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132-662: The Mandelbrot set ( / ˈ m æ n d əl b r oʊ t , - b r ɒ t / ) is a two-dimensional set with a relatively simple definition that exhibits great complexity, especially as it is magnified. It is popular for its aesthetic appeal and fractal structures. The set is defined in the complex plane as the complex numbers c {\displaystyle c} for which the function f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c} does not diverge to infinity when iterated starting at z = 0 {\displaystyle z=0} , i.e., for which

264-438: A n ) n = − ∞ ∞ {\textstyle {(a_{n})}_{n=-\infty }^{\infty }} is a bi-infinite sequence , and can also be written as ( … , a − 1 , a 0 , a 1 , a 2 , … ) {\textstyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )} . In cases where

396-464: A n ) . {\textstyle (a_{n}).} Here A is the domain, or index set, of the sequence. Sequences and their limits (see below) are important concepts for studying topological spaces. An important generalization of sequences is the concept of nets . A net is a function from a (possibly uncountable ) directed set to a topological space. The notational conventions for sequences normally apply to nets as well. The length of

528-413: A collection or family , especially when its elements are themselves sets. Roster or enumeration notation defines a set by listing its elements between curly brackets , separated by commas: This notation was introduced by Ernst Zermelo in 1908. In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in

660-429: A sequence , a tuple , or a permutation of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 4, 2} represent the same set. For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ' ... '. For instance, the set of the first thousand positive integers may be specified in roster notation as An infinite set

792-412: A set , it contains members (also called elements , or terms ). The number of elements (possibly infinite ) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to

924-521: A centerpiece of this field ever since. The Mandelbrot set is the uncountable set of values of c in the complex plane for which the orbit of the critical point z = 0 {\textstyle z=0} under iteration of the quadratic map remains bounded . Thus, a complex number c is a member of the Mandelbrot set if, when starting with z 0 = 0 {\displaystyle z_{0}=0} and applying

1056-512: A complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization , mathematical beauty , and motif . The Mandelbrot set has its origin in complex dynamics , a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The fractal was first defined and drawn in 1978 by Robert W. Brooks and Peter Matelski as part of

1188-410: A computer experiment in 1991, where he computed the number of iterations required for the series to diverge for z = − 3 4 + i ε {\displaystyle z=-{\tfrac {3}{4}}+i\varepsilon } ( − 3 4 {\displaystyle -{\tfrac {3}{4}}} being the location thereof). As the series does not diverge for

1320-406: A consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set . For instance, a value of c belongs to the Mandelbrot set if and only if the corresponding Julia set is connected. Thus, the Mandelbrot set may be seen as a map of the connected Julia sets. This principle

1452-466: A definition is called a semantic description . Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set F can be defined as follows: F = { n ∣ n  is an integer, and  0 ≤ n ≤ 19 } . {\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.} In this notation,

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1584-663: A distance from L {\displaystyle L} less than d {\displaystyle d} . For example, the sequence a n = n + 1 2 n 2 {\textstyle a_{n}={\frac {n+1}{2n^{2}}}} shown to the right converges to the value 0. On the other hand, the sequences b n = n 3 {\textstyle b_{n}=n^{3}} (which begins 1, 8, 27, ...) and c n = ( − 1 ) n {\displaystyle c_{n}=(-1)^{n}} (which begins −1, 1, −1, 1, ...) are both divergent. If

1716-599: A function of n . Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions. Not all sequences can be specified by a recurrence relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...). There are many different notions of sequences in mathematics, some of which ( e.g. , exact sequence ) are not covered by

1848-406: A hyperbolic component of period q bifurcates from the main cardioid at a point on the edge of the cardioid corresponding to an internal angle of 2 π p q {\displaystyle {\tfrac {2\pi p}{q}}} . The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the p / q -limb . Computer experiments suggest that

1980-416: A limit if the elements of the sequence become closer and closer to some value L {\displaystyle L} (called the limit of the sequence), and they become and remain arbitrarily close to L {\displaystyle L} , meaning that given a real number d {\displaystyle d} greater than zero, all but a finite number of the elements of the sequence have

2112-483: A natural number N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} we have If ( a n ) {\displaystyle (a_{n})} is a sequence of complex numbers rather than a sequence of real numbers, this last formula can still be used to define convergence, with the provision that | ⋅ | {\displaystyle |\cdot |} denotes

2244-415: A recurrence relation is Recamán's sequence , defined by the recurrence relation with initial term a 0 = 0. {\displaystyle a_{0}=0.} A linear recurrence with constant coefficients is a recurrence relation of the form where c 0 , … , c k {\displaystyle c_{0},\dots ,c_{k}} are constants . There

2376-420: A repelling period- q cycle. As we pass through the bifurcation parameter into the p q {\displaystyle {\tfrac {p}{q}}} -bulb, the attracting fixed point turns into a repelling fixed point (the α {\displaystyle \alpha } -fixed point), and the period- q cycle becomes attracting. Bulbs that are interior components of the Mandelbrot set in which

