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Archimedes' principle

2014-03-03T15:16:34Z

5.151.78.32: Undid revision 597966613 by 74.107.65.212 (talk)

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Atomic mass unit

2014-02-02T12:37:29Z

5.151.194.14: /* Terminology */

{{distinguish|Hundredth monkey effect}}
{{refimprove|date=January 2012}}
[[Image:monkey-typing.jpg|thumb|A [[chimpanzee]] (rather than the usual monkey) sitting at a typewriter. Given enough time, a hypothetical monkey (or in this case ape) typing at random would, as part of its output, [[almost surely]] produce all of [[William Shakespeare|Shakespeare's]] plays.]]

The '''infinite monkey theorem''' states that a monkey hitting keys at [[randomness|random]] on a [[typewriter keyboard]] for an infinite amount of time will [[almost surely]] type a given text, such as the complete works of [[William Shakespeare]].

In this context, "almost surely" is a mathematical term with a precise meaning, and the "monkey" is not an actual [[monkey]], but a [[metaphor]] for an abstract device that produces an endless [[random sequence]] of letters and symbols. One of the earliest instances of the use of the "monkey metaphor" is that of French mathematician [[Émile Borel]] in 1913,<ref name=":0" /> but the earliest instance may be even earlier. The relevance of the theorem is [[#Applications and criticisms|questionable]]—the [[probability]] of a universe full of monkeys typing a complete work such as Shakespeare's ''[[Hamlet]]'' is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the [[age of the universe]] is ''extremely'' low (but technically not zero).

Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to [[Aristotle]]'s ''[[On Generation and Corruption]]'' and [[Cicero]]'s ''[[De natura deorum]]'' (On the Nature of the Gods), through [[Blaise Pascal]] and [[Jonathan Swift]], and finally to modern statements with their iconic simians and typewriters. In the early 20th century, [[Émile Borel]] and [[Arthur Eddington]] used the theorem to illustrate the timescales implicit in the foundations of [[statistical mechanics]].

==Solution==

===Direct proof===
There is a straightforward proof of this theorem. As an introduction, recall that if two events are [[statistically independent]], then the probability of both happening equals the product of the probabilities of each one happening independently. For example, if the chance of rain in [[Moscow]] on a particular day in the future is 0.4 and the chance of an [[earthquake]] in [[San Francisco]] on that same day is 0.00003, then the chance of both happening on that day is {{nowrap|1=0.4 × 0.00003 = 0.000012}}, assuming that they are indeed independent.

Suppose the typewriter has 50 keys, and the word to be typed is ''banana''. If the keys are pressed randomly and independently, it means that each key has an equal chance of being pressed. Then, the chance that the first letter typed is 'b' is 1/50, and the chance that the second letter typed is ''a'' is also 1/50, and so on. Therefore, the chance of the first six letters spelling ''banana'' is
:(1/50) × (1/50) × (1/50) × (1/50) × (1/50) × (1/50) = (1/50)6 = 1/15 625 000 000 ,
less than one in 15 billion, but not zero, hence a possible outcome.

From the above, the chance of ''not'' typing ''banana'' in a given block of 6 letters is 1 − (1/50)6. Because each block is typed independently, the chance ''X''''n'' of not typing ''banana'' in any of the first ''n'' blocks of 6 letters is

:<math>X_n=\left(1-\frac{1}{50^6}\right)^n.</math>

As ''n'' grows, ''X''''n'' gets smaller. For an ''n'' of a million, ''X''''n'' is roughly 0.9999, but for an ''n'' of 10 billion ''X''''n'' is roughly 0.53 and for an ''n'' of 100 billion it is roughly 0.0017. As ''n'' approaches infinity, the probability ''X''''n'' [[limit of a function|approaches]] zero; that is, by making ''n'' large enough, ''X''''n'' can be made as small as is desired,<ref name="Isaac1995">{{cite book |last=Isaac |first=Richard E. |title=The Pleasures of Probability |year=1995 |publisher=Springer | isbn = 0-387-94415-X |pages=48–50}} Isaac generalizes this argument immediately to variable text and alphabet size; the common main conclusion is on p.50.</ref><ref group="note">This shows that the probability of typing "banana" in one of the predefined non-overlapping blocks of six letters tends to 1. In addition the word may appear across two blocks, so the estimate given is conservative.</ref> and the chance of typing ''banana'' approaches 100%.

The same argument shows why at least one of infinitely many monkeys will produce a text as quickly as it would be produced by a perfectly accurate human typist copying it from the original. In this case ''X''''n'' = (1 − (1/50)6)''n'' where ''X''''n'' represents the probability that none of the first ''n'' monkeys types ''banana'' correctly on their first try. When we consider 100 billion monkeys, the probability falls to 0.17%, and as the number of monkeys ''n'' increases, the value of ''X''''n'' – the probability of the monkeys failing to reproduce the given text – approaches zero arbitrarily closely. The limit, for ''n'' going to infinity, is zero.

