An arrow is a graphical symbol , such as ← or →, or a pictogram , used to point or indicate direction. In its simplest form, an arrow is a triangle , chevron , or concave kite , usually affixed to a line segment or rectangle , and in more complex forms a representation of an actual arrow (e.g. ➵ U+27B5). The direction indicated by an arrow is the one along the length of the line or rectangle toward the single pointed end.
36-698: Double Arrow or Double arrow may refer to: a subset of arrows in Unicode the British Rail Double Arrow logo, now officially known as the National Rail Double Arrow the Double Arrow Lodge , near Seeley Lake in the US state of Montana Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
72-565: A {\displaystyle ^{b}a} for tetration allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these tetration towers ). Finally, as an example, the fourth Ackermann number 4 ↑ 4 4 {\displaystyle 4\uparrow ^{4}4} could be represented as: Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an n -arrow operator ↑ n {\displaystyle \uparrow ^{n}}
108-531: A ↑ 0 b = a × b ) {\displaystyle (a\uparrow ^{0}b=a\times b)} as the base case and iterate from there. Then exponentiation becomes repeated multiplication. The formal definition would be for all integers a , b , n {\displaystyle a,b,n} with a ≥ 0 , n ≥ 0 , b ≥ 0 {\displaystyle a\geq 0,n\geq 0,b\geq 0} . Note, however, that Knuth did not define
144-440: A ↑ n b {\displaystyle a\uparrow ^{n}b} is a shorter alternative notation for n uparrows. Thus a ↑ 4 b = a ↑ ↑ ↑ ↑ b {\displaystyle a\uparrow ^{4}b=a\uparrow \uparrow \uparrow \uparrow b} . Attempting to write a ↑ ↑ b {\displaystyle a\uparrow \uparrow b} using
180-519: A b ) {\displaystyle (a\uparrow ^{1}b=a\uparrow b=a^{b})} as the base case, and tetration ( a ↑ 2 b = a ↑ ↑ b ) {\displaystyle (a\uparrow ^{2}b=a\uparrow \uparrow b)} as repeated exponentiation. This is equivalent to the hyperoperation sequence except it omits the three more basic operations of succession , addition and multiplication . One can alternatively choose multiplication (
216-511: A numerical value, and downward arrows indicate a decrease. In mathematical logic , a right-facing arrow indicates material conditional , and a left-right (bidirectional) arrow indicates if and only if , an upwards arrow indicates the NAND operator (negation of conjunction), an downwards arrow indicates the NOR operator (negation of disjunction). Use of arrow symbols in mathematical notation developed in
252-472: A visual technique to make a graffito stand out or give it a sense of movement. The graffiti theoretician RAMM:ΣLL:ZΣΣ described adornments, such as arrows, in wildstyle paintings as ornaments that ‘armed’ the letters of a piece. The Philadelphia graffiti artist Cool Earl began using arrows in 1967, although the New York graffiti writer SJK 171 may have been the first to do so. The graffiti artist Mare139
288-500: Is the manicule (pointing hand, 👈). Pedro Reinel in c. 1505 first used the fleur-de-lis as indicating north in a compass rose ; the convention of marking the eastern direction with a cross is older (medieval). Use of the arrow symbol does not appear to pre-date the 18th century. An early arrow symbol is found in an illustration of Bernard Forest de Bélidor 's treatise L'architecture hydraulique , printed in France in 1737. The arrow
324-498: Is a method of notation for very large integers , introduced by Donald Knuth in 1976. In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations . Goodstein also suggested the Greek names tetration , pentation , etc., for the extended operations beyond exponentiation . The sequence starts with a unary operation (the successor function with n = 0), and continues with
360-866: Is another. For example: The general definition of the up-arrow notation is as follows (for a ≥ 0 , n ≥ 1 , b ≥ 0 {\displaystyle a\geq 0,n\geq 1,b\geq 0} ): a ↑ n b = H n + 2 ( a , b ) = a [ n + 2 ] b . {\displaystyle a\uparrow ^{n}b=H_{n+2}(a,b)=a[n+2]b.} Here, ↑ n {\displaystyle \uparrow ^{n}} stands for n arrows, so for example 2 ↑ ↑ ↑ ↑ 3 = 2 ↑ 4 3. {\displaystyle 2\uparrow \uparrow \uparrow \uparrow 3=2\uparrow ^{4}3.} The square brackets are another notation for hyperoperations. The hyperoperations naturally extend
396-669: Is comparable to arbitrarily-long Conway chained arrow notation. These functions are all computable. Even faster computable functions, such as the Goodstein sequence and the TREE sequence require the usage of large ordinals, may occur in certain combinatorical and proof-theoretic contexts. There exist functions which grow uncomputably fast, such as the Busy Beaver , whose very nature will be completely out of reach from any up-arrow, or even any ordinal-based analysis. Without reference to hyperoperation