2508-456: A rotation number of 2 / 5 {\displaystyle 2/5} . Its distinctive antenna-like structure comprises a junction point from which five spokes emanate. Among these spokes, called the principal spoke is directly attached to the 2 / 5 {\displaystyle 2/5} bulb, and the 'smallest' non-principal spoke is positioned approximately 2 / 5 {\displaystyle 2/5} of

2640-400: A sequence are discussed after the examples. The prime numbers are the natural numbers greater than 1 that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence (2, 3, 5, 7, 11, 13, 17, ...). The prime numbers are widely used in mathematics , particularly in number theory where many results related to them exist. The Fibonacci numbers comprise

2772-440: A sequence converges, then the value it converges to is unique. This value is called the limit of the sequence. The limit of a convergent sequence ( a n ) {\displaystyle (a_{n})} is normally denoted lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} . If ( a n ) {\displaystyle (a_{n})}

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2904-404: A sequence is defined as the number of terms in the sequence. A sequence of a finite length n is also called an n -tuple . Finite sequences include the empty sequence  ( ) that has no elements. Normally, the term infinite sequence refers to a sequence that is infinite in one direction, and finite in the other—the sequence has a first element, but no final element. Such a sequence

3036-463: A sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting

3168-407: A sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy (see below). Each of the hyperbolic components has a center , which is a point c such that the inner Fatou domain for f c ( z ) {\displaystyle f_{c}(z)} has

3300-409: A sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways. Moreover, the subscripts and superscripts could have been left off in the third, fourth, and fifth notations, if the indexing set was understood to be

3432-450: A sequence of sequences: ( ( a m , n ) n ∈ N ) m ∈ N {\textstyle ((a_{m,n})_{n\in \mathbb {N} })_{m\in \mathbb {N} }} denotes a sequence whose m th term is the sequence ( a m , n ) n ∈ N {\textstyle (a_{m,n})_{n\in \mathbb {N} }} . An alternative to writing

3564-441: A set S , denoted | S | , is the number of members of S . For example, if B = {blue, white, red} , then | B | = 3 . Repeated members in roster notation are not counted, so | {blue, white, red, blue, white} | = 3 , too. More formally, two sets share the same cardinality if there exists a bijection between them. The cardinality of the empty set is zero. The list of elements of some sets

3696-465: A set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B ; more formally, a function is a special kind of relation , one that relates each element of A to exactly one element of B . A function is called An injective function is called an injection , a surjective function is called a surjection , and a bijective function is called a bijection or one-to-one correspondence . The cardinality of

3828-543: A study of Kleinian groups . On 1 March 1980, at IBM 's Thomas J. Watson Research Center in Yorktown Heights , New York , Benoit Mandelbrot first visualized the set. Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980. The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard (1985), who established many of its fundamental properties and named

3960-656: A super-attracting cycle—that is, that the attraction is infinite. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. Therefore, f c n ( 0 ) = 0 {\displaystyle f_{c}^{n}(0)=0} for some n . If we call this polynomial Q n ( c ) {\displaystyle Q^{n}(c)} (letting it depend on c instead of z ), we have that Q n + 1 ( c ) = Q n ( c ) 2 + c {\displaystyle Q^{n+1}(c)=Q^{n}(c)^{2}+c} and that

4092-425: A turn counterclockwise from the principal spoke, providing a distinctive identification as a 2 / 5 {\displaystyle 2/5} -bulb. This raises the question: how does one discern which among these spokes is the 'smallest'? In the theory of external rays developed by Douady and Hubbard. there are precisely two external rays landing at the root point of a satellite hyperbolic component of

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4224-409: A two-dimensional planar region. Curves with Hausdorff dimension 2, despite being (topologically) 1-dimensional, are oftentimes capable of having nonzero area (more formally, a nonzero planar Lebesgue measure ). Whether this is the case for the Mandelbrot set boundary is an unsolved problem. It has been shown that the generalized Mandelbrot set in higher-dimensional hypercomplex number spaces (i.e. when

4356-455: Is disconnected . This conjecture was based on computer pictures generated by programs that are unable to detect the thin filaments connecting different parts of M {\displaystyle M} . Upon further experiments, he revised his conjecture, deciding that M {\displaystyle M} should be connected. A topological proof of the connectedness was discovered in 2001 by Jeremy Kahn . The dynamical formula for

4488-565: Is monotonically decreasing if each consecutive term is less than or equal to the previous one, and is strictly monotonically decreasing if each is strictly less than the previous. If a sequence is either increasing or decreasing it is called a monotone sequence. This is a special case of the more general notion of a monotonic function . The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid any possible confusion with strictly increasing and strictly decreasing , respectively. If

