Fractals/Mathematics/group

Here one can find examples of relation between fractals and groups.Group theory at wikipedia Introduction "Group theory is very useful in that it finds commonalities among disparate things through the power of abstraction."mathilluminated : Unit 6 The Beauty of Symmetry There are important connections between the algebraic structure of self-similar groups and the dynamical properties of the polynomials . Group theory : geometric combinatorial Computational Definitions Alphabet Alphabet it is a finite set of symbols x : Automaton Automaton is the basic abstract mathematical model of sequential machine. Different types of automata : recognition automata, Turing machines Moore machines Mealy machines, cellular automata pushdown automata Automaton has two visual presentations: a flow table, which describes the transitions to the next states and the outputs, a state diagram. Class equivalence class of an element a in the set X is the subset of all elements x of the set X which are equivalent to a Expansion p-adic digit a natural number between 0 and p − 1 (inclusive). A p-adic integer is a sequence of p-adic digits : n-adic expansion of number wikipedia : p-adic number binary integer or dyadic integer or 2-adic integer : where a is an element of binary alphabet X ={0,1} = (0, .. ,2-1) Graph The Schreier graphs Moore diagram of the automaton ( or the state diagram for a Moore machine) it is a directed labeled graph with : the vertices ( nodes) identified with the states of automaton / generators of the fundamental group the edges (lines with arrows) show the state transition, labels : (input,output) are a directed pair of elements in X Group "A group is a collection of objects that obey a strict set of rules when acted upon by an operation."learner.org course : mathilluminated - symmetry A group is the algebraic structure : , where : is a non-empty set denotes a binary operation called the group operation: which must obey the following rules (or axioms) : Closure, Associativity Identity Inverse CommutativityElementary group theory Identity : " the group must have an element that serves as the Identity. The characteristic feature of the Identity is that when it is combined with any other member under the group operation, it leaves that member unchanged." learner.org course : mathilluminated - symmetry Inverse : " each member or element of the group must have an inverse. When a member is combined with its inverse under the group operation, the result is the Identity "learner.org course : mathilluminated - symmetry Closure : "This means that whenever two group members are combined under the group operation, the result is another member of the group"learner.org course : mathilluminated - symmetry Associativity : "if we take a list of three or more group members and combine them two at a time, it doesn’t matter which end of the list we start with"learner.org course : mathilluminated - symmetry Automaton group = Group generated by automaton FR = Functionally Recursive Group IMG = Iterated Monodromy Group IMGIterated monodromy group at wikipediaIntroduction to Iterated Monodromy Groups by Sébastien Godillon The iterated monodromy group acts by automorphism on the rooted tree of preimages where a vertex is connected by an edge with . Self-similar group Machine Finite State MachinesComputer Science Logo Style Volume 3: Beyond Programming Brian Harvey Mealy machinewikipedia : Mealy_machine Moore machine Polynomial postcritically finite polynomial : the orbit of the critical point is finite. It is when critical point is periodic or preperiodic.Alfredo Poirier : On Post Critically Finite Polynomials Part One: Critical Portraits Relation Equivalence relation ~ over/on the set X it is a binary relation relation on X which is reflexive, symmetric, and transitive it induces partition P of a set Xwikipedia : Equivalence relation into several disjoint subsets, called equivalence classes Sequence ks = kneading sequence(s) Word Word w