ? and d .


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Celationship (et!een ? and d .


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The fo&r most (ea&tif&l fractals inthis &niverse)4elical =ractal

3irc&lar =ractal

discrete version

piecing &p individ&al levels)

S&per 3irc&lar =ractal

contin&o&s version

individ&al fractal

S&per 4elical =ractal

=o&r different fractals !ith thesame manifestations)


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Ac1no!ledgements.

Dr. =edor D&*hin. Assistant ProfessorAndre! ames Jric1er.

Assistant Professor3he! $oc1 K&e.


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Ceference.

• ?arrett HLNeill, lementary Differential Beometry. P&(lished (ylsevier Academic Press, second edition.

Helical Fractal

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