4620-474: Is a divergent sequence, then the expression lim n → ∞ a n {\textstyle \lim _{n\to \infty }a_{n}} is meaningless. A sequence of real numbers ( a n ) {\displaystyle (a_{n})} converges to a real number L {\displaystyle L} if, for all ε > 0 {\displaystyle \varepsilon >0} , there exists

4752-468: Is a general method for expressing the general term a n {\displaystyle a_{n}} of such a sequence as a function of n ; see Linear recurrence . In the case of the Fibonacci sequence, one has c 0 = 0 , c 1 = c 2 = 1 , {\displaystyle c_{0}=0,c_{1}=c_{2}=1,} and the resulting function of n

4884-403: Is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A is a subset of B , then the region representing A is completely inside the region representing B . If two sets have no elements in common, the regions do not overlap. A Venn diagram , in contrast, is a graphical representation of n sets in which

5016-422: Is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is and the set of all integers is Another way to define a set is to use a rule to determine what the elements are: Such

5148-506: Is a set with exactly one element; such a set may also be called a unit set . Any such set can be written as { x }, where x is the element. The set { x } and the element x mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat. If every element of set A is also in B , then A is described as being a subset of B , or contained in B , written A ⊆ B , or B ⊇ A . The latter notation may be read B contains A , B includes A , or B

5280-411: Is a simple classical example, defined by the recurrence relation with initial terms a 0 = 0 {\displaystyle a_{0}=0} and a 1 = 1 {\displaystyle a_{1}=1} . From this, a simple computation shows that the first ten terms of this sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34. A complicated example of a sequence defined by

5412-401: Is a strictly increasing sequence of positive integers. Some other types of sequences that are easy to define include: An important property of a sequence is convergence . If a sequence converges, it converges to a particular value known as the limit . If a sequence converges to some limit, then it is convergent . A sequence that does not converge is divergent . Informally, a sequence has

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5544-435: Is a superset of A . The relationship between sets established by ⊆ is called inclusion or containment . Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B . If A is a subset of B , but A is not equal to B , then A is called a proper subset of B . This can be written A ⊊ B . Likewise, B ⊋ A means B is a proper superset of A , i.e. B contains A , and

5676-431: Is also conjectured to be self-similar around generalized Feigenbaum points (e.g., −1.401155 or −0.1528 + 1.0397 i ), in the sense of converging to a limit set. The Mandelbrot set in general is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. These copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to

5808-464: Is bi-infinite. This sequence could be denoted ( 2 n ) n = − ∞ ∞ {\textstyle {(2n)}_{n=-\infty }^{\infty }} . A sequence is said to be monotonically increasing if each term is greater than or equal to the one before it. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }}

5940-409: Is called a lower bound . If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded . A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. For instance, the sequence of positive even integers (2, 4, 6, ...) is a subsequence of

6072-465: Is called a singly infinite sequence or a one-sided infinite sequence when disambiguation is necessary. In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence , two-way infinite sequence , or doubly infinite sequence . A function from the set Z of all integers into a set, such as for instance the sequence of all even integers ( ..., −4, −2, 0, 2, 4, 6, 8, ... ),

6204-418: Is called an index , and the set of values that it can take is called the index set . It is often useful to combine this notation with the technique of treating the elements of a sequence as individual variables. This yields expressions like ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , which denotes a sequence whose n th element

6336-416: Is easily discernible by inspection. Other examples are sequences of functions , whose elements are functions instead of numbers. The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences. Other notations can be useful for sequences whose pattern cannot be easily guessed or for sequences that do not have a pattern such as the digits of π . One such notation

6468-399: Is endless, or infinite . For example, the set N {\displaystyle \mathbb {N} } of natural numbers is infinite. In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality . Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that

6600-624: Is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane. Similarly, Yoccoz first proved the local connectivity of Julia sets, before establishing it for the Mandelbrot set at the corresponding parameters. For every rational number p q {\displaystyle {\tfrac {p}{q}}} , where p and q are relatively prime ,

6732-407: Is given by This gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set. Douady and Hubbard showed that the Mandelbrot set is connected . They constructed an explicit conformal isomorphism between the complement of the Mandelbrot set and the complement of the closed unit disk . Mandelbrot had originally conjectured that the Mandelbrot set

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6864-406: Is given by Binet's formula . A holonomic sequence is a sequence defined by a recurrence relation of the form where c 1 , … , c k {\displaystyle c_{1},\dots ,c_{k}} are polynomials in n . For most holonomic sequences, there is no explicit formula for expressing a n {\displaystyle a_{n}} as

6996-495: Is given by Sloane's OEIS :  A000740 . It is conjectured that the Mandelbrot set is locally connected . This conjecture is known as MLC (for Mandelbrot locally connected ). By the work of Adrien Douady and John H. Hubbard , this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important hyperbolicity conjecture mentioned above. The work of Jean-Christophe Yoccoz established local connectivity of