over alphabet X is any sequence of symbols from alphabet X. Word can be : infinite finite empty = the word of length zero : denotes word beginning with Examples "The iterated monodromy groups of quadratic rational maps with size of postcritical set at most 3, arranged in a table. The algebraic structure of most of them is not yet well understood." From fractal groups to fractal sets Authors: Laurent Bartholdi, Rostislav I. Grigorchuk, Volodymyr V. Nekrashevych f(z) standard actions nucleus machine groupcomment The adding machineBinary adding group Basilica group conjugate to the Chebyshev polynomial T2 From self-similar groups to self-similar sets and spectra by Rostislav Grigorchuk, Volodymyr Nekrashevych, Zoran Sunic the Hanoi Towers group the Sierpinski gasket Thompson-Like GroupGroups for Dendrite Julia Sets by Will Smith intermediate growth Where : is a permutation Software CoxeterTilings in JS : Coxeter group kaleidoscope Group Explorer Group Explorer is mathematical visualization software for the abstract algebra classroom.Explorer GAP GAPGAP software at wikipedia is a CAS software.CAS at wikipedia To run : /usr/share/gap/bin/gap.sh If the system failed to load packages install libraries, packages and compile them ( nq, pargap, fr). Run test : Read( Filename( DirectoriesLibrary( "tst" ), "testinstall.g" ) ); Load package fr by Laurent Bartholdi FR package by Laurent Bartholdi for GAP CAS LoadPackage("fr"); Run fr test : Read( Filename( DirectoriesLibrary( "pkg/fr/tst" ), "testall.g" ) ); GAP and fr package can use external programs like Mandel, ImageMagic, graphviz or rsvg-view to draw and display images. Draw DrawDraw function from fr package by Laurent Bartholdi Draw(NucleusMachine(BasilicaGroup)); creates graph description of the m (Mealy machine or element m ). It is converted to Postscript using the program dot from the graphviz packagegraphviz - - Graph Visualization Software displays image in a separate X window using the command lin program display ( from ImageMagic) or rsvg-view.librsvg - free, open source SVG rendering library This works on UNIX systems. One can right click on image to see local menu of display program. If a second argument of Draw function ( filename) is present, the graph is saved, in dot format, under that filename : Draw(NucleusMachine(BasilicaGroup),"a.dot"); Saves output to a.dot file. Dot file is a text file describing graph in dot language. digraph MealyMachine { a [shape=circle] b [shape=circle] c [shape=circle] d [shape=circle] e [shape=circle] f [shape=circle] g [shape=circle] a -> a [label="1/1",color=red]; a -> a [label="2/2",color=blue]; b -> a [label="1/1",color=red]; b -> d [label="2/2",color=blue]; c -> a [label="1/1",color=red]; c -> e [label="2/2",color=blue]; d -> a [label="1/2",color=red]; d -> b [label="2/1",color=blue]; e -> c [label="1/2",color=red]; e -> a [label="2/1",color=blue]; f -> a [label="1/2",color=red]; f -> g [label="2/1",color=blue]; g -> f [label="1/2",color=red]; g -> a [label="2/1",color=blue]; } This a.dot file can be converted to other formats using command line program dot. For example in ps file : dot -Tps a.dot -o a.ps or png file : dot -Tpng a.dot -o a.png or svg : dot -Tsvg a.dot -o a.svg External angle Function from FR package : ExternalAngle(machine) Returns: The external angle identifying machine . In case machine is the IMG machine of a unicritical polynomial, this function computes the external angle landing at the critical value. gap> z := Indeterminate(COMPLEX_FIELD,"z"); z gap> r := P1MapRational(z^2-1); # Basilica Julia set <P1 mapping of degree 2> gap> m:=IMGMachine(r); <FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f2*f1*f3 ]> gap> ExternalAngle(m); {2/3} Elements(last); # More precisely, it computes the equivalence class of that external angle under ExternalAnglesRelation [ 1/3, 2/3 ] Another example : gap> m:= PolynomialIMGMachine(2,[1/7]); # the machine descibing the rabbit : degree=2, gap> ExternalAngle(m); {2/7} gap> Elements(last); [ 