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Topic: Tetration

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	Home of Tetration (Site not responding. Last check: 2007-10-13)
		Tetration is an obscure mathematical operation that currently falls under the category of Pure Mathematics, because it has almost no application.
		Even so, many mathematicians have been interested in tetration, because of the historical significance of its relatives in the hyper-operation sequence, and because it is easy to make very large numbers with tetration.
		Now that we know what tetration is, it is easy to see why such an obscure mathematical operation would be studied for hundreds of years, even without any substantial progress.
	tetration.itgo.com (301 words)

	Tetration - Wikipedia, the free encyclopedia
		Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation.
		The portmanteau word tetration was coined by Reuben Louis Goodstein from tetra- (four) and iteration.
		Tetration is used for the notation of very large numbers.
	en.wikipedia.org /wiki/Tetration (1055 words)

	tetration.org
		Tetration is also referred to as iterated exponentials, hyperpowers and the dynamics of the exponential map.
		At least this used to be true; the last decade has seen a series of articles on continuous iteration, first the closed form solution for the continuous iteration of the logistics equation, and then a general proof for the continuous iteration of real functions.
		Because tetration is based on exponentiation and exponentiation is so fundamental to mathematical physics the question of tetration's relevance to physics with be explored.
	www.tetration.org /Tetration (475 words)

	large numbers
		Just as writing out a number "in full," or in place-value notation, becomes unwieldy with numbers as big as a googol, so exponentiation, in turn, endangers the world's forests if it tries to take on seriously large numbers.
		A more effective shorthand is tetration – "tetra" (from the Greek meaning "four") because it is the fourth dyadic operation in the series: addition, multiplication, exponentiation, tetration.
		Tetration goes by various other names including superpower, superdegree, and, the one used most commonly in mathematical circles and also here from now on, hyper4.
	www.daviddarling.info /encyclopedia/L/large_numbers.html (1109 words)

	Array Notation (Site not responding. Last check: 2007-10-13)
		Tetration is also known as power towers, it is also represented as a^^b = a {4} b (under old version) = a {2} b (under new version) = a^(a^(a^(a^.....^a)...)) a to the power of itself b times.
		Tetrational arrays consist of super dimensional arrays, trimensional, quadramensional, etc in the same way dimensional arrays consists of planar, realmic, flunic, etc arrays.
		{A / A / A / A (/2) A / A / A (/3,4,5,2) A } - this is a tetrational legion array.
	members.cox.net /hedrondude/array.htm (2539 words)

	A New Kind of Science: The NKS Forum - Hyper-operations
		The inverse operation of zeration (commutative) generates a new class of numbers (the Rubtsov’s “delta” numbers) that can be put in bi-jection with the set of the logarithms of negative numbers.
		Nevertheless, its study for negative arguments implies again the logarithms of negative numbers and, therefore, is connected with the analysis of the inverse of “zeration”.
		Tetration and higher-level hyper-operations give an idea of “immensity” and could be a tool for a new approach, for instance, to the theory of infinite ordinals
	forum.wolframscience.com /showthread.php?s=&threadid=579 (590 words)

	Hyper4
		The hyper4 has not been extended to real numbers as addition (1st level dyadic operation), multiplication (2nd level) and exponentiation (3rd level) have.
		Alternate names for hyper4 can include: tetration, superpower, superdegree, and powerlog.
		This concept can be extended to the fifth degree and beyond as well, however each definition becomes recursive upon the last one...
	www.ebroadcast.com.au /lookup/encyclopedia/hy/Hyper4.html (115 words)

	Big Numbers
		The word "tetration" comes from "tetra" (meaning "four") because it is the fourth operation in this series: addition, multiplication, exponentiation, tetration.
		Tetration of even quite small numbers produces extremely large numbers.
		Mathematicians and scientists almost never use tetration, and probably a large majority have never even heard of it.
	thinkzone.wlonk.com /MathFun/BigNum.htm (256 words)

	1e
		Originally was decided to try to receive an assotiation for a numerical evaluation of the superradical of positive whole tetrations.
		The given task can be decided, determining a tetration through a degree and to take advantage of one of methods of a numerical solution of the equation
		From expression (1.6) it is easy to receive a value of a tetration at negative tetrations.
	numbers.newmail.ru /english/01/1e.html (275 words)