7128-503: Is given by the variable a n {\displaystyle a_{n}} . For example: One can consider multiple sequences at the same time by using different variables; e.g. ( b n ) n ∈ N {\textstyle (b_{n})_{n\in \mathbb {N} }} could be a different sequence than ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} . One can even consider

7260-482: Is in B ". The statement " y is not an element of B " is written as y ∉ B , which can also be read as " y is not in B ". For example, with respect to the sets A = {1, 2, 3, 4} , B = {blue, white, red} , and F = { n | n is an integer, and 0 ≤ n ≤ 19} , The empty set (or null set ) is the unique set that has no members. It is denoted ∅ , ∅ {\displaystyle \emptyset } , { }, ϕ , or ϕ . A singleton set

7392-431: Is in contrast to the definition of sequences of elements as functions of their positions. To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation. The Fibonacci sequence

7524-415: Is measured by determining the length of the arc between the two angles. If the root point of the main cardioid is the cusp at c = 1 / 4 {\displaystyle c=1/4} , then the main cardioid is the 0 / 1 {\displaystyle 0/1} -bulb. The root point of any other bulb is just the point where this bulb is attached to the main cardioid. This prompts

7656-404: Is monotonically increasing if and only if a n + 1 ≥ a n {\textstyle a_{n+1}\geq a_{n}} for all n ∈ N . {\displaystyle n\in \mathbf {N} .} If each consecutive term is strictly greater than (>) the previous term then the sequence is called strictly monotonically increasing . A sequence

7788-428: Is not equal to A . A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset), while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B . Examples: The empty set is a subset of every set, and every set is a subset of itself: An Euler diagram

7920-469: Is one period-q bulb called the p q {\displaystyle {\frac {p}{q}}} bulb, which is tangent to the main cardioid at the parameter and which contains parameters with q {\displaystyle q} -cycles having combinatorial rotation number p q {\displaystyle {\frac {p}{q}}} . More precisely, the q {\displaystyle q} periodic Fatou components containing

8052-500: Is replaced by the expression dist ⁡ ( a n , L ) {\displaystyle \operatorname {dist} (a_{n},L)} , which denotes the distance between a n {\displaystyle a_{n}} and L {\displaystyle L} . If ( a n ) {\displaystyle (a_{n})} and ( b n ) {\displaystyle (b_{n})} are convergent sequences, then

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8184-479: Is the complex number with the largest magnitude within the set, but otherwise the threshold is arbitrary). If c {\displaystyle c} is held constant and the initial value of z {\displaystyle z} is varied instead, the corresponding Julia set for the point c {\displaystyle c} is obtained. The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of

8316-568: Is to write down a general formula for computing the n th term as a function of n , enclose it in parentheses, and include a subscript indicating the set of values that n can take. For example, in this notation the sequence of even numbers could be written as ( 2 n ) n ∈ N {\textstyle (2n)_{n\in \mathbb {N} }} . The sequence of squares could be written as ( n 2 ) n ∈ N {\textstyle (n^{2})_{n\in \mathbb {N} }} . The variable n

8448-440: Is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of S with the elements of P ( S ) will leave some elements of P ( S ) unpaired. (There is never a bijection from S onto P ( S ) .) A partition of a set S is a set of nonempty subsets of S , such that every element x in S is in exactly one of these subsets. That is,

8580-691: The 1 / 3 {\displaystyle 1/3} and 1 / 2 {\displaystyle 1/2} -bulbs is the 2 / 5 {\displaystyle 2/5} -bulb, again given by Farey addition. 1 3 {\displaystyle {\frac {1}{3}}} ⊕ {\displaystyle \oplus } 1 2 {\displaystyle {\frac {1}{2}}} = {\displaystyle =} 2 5 {\displaystyle {\frac {2}{5}}} Set (mathematics) Sets are uniquely characterized by their elements; this means that two sets that have precisely

8712-441: The codomain of the sequence is fixed by context, for example by requiring it to be the set R of real numbers, the set C of complex numbers, or a topological space . Although sequences are a type of function, they are usually distinguished notationally from functions in that the input is written as a subscript rather than in parentheses, that is, a n rather than a ( n ) . There are terminological differences as well:

8844-427: The convergence properties of sequences. In particular, sequences are the basis for series , which are important in differential equations and analysis . Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers . There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify

8976-500: The diameter of the limb tends to zero like 1 q 2 {\displaystyle {\tfrac {1}{q^{2}}}} . The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like 1 q {\displaystyle {\tfrac {1}{q}}} . A period- q limb will have q − 1 {\displaystyle q-1} "antennae" at

9108-411: The limit of a sequence of rational numbers (e.g. via its decimal expansion , also see completeness of the real numbers ). As another example, π is the limit of the sequence (3, 3.1, 3.14, 3.141, 3.1415, ...), which is increasing. A related sequence is the sequence of decimal digits of π , that is, (3, 1, 4, 1, 5, 9, ...). Unlike the preceding sequence, this sequence does not have any pattern that