1/7, 2/7 ] Machine PolynomialIMGMachine PolynomialIMGMachine(d, per[, pre[, formal]]) This function creates a IMG machine that describes a topological polynomial. The polynomial is described symbolically in the language of external angles. d is the degree of the polynomial. per is the list of angles pre is the list of preangles. angles are rational numbers, considered modulo 1. Each entry in per or pre is either a rational (interpreted as an angle), or a list of angles [a1 , . . . , ai ] such that da1 = . . . = dai . The angles in per are angles landing at the root of a Fatou component, and the angles in pre land at the other points of Julia set. gap> m:=PolynomialIMGMachine(2,[1/3],[]); # the Basilica <FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f3*f2*f1 ]> gap> Display(m); G | 1 2 ----+---------+---------+ f1 | f1^-1,2 f2*f1,1 f2 | f1,1 <id>,2 f3 | f3,2 <id>,1 ----+---------+---------+ Adding element: FRElement(...,f3) Relator: f3*f2*f1 gap> Display(PolynomialIMGMachine(2,[],[1/6])); # z^2+I G | 1 2 ----+---------------+---------+ f1 | f1^-1*f2^-1,2 f2*f1,1 f2 | f1,1 f3,2 f3 | f2,1 <id>,2 f4 | f4,2 <id>,1 ----+---------------+---------+ Adding element: FRElement(...,f4) Relator: f4*f3*f2*f1 PostCriticalMachine PostCriticalMachine(f) Returns: The Mealy machine of f ’s post-critical orbit. This function constructs a Mealy machine P on the alphabet [1], which describes the post-critical set of f . gap> z := Indeterminate(Rationals,"z");; gap> m := PostCriticalMachine(z^2); <Mealy machine on alphabet [ 1 ] with 2 states> gap> Display(m); | 1 ---+-----+ a | a,1 b | b,1 ---+-----+ gap> Correspondence(m); [ 0, infinity ] It is in fact an oriented graph with constant out-degree 1. Draw(m); gap> m := PostCriticalMachine(z^2-1);; Display(m); Correspondence(m); | 1 ---+-----+ a | c,1 b | b,1 c | a,1 ---+-----+ [ -1, infinity, 0 ] Kneading Sequence KneadingSequence(angle) "This function converts a rational angle to a kneading sequence,Unimodal Maps and Kneading Theory by Warren Weckesser to describe a quadratic polynomial.ADMISSIBILITY OF KNEADING SEQUENCES AND STRUCTURE OF HUBBARD TREES FOR QUADRATIC POLYNOMIALS by HENK BRUIN AND DIERK SCHLEICHER" ( from fr doc ) KneadingSequence(1/7); gives : [ 1, 1 ] "If angle is in [1/7, 2/7] and the option marked is set, the kneading sequence is decorated with markings in A,B,C." ( from fr doc ) KneadingSequence(1/5:marked); gives : [ "A1", "B1", "B0" ] Rays of root points ExternalAnglesRelation(degree, n) " This function returns ... all pairs of external angles that land at a common point of period up to n under angle multiplication by by degree ." ( from fr doc) In other words it gives angles in turns of external rays landing on root points of period n hyperbolic components of Mandelbrot set.Parameter rays of root points of period p components of Mandelbrot set For complex quadratic polynomials ( degree = 2) and period 3 : ExternalAnglesRelation(2,3); <equivalence relation on Rationals > It needs one more command : EquivalenceRelationPartition(last); and gives : [ [ 1/7, 2/7 ], [ 1/3, 2/3 ], [ 3/7, 4/7 ], [ 5/7, 6/7 ] ] This list has 4 elements : 3 pairs of period 3 angles i/7 1 pair of period 2 angles i/3 Internal Adress "This function returns internal addresses for all periodic points of period up to n under angle doubling. These internal addresses describe the prominent hyperbolic components along the path from the landing point to the main cardioid in the Mandelbrot set." (from fr doc ) Compare it with angled internall adresses Internal addresses in the Mandelbrot set and irreducibility of polynomials by Dierk Schleicher Note that angle is a fraction with denominator ( odd number ) : where n is period and d denominator Period 2 right|thumb|Basilica Julia set for with external rays 1/3 and 2/3 landing on repelling fixed point alpha AllInternalAddresses(2); [ [ ], [ [ 1/3, 2/3, 2 ] ] ] For period 1 list is a empty. [] For period 2 it gives : [ [ 1/3, 2/3, 2 ] ] which contain one sublist : [1/3, 2/3, 