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#1732855943020432-618: Is comparable to tetrational growth and is upper-bounded by a function involving the first four hyperoperators;. Then, f ω ( x ) {\displaystyle f_{\omega }(x)} is comparable to the Ackermann function , f ω + 1 ( x ) {\displaystyle f_{\omega +1}(x)} is already beyond the reach of indexed arrows but can be used to approximate Graham's number , and f ω 2 ( x ) {\displaystyle f_{\omega ^{2}}(x)}
468-461: Is defined as iterated exponentiation, which Knuth denoted by a “double arrow”: For example, Expressions are evaluated from right to left, as the operators are defined to be right-associative . According to this definition, This already leads to some fairly large numbers, but the hyperoperator sequence does not stop here. Pentation , defined as iterated tetration, is represented by the “triple arrow”: Hexation , defined as iterated pentation,
504-440: Is here used to illustrate the direction of the flow of water and of the water wheel's rotation. At about the same time, arrow symbols were used to indicate the flow of rivers in maps. A trend toward abstraction, in which the arrow's fletching is removed, can be observed in the mid-to-late 19th century. The arrow can be seen in the work of Paul Klee . In a further refinement of the symbol, John Richard Green's A Short History of
540-669: Is known for creating 3D sculptures of arrows. In Unicode , the block Arrows occupies the hexadecimal range U+2190–U+21FF, as described below. Additional arrows can be found in the Combining Diacritical Marks , Combining Diacritical Marks Extended , Combining Diacritical Marks for Symbols , Halfwidth and Fullwidth Forms , Miscellaneous Mathematical Symbols-B , Miscellaneous Symbols and Pictographs , Miscellaneous Technical , Modifier Tone Letters and Spacing Modifier Letters Unicode blocks . Knuth%27s up-arrow notation In mathematics , Knuth's up-arrow notation
576-417: Is represented by the “quadruple arrow”: and so on. The general rule is that an n {\displaystyle n} -arrow operator expands into a right-associative series of ( n − 1 {\displaystyle n-1} )-arrow operators. Symbolically, Examples: In expressions such as a b {\displaystyle a^{b}} , the notation for exponentiation
612-766: Is understood to be a ↑ ( b ↑ c ) {\displaystyle a\uparrow (b\uparrow c)} , instead of ( a ↑ b ) ↑ c {\displaystyle (a\uparrow b)\uparrow c} . If ambiguity is not an issue parentheses are sometimes dropped. Computing 0 ↑ n b = H n + 2 ( 0 , b ) = 0 [ n + 2 ] b {\displaystyle 0\uparrow ^{n}b=H_{n+2}(0,b)=0[n+2]b} results in Computing 2 ↑ n b {\displaystyle 2\uparrow ^{n}b} can be restated in terms of an infinite table. We place
648-933: Is useful (and also for descriptions with a variable number of arrows), or equivalently, hyper operators . Some numbers are so large that even that notation is not sufficient. The Conway chained arrow notation can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful. 6 ↑ ↑ 4 {\displaystyle 6\uparrow \uparrow 4} = 6 6 . . . 6 ⏟ 4 {\displaystyle \underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}} , Since 6 ↑ ↑ 4 {\displaystyle 6\uparrow \uparrow 4} = 6 6 6 6 {\displaystyle 6^{6^{6^{6}}}} = 6 6 46 , 656 {\displaystyle 6^{6^{46,656}}} , Thus
684-434: Is usually to write the exponent b {\displaystyle b} as a superscript to the base number a {\displaystyle a} . But many environments — such as programming languages and plain-text e-mail — do not support superscript typesetting. People have adopted the linear notation a ↑ b {\displaystyle a\uparrow b} for such environments;
720-431: The arithmetic operations of addition and multiplication as follows. Addition by a natural number is defined as iterated incrementation: Multiplication by a natural number is defined as iterated addition : For example, Exponentiation for a natural power b {\displaystyle b} is defined as iterated multiplication, which Knuth denoted by a single up-arrow: For example, Tetration
756-415: The binary operations of addition ( n = 1), multiplication ( n = 2), exponentiation ( n = 3), tetration ( n = 4), pentation ( n = 5), etc. Various notations have been used to represent hyperoperations. One such notation is H n ( a , b ) {\displaystyle H_{n}(a,b)} . Knuth's up-arrow notation ↑ {\displaystyle \uparrow }
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#1732855943020792-593: The fast-growing hierarchy . The fast-growing hierarchy uses successive function iteration and diagonalization to systematically create faster-growing functions from some base function f ( x ) {\displaystyle f(x)} . For the standard fast-growing hierarchy using f 0 ( x ) = x + 1 {\displaystyle f_{0}(x)=x+1} , f 2 ( x ) {\displaystyle f_{2}(x)} already exhibits exponential growth, f 3 ( x ) {\displaystyle f_{3}(x)}
828-423: The "nil-arrow" ( ↑ 0 {\displaystyle \uparrow ^{0}} ). One could extend the notation to negative indices (n ≥ -2) in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing: The up-arrow operation is a right-associative operation , that is, a ↑ b ↑ c {\displaystyle a\uparrow b\uparrow c}