9240-842: The n loops divide the plane into 2 zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are A , B , and C , there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist). There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them. Many of these important sets are represented in mathematical texts using bold (e.g. Z {\displaystyle \mathbf {Z} } ) or blackboard bold (e.g. Z {\displaystyle \mathbb {Z} } ) typeface. These include Each of

9372-420: The natural numbers . In the second and third bullets, there is a well-defined sequence ( a k ) k = 1 ∞ {\textstyle {(a_{k})}_{k=1}^{\infty }} , but it is not the same as the sequence denoted by the expression. Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion . This

9504-500: The real and imaginary parts of c {\displaystyle c} as image coordinates on the complex plane , pixels may then be colored according to how soon the sequence | f c ( 0 ) | , | f c ( f c ( 0 ) ) | , … {\displaystyle |f_{c}(0)|,|f_{c}(f_{c}(0))|,\dotsc } crosses an arbitrarily chosen threshold (the threshold must be at least 2, as −2

9636-470: The uniformisation of the complement of the Mandelbrot set, arising from Douady and Hubbard's proof of the connectedness of M {\displaystyle M} , gives rise to external rays of the Mandelbrot set. These rays can be used to study the Mandelbrot set in combinatorial terms and form the backbone of the Yoccoz parapuzzle . The boundary of the Mandelbrot set is the bifurcation locus of

9768-462: The vertical bar "|" means "such that", and the description can be interpreted as " F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar. Philosophy uses specific terms to classify types of definitions: If B is a set and x is an element of B , this is written in shorthand as x ∈ B , which can also be read as " x belongs to B ", or " x

9900-414: The "escape time algorithm" mentioned below. The main cardioid is the period 1 continent. It is the region of parameters c {\displaystyle c} for which the map has an attracting fixed point . It consists of all parameters of the form for some μ {\displaystyle \mu } in the open unit disk . To the left of the main cardioid, attached to it at

10032-466: The Mandelbrot Set are distinguishable by both their attracting cycles and the geometric features of their structure. Each bulb is characterized by an antenna attached to it, emanating from a junction point and displaying a certain number of spokes indicative of its period. For instance, the 2 / 5 {\displaystyle 2/5} bulb is identified by its attracting cycle with

10164-406: The Mandelbrot Set. Each of these rays possesses an external angle that undergoes doubling under the angle doubling map θ ↦ {\displaystyle \theta \mapsto } 2 θ {\displaystyle 2\theta } . According to this theorem, when two rays land at the same point, no other rays between them can intersect. Thus, the 'size' of this region

10296-516: The Mandelbrot set are often referred to as "queer" or ghost components. For real quadratic polynomials, this question was proved in the 1990s independently by Lyubich and by Graczyk and Świątek. (Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the Feigenbaum diagram . So this result states that such windows exist near every parameter in the diagram.) Not every hyperbolic component can be reached by

10428-480: The Mandelbrot set at all finitely renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies. Since then, local connectivity has been proved at many other points of M {\displaystyle M} , but the full conjecture is still open. The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points . It

10560-407: The Mandelbrot set if and only if | z n | ≤ 2 {\displaystyle |z_{n}|\leq 2} for all n ≥ 0 {\displaystyle n\geq 0} . In other words, the absolute value of z n {\displaystyle z_{n}} must remain at or below 2 for c {\displaystyle c} to be in

10692-413: The Mandelbrot set, M {\displaystyle M} , and if that absolute value exceeds 2, the sequence will escape to infinity. Since c = z 1 {\displaystyle c=z_{1}} , it follows that | c | ≤ 2 {\displaystyle |c|\leq 2} , establishing that c {\displaystyle c} will always be in

10824-399: The above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it. Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, Q + {\displaystyle \mathbf {Q} ^{+}} represents the set of positive rational numbers. A function (or mapping ) from

10956-417: The addition of the ring and intersection as the multiplication of the ring. Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra , such as groups , fields and rings , are sets closed under one or more operations. Sequence#Bounded In mathematics , a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like

11088-417: The attracting cycle all touch at a common point (commonly called the α {\displaystyle \alpha } -fixed point ). If we label these components U 0 , … , U q − 1 {\displaystyle U_{0},\dots ,U_{q-1}} in counterclockwise orientation, then f c {\displaystyle f_{c}} maps

11220-400: The cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane , and indeed of any finite-dimensional Euclidean space . The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and

11352-470: The cardinality of a straight line. In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice . (ZFC is the most widely-studied version of axiomatic set theory.) The power set of a set S is the set of all subsets of S . The empty set and S itself are elements of the power set of S , because these are both subsets of S . For example,

11484-421: The closed disk of radius 2 around the origin. The intersection of M {\displaystyle M} with the real axis is the interval [ − 2 , 1 4 ] {\displaystyle \left[-2,{\frac {1}{4}}\right]} . The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family , The correspondence