	[Hyper-operations. Progress report. Tetration.] - A New Kind of Science: The NKS Forum
		The hyper-calculator allows to obtain the natural (base e) super-logarithm and the natural tower (tetration, base e) of a number, as well as the square tower and the square super-root of a number.
		Please also note that tetration is used in the proposed RRH© number notation hyper-format (s=4) and that any positive real number can be represented as p*(b#n), with this format.
		The extension of tetration to the reals is tightly connected with the analytic continuation of operators, as performed, for instance, within the “Fractional Calculus”.
	forum.wolframscience.com /archive/topic/956-1.html (1186 words)

	tetration.org - Home Page
		Tetration is defined as iterated exponentiation but while exponentiation is essential to a large body of mathematics, little is known about tetration due to its chaotic properties.
		Mathematicians have been researching tetration since at least the time of Euler but it is only at the end of the twentieth century that the combination of advances in dynamical systems and access to powerful computers is making real progress possible.
		Stephen Wolfram at Wolfram Research for conversations on the connection between tetration and dynamics, for his work in general, and for Mathematica.
	www.tetration.org (421 words)

	I Ching PLUS : MATHEMATICS AND THE I CHING
		Mathematics and the I Ching touches on such concepts as recursion, exponentiation, tetration, scalars and vectors, probabilities, randomness, binomial theorem (Pascal's Triangle), complexity/chaos (in the form of Sierpinski's gasket), and the universal category of number types, their qualitative nature, we use in Mathematics.
		Our brain seems to use tetration where it (1) recruits the current set of 8 qualities and then (2) uses them as sources of analogy to describe differences WITHIN the existing 8 qualities, giving us 8 octets, sets of 8 hexagrams that can be used to describe differences WITHIN one quality.
		The method of self-referencing, through exponentiation and tetration (the 'other' forms of self-referencing are slower in development - they are the use of addition and multiplication) is a method our brain seems to use.
	pages.prodigy.net /lofting/icmaths.html (2592 words)

	Googolplex - Free net encyclopedia
		In pure mathematics, the magnitude of a googolplex is not as large as some of the specially defined extraordinarily large numbers, such as those written with tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation.
		This last number can be expressed more concisely as ${\ ^{6}9}$ using tetration, or $9\uparrow\uparrow6$ using up-arrow notation.
		Some sequences grow very quickly; for instance, the first two Ackermann numbers are 1 and 4, but then the third is ${\ ^{7625597484987}3}$ , a power tower of threes more than seven trillion high.
	www.netipedia.com /index.php/Googolplex (706 words)

	Mailgate: sci.math.symbolic: Re: TETRATION
		In article , ed wrote: > If TETRATION is defined as recursive exponentiation, for example: > > 2 tetrated to the 2nd = 2 ^ 2 = 4 and i.e.
		Tet(2)=2^2 > 2 tetrated to the 3rd = 2 ^ (2 ^ 2) = 2 ^ 4 = 16 i.e.
		Yes, but the real problem is to pick out the _right_ definition among all possible ones.
	mailgate.supereva.it /sci/sci.math.symbolic/msg05546.html (332 words)

	addition, multiplication, exponentiation, then ??? (Site not responding. Last check: 2007-10-13)
		Tetration can be extended to real number values of 'x':
		The extension of tetration for real 'y' is a subject of active research:
		Tetration can also be extended to complex number values of 'x':
	www.computershack.info /addition-multiplication-exponentiation-then-.html (2053 words)

	How do you differentiate a tetration function? - Farlops Industries
		Many people have heard about addition, multiplication (which is interated addition) and exponentiation (which is iterated multiplication) but tetration is the obscure forth operation in this sequence--interated exponentiation.
		There are others that follow in this sequence of operations--repeated tetration, and so on--but let's just wrap our brains around this first.
		Anyway, again, I was searching the Web for information about tetration and I came across Aaronson's interesting collection of essays on computer science, mathematics, AI and other matters.
	www.farlops.com /2001/04/tetration.html (211 words)

	Re: Tetration operator in functional programming
		There is also a notion of "dependent tetration" which is even more mysterious.
		Just as Sigma(a:t).b corresponds to "dependent sum" (with types or numbers) and Pi(a:t).b corresponds to "dependent product", there is a Tetra(a:t).b corresponding to "dependent tetration".
		This operation is mysterious because exponentiation, the operator upon which tetration is defined, is the first operator in the Ackermann hierarchy which is not symmetric.
	www.mail-archive.com /haskell@haskell.org/msg07354.html (291 words)

