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Regular Paperfolding Sequence
In mathematics the regular paperfolding sequence, also known as the dragon curve sequence, is an infinite sequence of 0s and 1s. It is obtained from the repeating partial sequence by filling in the question marks by another copy of the whole sequence. The first few terms of the resulting sequence are: If a strip of paper is folded repeatedly in half in the same direction, i times, it will get 2^i-1 folds, whose direction (left or right) is given by the pattern of 0's and 1's in the first 2^i-1 terms of the regular paperfolding sequence. Opening out each fold to create a right-angled corner (or, equivalently, making a sequence of left and right turns through a regular grid, following the pattern of the paperfolding sequence) produces a sequence of polygonal chains that approaches the dragon curve fractal: Properties The value of any given term t_n in the regular paperfolding sequence, starting with n=1, can be found recursively as follows. Divide n by two, as many times as possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dragon Curve
A dragon curve is any member of a family of self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The dragon curve is probably most commonly thought of as the shape that is generated from repeatedly folding a strip of paper in half, although there are other curves that are called dragon curves that are generated differently. Heighway dragon The Heighway dragon (also known as the Harter–Heighway dragon or the Jurassic Park dragon) was first investigated by NASA physicists John Heighway, Bruce Banks, and William Harter. It was described by Martin Gardner in his Scientific American column ''Mathematical Games'' in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth. It appeared on the section title pages of the Michael Crichton novel '' Jurassic Park''. Construction The Heighway dragon can be constructed from a base line segment by repeatedly replacing each segment by two segments with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygonal Chain
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. Name A polygonal chain may also be called a polygonal curve, polygonal path, polyline,. piecewise linear curve, broken line or, in geographic information systems, a linestring or linear ring. Variations A simple polygonal chain is one in which only consecutive (or the first and the last) segments intersect and only at their endpoints. A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment. A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 Order Dragon Curve
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3 Order Dragon Curve
3 (three) is a number, numeral and digit. It is the natural number following 2 and preceding 4, and is the smallest odd prime number and the only prime preceding a square number. It has religious or cultural significance in many societies. Evolution of the Arabic digit The use of three lines to denote the number 3 occurred in many writing systems, including some (like Roman and Chinese numerals) that are still in use. That was also the original representation of 3 in the Brahmic (Indian) numerical notation, its earliest forms aligned vertically. However, during the Gupta Empire the sign was modified by the addition of a curve on each line. The Nāgarī script rotated the lines clockwise, so they appeared horizontally, and ended each line with a short downward stroke on the right. In cursive script, the three strokes were eventually connected to form a glyph resembling a with an additional stroke at the bottom: ३. The Indian digits spread to the Caliphate in the 9th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4 Order Dragon Curve
4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures. In mathematics Four is the smallest composite number, its proper divisors being and . Four is the sum and product of two with itself: 2 + 2 = 4 = 2 x 2, the only number b such that a + a = b = a x a, which also makes four the smallest squared prime number p^. In Knuth's up-arrow notation, , and so forth, for any number of up arrows. By consequence, four is the only square one more than a prime number, specifically three. The sum of the first four prime numbers two + three + five + seven is the only sum of four consecutive prime numbers that yields an odd prime number, seventeen, which is the fourth super-prime. Four lies between the first proper pair of twin primes, three and five, which are the first two Fermat primes, like seventeen, which is the third. On the other han ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5 Order Dragon Curve
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has attained significance throughout history in part because typical humans have five digits on each hand. In mathematics 5 is the third smallest prime number, and the second super-prime. It is the first safe prime, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second Fermat prime and the third Mersenne prime exponent, as well as the third Catalan number, and the third Sophie Germain prime. Notably, 5 is equal to the sum of the ''only'' consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, ( 3, 5) and (5, 7). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
String Substitution
In computer science, in the area of formal language theory, frequent use is made of a variety of string functions; however, the notation used is different from that used for computer programming, and some commonly used functions in the theoretical realm are rarely used when programming. This article defines some of these basic terms. Strings and languages A string is a finite sequence of characters. The empty string is denoted by \varepsilon. The concatenation of two string s and t is denoted by s \cdot t, or shorter by s t. Concatenating with the empty string makes no difference: s \cdot \varepsilon = s = \varepsilon \cdot s. Concatenation of strings is associative: s \cdot (t \cdot u) = (s \cdot t) \cdot u. For example, (\langle b \rangle \cdot \langle l \rangle) \cdot (\varepsilon \cdot \langle ah \rangle) = \langle bl \rangle \cdot \langle ah \rangle = \langle blah \rangle. A language is a finite or infinite set of strings. Besides the usual set operations like union, inter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (excep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Providence, RI
Providence is the capital and most populous city of the U.S. state of Rhode Island. One of the oldest cities in New England, it was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony. He named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city developed as a busy port as it is situated at the mouth of the Providence River in Providence County, at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool, jewelry, and silverware industries. Today, the city of Providence is home to eight hospitals and eight institutions of higher learning which have shifted the city's economy into service industries, though it still retains some manufacturing activity. At the 2020 census, Providence had a populatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