11616-441: The complex modulus, i.e. | z | = z ∗ z {\displaystyle |z|={\sqrt {z^{*}z}}} . If ( a n ) {\displaystyle (a_{n})} is a sequence of points in a metric space , then the formula can be used to define convergence, if the expression | a n − L | {\displaystyle |a_{n}-L|}

11748-428: The complex plane as a curve in the real Cartesian plane of degree 2 n + 1 {\displaystyle 2^{n+1}} in x and y . Each curve n > 0 {\displaystyle n>0} is the mapping of an initial circle of radius 2 under p n {\displaystyle p_{n}} . These algebraic curves appear in images of the Mandelbrot set computed using

11880-424: The component U j {\displaystyle U_{j}} to the component U j + p ( mod ⁡ q ) {\displaystyle U_{j+p\,(\operatorname {mod} q)}} . The change of behavior occurring at c p q {\displaystyle c_{\frac {p}{q}}} is known as a bifurcation : the attracting fixed point "collides" with

12012-499: The corresponding polynomial forms a connected set . In the same way, the boundary of the Mandelbrot set can be defined as the bifurcation locus of this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial (when it is iterated repeatedly) changes drastically. The Mandelbrot set is a compact set , since it is closed and contained in the closed disk of radius 2 centred on zero . A point c {\displaystyle c} belongs to

12144-432: The definitions and notations introduced below. In this article, a sequence is formally defined as a function whose domain is an interval of integers . This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring

12276-510: The degree of Q n ( c ) {\displaystyle Q^{n}(c)} is 2 n − 1 {\displaystyle 2^{n-1}} . Therefore, constructing the centers of the hyperbolic components is possible by successively solving the equations Q n ( c ) = 0 , n = 1 , 2 , 3 , . . . {\displaystyle Q^{n}(c)=0,n=1,2,3,...} . The number of new centers produced in each step

12408-697: The domain of a sequence in the subscript is to indicate the range of values that the index can take by listing its highest and lowest legal values. For example, the notation ( k 2 ) ) k = 1 10 {\textstyle (k^{2}){\vphantom {)}}_{k=1}^{10}} denotes the ten-term sequence of squares ( 1 , 4 , 9 , … , 100 ) {\displaystyle (1,4,9,\ldots ,100)} . The limits ∞ {\displaystyle \infty } and − ∞ {\displaystyle -\infty } are allowed, but they do not represent valid values for

12540-433: The domain of a sequence to be the set of natural numbers . This narrower definition has the disadvantage that it rules out finite sequences and bi-infinite sequences, both of which are usually called sequences in standard mathematical practice. Another disadvantage is that, if one removes the first terms of a sequence, one needs reindexing the remainder terms for fitting this definition. In some contexts, to shorten exposition,

12672-477: The elements at each position. The notion of a sequence can be generalized to an indexed family , defined as a function from an arbitrary index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite , as in these examples, or infinite , such as

12804-422: The elements outside the union of A and B are the elements that are outside A and outside B ). The cardinality of A × B is the product of the cardinalities of A and B . This is an elementary fact when A and B are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true. The power set of any set becomes a Boolean ring with symmetric difference as

12936-673: The exact value of z = − 3 4 {\displaystyle z=-{\tfrac {3}{4}}} , the number of iterations required increases with a small ε {\displaystyle \varepsilon } . It turns out that multiplying the value of ε {\displaystyle \varepsilon } with the number of iterations required yields an approximation of π {\displaystyle \pi } that becomes better for smaller ε {\displaystyle \varepsilon } . For example, for ε {\displaystyle \varepsilon } = 0.0000001,

13068-562: The family of quadratic polynomials. In other words, the boundary of the Mandelbrot set is the set of all parameters c {\displaystyle c} for which the dynamics of the quadratic map z n = z n − 1 2 + c {\displaystyle z_{n}=z_{n-1}^{2}+c} exhibits sensitive dependence on c , {\displaystyle c,} i.e. changes abruptly under arbitrarily small changes of c . {\displaystyle c.} It can be constructed as

13200-456: The graph of exponentiation) are also not computable in the BSS model. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis , which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true. As

13332-474: The index, only the supremum or infimum of such values, respectively. For example, the sequence ( a n ) n = 1 ∞ {\textstyle {(a_{n})}_{n=1}^{\infty }} is the same as the sequence ( a n ) n ∈ N {\textstyle (a_{n})_{n\in \mathbb {N} }} , and does not contain an additional term "at infinity". The sequence (

13464-444: The inquiry: which is the largest bulb between the root points of the 0 / 1 {\displaystyle 0/1} and 1 / 2 {\displaystyle 1/2} -bulbs? It is clearly the 1 / 3 {\displaystyle 1/3} -bulb. And note that 1 / 3 {\displaystyle 1/3} is obtained from the previous two fractions by Farey addition , i.e., adding

13596-434: The integer sequence whose elements are the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...). Other examples of sequences include those made up of rational numbers , real numbers and complex numbers . The sequence (.9, .99, .999, .9999, ...), for instance, approaches the number 1. In fact, every real number can be written as