864-524: The English People of 1874 contained maps by cartographer Emil Reich, which indicated army movements by curved lines, with solid triangular arrowheads placed intermittently along the lines. Arrows are universally recognised for indicating directions. They are widely used on signage and for wayfinding , and are often used in road surface markings . A two-way road may be indicated by "↕" or "⇅". Upward arrows are often used to indicate an increase in
900-442: The familiar superscript notation gives a power tower . If b {\displaystyle b} is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower. Continuing with this notation, a ↑ ↑ ↑ b {\displaystyle a\uparrow \uparrow \uparrow b} could be written with a stack of such power towers, each describing
936-997: The first half of the 20th century. David Hilbert in 1922 introduced the arrow symbol representing logical implication . The double-headed arrow representing logical equivalence was introduced by Albrecht Becker in Die Aristotelische Theorie der Möglichkeitsschlüsse , Berlin, 1933. Knuth's up-arrow notation uses multiple up arrows, such as ⇈, for iterated, or repeated, exponentiation ( tetration ). The quantum theory of electron spin uses either upward or downward arrows. A vector may be denoted with an overhead arrow, such as in x → {\displaystyle {\vec {x}}} or A B ⟶ {\textstyle {\stackrel {\longrightarrow }{AB}}} . Arrows are regularly used in contemporary graffiti designs, incorporated as an element in both simplistic tags and complex wildstyle pieces . Arrows are used as
972-421: The left column with values 4. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken. We place the numbers 10 b {\displaystyle 10^{b}} in the top row, and fill the left column with values 10. To determine a number in the table, take the number immediately to
1008-432: The left, then look up the required number in the previous row, at the position given by the number just taken. For 2 ≤ b ≤ 9 the numerical order of the numbers 10 ↑ n b {\displaystyle 10\uparrow ^{n}b} is the lexicographical order with n as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for
1044-522: The number of power towers in the stack to its left: And more generally: This might be carried out indefinitely to represent a ↑ n b {\displaystyle a\uparrow ^{n}b} as iterated exponentiation of iterated exponentiation for any a {\displaystyle a} , n {\displaystyle n} , and b {\displaystyle b} (although it clearly becomes rather cumbersome). The Rudy Rucker notation b
1080-588: The numbers 2 b {\displaystyle 2^{b}} in the top row, and fill the left column with values 2. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken. The table is the same as that of the Ackermann function , except for a shift in n {\displaystyle n} and b {\displaystyle b} , and an addition of 3 to all values. We place
1116-420: The numbers 3 b {\displaystyle 3^{b}} in the top row, and fill the left column with values 3. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken. We place the numbers 4 b {\displaystyle 4^{b}} in the top row, and fill
Double Arrow - Misplaced Pages Continue
1152-982: The result comes out with 6 6 . . . 6 ⏟ 4 {\displaystyle \underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}} 10 ↑ ( 3 × 10 ↑ ( 3 × 10 ↑ 15 ) + 3 ) {\displaystyle 10\uparrow (3\times 10\uparrow (3\times 10\uparrow 15)+3)} = 100000...000 ⏟ 300000...003 ⏟ 300000...000 ⏟ 15 {\displaystyle \underbrace {100000...000} _{\underbrace {300000...003} _{\underbrace {300000...000} _{15}}}} or 10 3 × 10 3 × 10 15 + 3 {\displaystyle 10^{3\times 10^{3\times 10^{15}}+3}} (Petillion) Even faster-growing functions can be categorized using an ordinal analysis called
1188-427: The size of the one above it. Again, if b {\displaystyle b} is a variable or is too large, the stack might be written using dots and a note indicating its height. Furthermore, a ↑ ↑ ↑ ↑ b {\displaystyle a\uparrow \uparrow \uparrow \uparrow b} might be written using several columns of such stacks of power towers, each column describing
1224-543: The title Double Arrow . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Double_Arrow&oldid=989394946 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Arrows in Unicode An older (medieval) convention
1260-409: The up-arrow operators can be formally defined by for all integers a , b , n {\displaystyle a,b,n} with a ≥ 0 , n ≥ 1 , b ≥ 0 {\displaystyle a\geq 0,n\geq 1,b\geq 0} . This definition uses exponentiation ( a ↑ 1 b = a ↑ b =
1296-415: The up-arrow suggests 'raising to the power of'. If the character set does not contain an up arrow, the caret (^) is used instead. The superscript notation a b {\displaystyle a^{b}} doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation a ↑ b {\displaystyle a\uparrow b} instead.
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