	PreCalculus
		Growth rates may also be faster than exponential.
		Tetration is iterated exponentiation, the first hyper operator after exponentiation.
		The Ackermann function is a simple example of a recursive computer function that takes two natural numbers as arguments and yields a natural number.
	home.att.net /~srschmitt/precalc/precalc-13-02-00.html (727 words)

	What is the largest possible number you can make using only two digits?
		There is a function called tetration that is more powerful than powers.
		There can also be functions that follow this pattern so that the next one would be 9 tetrated to 9 nine times.
		In fact you wouldn't even need to use tetration to get infinity, you only need to use one digit.
	www.funtrivia.com /askft/Question57623.html (196 words)

	Really Big Numbers
		Consider tetration (verb to tetrate), from the Greek for four (tetra) as it is fourth in order: addition, multiplication, exponentiation, tetration.
		As far as I understand, "rban(n) = n to the nth tetration n times" means n^(n quintated with n).
		Anyway, as far as I understand it, ban and rban are "simple" compositions of tetration and quintation, which means that they are still primitive recursive.
	c2.com /cgi/wiki?ReallyBigNumbers (3801 words)

	perplexus.info :: Numbers : Make the most of these digits
		Unfortunately, in his example he uses the iteration of the asterisk, which I understand is why his solution does not fall within the solution guidelines.
		For pentation and hexation, I do not know if either can be represented without an iteration of a known symbol, yet, with tetration, like exponentation, superscription can be used.
		Also used by Nick as a symbol to be used, yet first offered in this problem's posts by Happy (cid=352) is the factorial, x!.
	perplexus.info /show.php?pid=99&cid=33334 (548 words)

	Tetration Of J - an Astronomy Net God & Science Forum Message
		Tetration Of J - an Astronomy Net God & Science Forum Message
		Following Wolfram, I tried to calculation the tetration of the imaginary J to the order N
		Forum posts are Copyright their authors as specified in the heading above the post.
	www.astronomy.net /forums/god/messages/31250.shtml (122 words)

	Transfinite Numbers
		"b tetrated to the a." The name tetration is used since tetra is the Latin root for four, and tetration occurs in fourth place in the logical progression: addition, multiplication, exponentiation, tetration.
		You don't ordinarily hear much about tetration because it is so powerful an operation that tetrating even very small numbers with each other produces inordinately large numbers.
		If we allow tetration as a standard operation and let PPN be the set of all pseudo-polynomials formed by using natural number coefficients and tetration), as well as exponentiation), then it is not difficult to see that PPN is
	dbanach.com /infin.htm (2871 words)

	Tetration of the Square Root of Two (Site not responding. Last check: 2007-10-13)
		that it only works for tetration of the square root of two.
		Extending this to tetration of other values A than sqrt(2) looks quite hard
		I would like to include your tetration function in my project.
	www.groupsrv.com /science/about9652.html (1001 words)

	Definitions And Explanations - an Astronomy Net God & Science Forum Message (Site not responding. Last check: 2007-10-13)
		Tetration is simply the next logical step in the progression of algebraic operations.
		If n happens to equal x (x^x), we then express this in the notation for tetration (difficult
		Tetration of J - secretary - December 29, 2003 - 16:39 UTC
	www.astronomy.net /forums/god/messages/31222.shtml (401 words)

	Reference.com/Encyclopedia/Tetration
		However, no further values can be derived by further iteration in this fashion, as
		Since complex numbers can be raised to powers, tetration can be applied to numbers of the form
		, tetration is achieved by using the principal branch of the natural logarithm, and noting the relation:
	www.reference.com /browse/wiki/Tetration (1172 words)

	Re: Tetration operator in functional programming
		It should reflect arithmetic laws of tetration, which for example relate (b ^^ (a1*a2)) with b^^a1 and b^^a2 or so, but are there any?
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	www.cse.unsw.edu.au /~dons/haskell-1990-2000/msg07546.html (476 words)

	Who can name the bigger number? \| MetaFilter
		The pi algorithm and your algorithm are both using the same tetration operator, so it comes down to the choice of k (although at a guess I would think that it has to be at least k > the largest mersenne prime squared).
		The pi algorithm doesn't really use tetration, although the output number is bounded above by 9 tetrated by k.
		Your first algorithm is 9^(largest mersenne prime tetrated by the largest mersenne prime).
	www.metafilter.com /mefi/50105 (7763 words)

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