2] It describe period 2 hyperbolic component of Mandelbrot set with external rays 1/3 and 2/3 landing on its root point Parameter rays of root points of period p components of Mandelbrot set Period 3 right|thumb|Rabbit Julia set with 3 external rays : landing on fixed point right|thumb|lamination of Rabbit Julia set AllInternalAddresses(3); [ [ ], [ [ 1/3, 2/3, 2 ] ], [ [ 1/7, 2/7, 3 ], [ 3/7, 4/7, 3, 1/3, 2/3, 2 ], [ 5/7, 6/7, 3 ] ] ] For period 3 we have previous period 2 adress and new list for period 3 : [ [ 1/7, 2/7, 3 ], [ 3/7, 4/7, 3, 1/3, 2/3, 2 ], [ 5/7, 6/7, 3 ] ] It has 3 elements ( sublists). First element : [ 1/7, 2/7, 3 ] describes period 3 hyberbolic component with rays 1/7 and 2/7 landing on its root c=0.64951905283833*%i-0.125, which lays on the boundary of main cardioid with internal angle = 1/3 Second element : [ 3/7, 4/7, 3, 1/3, 2/3, 2 ] describes period 3 component with rays 3/7 and 4/7 landing on its root point. To find it one must go thru period 2 component. Because for c in wake (3/7, 4/7) dynamic rays 3/7 and 4/7 land together at a repelling periodic point then it also describes the airplane Julia set The Julia midgets scaling. The "Airplane" structure by Evgeny DemidovThe n-Category Café discussion on Benoit Mandelbrot" with landing angles [1/3, 2/3] and period 2. Third element : [ 5/7, 6/7, 3 ] describe period 3 hyberbolic component with rays 5/7 and 6/7 landing on its root point ( it is c=-0.64951905283833*%i-0.125, which lays on the boundary of main cardioid with internal angle = 2/3 ). Spider algorithm The Spider algorithm spider algorithm constructs polynomials with assigned combinatorics. Papers : Spider Algorithm - paper by John H. Hubbard and Dierk SchleicherAlgorithm - paper by John H. Hubbard and Dierk Schleicher online article by Claude Heiland-Allen Resurrecting Spiders by Claude Heiland-Allen original paper by Yuval Fisheroriginal paper by Yuval Fisher papers by Tore Moller JonassenTore Møller Jonassen papers by Gregory A. Kelseypapers by Gregory A. Kelsey THE MEDUSA ALGORITHM FOR POLYNOMIAL MATINGS by SUZANNE HRUSKA BOYD AND CHRISTIAN HENRIKSEN See also programs : program spider by Yuval Fisher : Mandel by Wolf Jung Goole search : "spider algorithm" polynomial RationalFunction RationalFunction(m) It returns a rational function f whose associated machine is m or a record describing the Thurston obstruction to realizability of f. gap> m := PolynomialIMGMachine(2,[1/3],[]); # the basilica <FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f3*f2*f1 ]> gap> RationalFunction(m); z^2-1. gap> m:=PolynomialIMGMachine(2,[1/7]); # the rabbit <FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f4) on Group( [ f1, f2, f3, f4 ] )/[ f4*f3*f2*f1 ]> gap> RationalFunction(m); z^2 + (-0.12256116687667946+I*0.74486176661942738) Delaunay triangulation gap> LoadPackage("fr"); gap> z := Indeterminate(COMPLEX_FIELD); x_1 gap> f := z^2-1; x_1^2-1. gap> m := IMGMachine(f); <FR machine with alphabet [ 1, 2 ] and adder FRElement(...,f3) on Group( [ f1,\ f2, f3 ] )/[ f2*f1*f3 ]> gap> Spider(m); <marked sphere on <triangulation with 6 vertices, 24 edges and 8 faces> marked by [ f1, f2, f3 ] -> [ f1^-1*f2^-1, f1, f2 ]> gap> Draw(last:julia); or gap> LoadPackage("fr"); gap>z := Indeterminate(COMPLEX_FIELD,"z"); gap> r := P1MapRational(z^2-1); <P1 mapping of degree 2> gap> IMGMachine(r); #I Post-critical points at [ P1Point("-1+0i"), P1Point("0+0i"), P1Point("P1infinity") ] <FR machine with alphabet [ 1 .. 2 ] and adder FRElement(...,f3) on Group( [ f1, f2, f3 ] )/[ f2*f1*f3 ]> gap> Spider(last); <marked sphere on <triangulation with 6 vertices, 24 edges and 8 faces> marked by [ f1, f2, f3 ] -> [ f1^-1*f2^-1, f\ 1, f2 ]> gap> Draw(last:julia); Draws dynamical plane on the sphere with marked Delaunay triangulation. One can rotate sphere with mouse in real time. References Books: Adventures in Group Theory Rubik's Cube, Merlin's Machine, and Other Mathematical Toys by David Joyner See also visual group theory by Nathan Carter