13728-437: The iteration repeatedly, the absolute value of z n {\displaystyle z_{n}} remains bounded for all n > 0 {\displaystyle n>0} . For example, for c = 1, the sequence is 0, 1, 2, 5, 26, ..., which tends to infinity , so 1 is not an element of the Mandelbrot set. On the other hand, for c = − 1 {\displaystyle c=-1} ,

13860-578: The limit set of a sequence of plane algebraic curves , the Mandelbrot curves , of the general type known as polynomial lemniscates . The Mandelbrot curves are defined by setting p 0 = z ,   p n + 1 = p n 2 + z {\displaystyle p_{0}=z,\ p_{n+1}=p_{n}^{2}+z} , and then interpreting the set of points | p n ( z ) | = 2 {\displaystyle |p_{n}(z)|=2} in

13992-404: The location within the boundary and the corresponding dynamical behavior for parameters drawn from associated bulbs emerges. The iteration of the quadratic polynomial f c ( z ) = z 2 + c {\displaystyle f_{c}(z)=z^{2}+c} , where c {\displaystyle c}  is a parameter drawn from one of the bulbs attached to

14124-459: The main body of the set. The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura . The fact that this is greater by a whole integer than its topological dimension, which is 1, reflects the extreme fractal nature of the Mandelbrot set boundary. Roughly speaking, Shishikura's result states that the Mandelbrot set boundary is so "wiggly" that it locally fills space as efficiently as

14256-773: The main cardioid called period-q bulbs (where ϕ {\displaystyle \phi } denotes the Euler phi function ), which consist of parameters c {\displaystyle c} for which f c {\displaystyle f_{c}} has an attracting cycle of period q {\displaystyle q} . More specifically, for each primitive q {\displaystyle q} th root of unity r = e 2 π i p q {\displaystyle r=e^{2\pi i{\frac {p}{q}}}} (where 0 < p q < 1 {\displaystyle 0<{\frac {p}{q}}<1} ), there

14388-495: The main cardioid within the Mandelbrot Set, gives rise to maps featuring attracting cycles of a specified period q {\displaystyle q}  and a rotation number p / q {\displaystyle p/q} . In this context, the attracting cycle of  exhibits rotational motion around a central fixed point, completing an average of p / q {\displaystyle p/q}  revolutions at each iteration. The bulbs within

14520-490: The maps f c {\displaystyle f_{c}} have an attracting periodic cycle are called hyperbolic components . It is conjectured that these are the only interior regions of M {\displaystyle M} and that they are dense in M {\displaystyle M} . This problem, known as density of hyperbolicity , is one of the most important open problems in complex dynamics . Hypothetical non-hyperbolic components of

14652-441: The number of iterations is 31415928 and the product is 3.1415928. In 2001, Aaron Klebanoff proved Boll's discovery. The Mandelbrot Set features a fundamental cardioid shape adorned with numerous bulbs directly attached to it. Understanding the arrangement of these bulbs requires a detailed examination of the Mandelbrot Set's boundary. As one zooms into specific portions with a geometric perspective, precise deducible information about

14784-402: The numerators and adding the denominators 0 1 {\displaystyle {\frac {0}{1}}} ⊕ {\displaystyle \oplus } 1 2 {\displaystyle {\frac {1}{2}}} = {\displaystyle =} 1 3 {\displaystyle {\frac {1}{3}}} Similarly, the largest bulb between

14916-606: The point c = − 3 / 4 {\displaystyle c=-3/4} , a circular bulb, the period-2 bulb is visible. The bulb consists of c {\displaystyle c} for which f c {\displaystyle f_{c}} has an attracting cycle of period 2 . It is the filled circle of radius 1/4 centered around −1. More generally, for every positive integer q > 2 {\displaystyle q>2} , there are ϕ ( q ) {\displaystyle \phi (q)} circular bulbs tangent to

15048-628: The positive integers (1, 2, 3, ...). The positions of some elements change when other elements are deleted. However, the relative positions are preserved. Formally, a subsequence of the sequence ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} is any sequence of the form ( a n k ) k ∈ N {\textstyle (a_{n_{k}})_{k\in \mathbb {N} }} , where ( n k ) k ∈ N {\displaystyle (n_{k})_{k\in \mathbb {N} }}

15180-482: The power α {\displaystyle \alpha } of the iterated variable z {\displaystyle z} tends to infinity) is convergent to the unit ( α {\displaystyle \alpha } -1)-sphere. In the Blum–Shub–Smale model of real computation , the Mandelbrot set is not computable, but its complement is computably enumerable . Many simple objects (e.g.,

15312-410: The power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} . The power set of a set S is commonly written as P ( S ) or 2 . If S has n elements, then P ( S ) has 2 elements. For example, {1, 2, 3} has three elements, and its power set has 2 = 8 elements, as shown above. If S is infinite (whether countable or uncountable ), then P ( S )

15444-537: The same elements are equal (they are the same set). This property is called extensionality . In particular, this implies that there is only one empty set. Sets are ubiquitous in modern mathematics. Indeed, set theory , more specifically Zermelo–Fraenkel set theory , has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Mathematical texts commonly denote sets by capital letters in italic , such as A , B , C . A set may also be called

15576-465: The sequence f c ( 0 ) {\displaystyle f_{c}(0)} , f c ( f c ( 0 ) ) {\displaystyle f_{c}(f_{c}(0))} , etc., remains bounded in absolute value . This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups . Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of

15708-468: The sequence is 0, −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set. The Mandelbrot set can also be defined as the connectedness locus of the family of quadratic polynomials f ( z ) = z 2 + c {\displaystyle f(z)=z^{2}+c} , the subset of the space of parameters c {\displaystyle c} for which the Julia set of

15840-562: The sequence of all even positive integers (2, 4, 6, ...). The position of an element in a sequence is its rank or index ; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In mathematical analysis , a sequence is often denoted by letters in the form of a n {\displaystyle a_{n}} , b n {\displaystyle b_{n}} and c n {\displaystyle c_{n}} , where

15972-426: The sequence of real numbers ( a n ) is such that all the terms are less than some real number M , then the sequence is said to be bounded from above . In other words, this means that there exists M such that for all n , a n ≤ M . Any such M is called an upper bound . Likewise, if, for some real m , a n ≥ m for all n greater than some N , then the sequence is bounded from below and any such m

16104-422: The set boundary being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point c {\displaystyle c} , whether the sequence f c ( 0 ) , f c ( f c ( 0 ) ) , … {\displaystyle f_{c}(0),f_{c}(f_{c}(0)),\dotsc } goes to infinity . Treating

16236-553: The set in honor of Mandelbrot for his influential work in fractal geometry . The mathematicians Heinz-Otto Peitgen and Peter Richter became well known for promoting the set with photographs, books (1986), and an internationally touring exhibit of the German Goethe-Institut (1985). The cover article of the August 1985 Scientific American introduced the algorithm for computing the Mandelbrot set. The cover

16368-620: The set of real numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of N {\displaystyle \mathbb {N} } are called countable sets ; these are either finite sets or countably infinite sets (sets of the same cardinality as N {\displaystyle \mathbb {N} } ); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of N {\displaystyle \mathbb {N} } are called uncountable sets . However, it can be shown that

16500-460: The set of indexing numbers is understood, the subscripts and superscripts are often left off. That is, one simply writes ( a k ) {\textstyle (a_{k})} for an arbitrary sequence. Often, the index k is understood to run from 1 to ∞. However, sequences are frequently indexed starting from zero, as in In some cases, the elements of the sequence are related naturally to

16632-400: The set while working at IBM 's Thomas J. Watson Research Center in Yorktown Heights, New York . Images of the Mandelbrot set exhibit an infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a fractal curve . The "style" of this recursive detail depends on the region of

16764-482: The subscript n refers to the n th element of the sequence; for example, the n th element of the Fibonacci sequence F {\displaystyle F} is generally denoted as F n {\displaystyle F_{n}} . In computing and computer science , finite sequences are usually called strings , words or lists , with the specific technical term chosen depending on

16896-489: The subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S . Suppose that a universal set U (a set containing all elements being discussed) has been fixed, and that A is a subset of U . Given any two sets A and B , Examples: The operations above satisfy many identities. For example, one of De Morgan's laws states that ( A ∪ B )′ = A ′ ∩ B ′ (that is,

17028-419: The top of its limb. The period of a given bulb is determined by counting these antennas. The numerator of the rotation number, p , is found by numbering each antenna counterclockwise from the limb from 1 to q − 1 {\displaystyle q-1} and finding which antenna is the shortest. In an attempt to demonstrate that the thickness of the p / q -limb is zero, David Boll carried out

17160-494: The type of object the sequence enumerates and the different ways to represent the sequence in computer memory . Infinite sequences are called streams . The empty sequence ( ) is included in most notions of sequence. It may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions , spaces , and other mathematical structures using

17292-546: The value of a sequence at the lowest input (often 1) is called the "first element" of the sequence, the value at the second smallest input (often 2) is called the "second element", etc. Also, while a function abstracted from its input is usually denoted by a single letter, e.g. f , a sequence abstracted from its input is usually written by a notation such as ( a n ) n ∈ A {\textstyle (a_{n})_{n\in A}} , or just as (

17424-481: Was created by Peitgen, Richter and Saupe at the University of Bremen . The Mandelbrot set became prominent in the mid-1980s as a computer-graphics demo , when personal computers became powerful enough to plot and display the set in high resolution. The work of Douady and Hubbard occurred during an increase in interest in complex dynamics and abstract mathematics , and the study of the Mandelbrot set has been

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