However, for physically meaningful numbers of monkeys typing for physically meaningful lengths of time the results are reversed. If there are as many monkeys as there are atoms in the [[observable universe]] typing extremely fast for trillions of times the life of the universe, the probability of the monkeys replicating even a ''single page'' of Shakespeare is effectively zero. See [[#Probabilities|Probabilities]], below.

===Infinite strings===
The two statements above can be stated more generally and compactly in terms of [[string (computer science)|strings]], which are sequences of characters chosen from some finite alphabet:
* Given an infinite string where each character is chosen [[Uniform distribution (discrete)|uniformly at random]], any given finite string almost surely occurs as a substring at some position.
* Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.

Both follow easily from the second [[Borel–Cantelli lemma]]. For the second theorem, let ''E''''k'' be the [[event (probability theory)|event]] that the ''k''th string begins with the given text. Because this has some fixed nonzero probability ''p'' of occurring, the ''E''''k'' are independent, and the below sum diverges,
:<math>\sum_{k=1}^\infty P(E_k) = \sum_{k=1}^\infty p = \infty,</math>
the probability that infinitely many of the ''E''''k'' occur is 1. The first theorem is shown similarly; one can divide the random string into nonoverlapping blocks matching the size of the desired text, and make ''E''''k'' the event where the ''k''th block equals the desired string.<ref group="note">The first theorem is proven by a similar if more indirect route in {{cite book |last=Gut |first=Allan |title=Probability: A Graduate Course |year=2005 |publisher=Springer |isbn=0-387-22833-0 |pages=97–100}}</ref>

===Probabilities===
Ignoring punctuation, spacing, and capitalization, a monkey typing letters uniformly at random has a chance of one in 26 of correctly typing the first letter of ''[[Hamlet]].'' It has a chance of one in 676 (26 × 26) of typing the first two letters. Because the probability shrinks [[exponential growth|exponentially]], at 20 letters it already has only a chance of one in 2620 = 19,928,148,895,209,409,152,340,197,376 (almost 2 × 1028). In the case of the entire text of ''Hamlet'', the probabilities are so vanishingly small they can barely be conceived in human terms. The text of Hamlet contains approximately 130,000 letters.<ref group="note">Using the Hamlet text [http://www.gutenberg.org/dirs/etext99/1ws2611.txt from gutenberg], there are 132680 alphabetical letters and 199749 characters overall</ref> Thus there is a probability of one in 3.4 × 10183,946 to get the text right at the first trial. The average number of letters that needs to be typed until the text appears is also 3.4 × 10183,946,<ref group="note">For any required string of 130,000 letters from the set a-z, the average number of letters that needs to be typed until the string appears is (rounded) 3.4 × 10183,946, except in the case that all letters of the required string are equal, in which case the value is about 4% more, 3.6 × 10183,946. In that case failure to have the correct string starting from a particular position reduces with about 4% the probability of a correct string starting from the next position (i.e., for overlapping positions the events of having the correct string are not independent; in this case there is a positive correlation between the two successes, so the chance of success after a failure is smaller than the chance of success in general). The figure 3.4 × 10183,946 is derived from ''n'' = 26130000 by taking the logarithm of both sides: log10(''n'') = 1300000×log10(26) = 183946.5352, therefore ''n'' = 100.5352 × 10183946 = 3.429 × 10183946.</ref> or including punctuation, 4.4 × 10360,783.<ref group="note">26 letters ×2 for capitalisation, 12 for punctuation characters = 64, 199749×log10(64) = 4.4 × 10360,783 (assuming capital letters are separate keys, as opposed to a key combination, which would make the problem vastly harder).</ref>

Even if every atom in the observable universe were a monkey with a typewriter, typing from the [[Big Bang]] until the [[end of the universe]], they would still need a ridiculously longer time - more than three hundred and sixty thousand ''orders of magnitude'' longer - to have even a 1 in 10500 chance of success. To put it another way, for a one in a trillion chance of success, there would need to be 10360,641 universes full of atomic monkeys.<ref group="note">Assume all atoms are hydrogen, there are ~1080 [[Observable universe#Matter content - number of atoms|atoms in the universe]] (the actual number would be lower since only 74% of atoms are hydrogen). Assume the monkeys write for 1038 years (1020 years is when [[Future of an expanding universe#Stellar remnants escape galaxies or fall into black holes|all stellar remnants will have either been ejected from their galaxies or fallen into black holes]], 1038 years is when all but 0.1% of [[Future of an expanding universe#All nucleons decay|protons have decayed]]). Assuming the monkeys type non-stop at a blisteringly fast 400 words per minute, that's about 2000 characters per minute (Shakespeare's average word length is a bit under 5 letters). There are about half a million minutes in a year, this means each monkey types a billion characters per year. This gives a total of 1080×1038×109=10127 letters typed - which is still zero in comparison to 10360,783. For a one in a trillion chance, multiply the letters typed by a trillion: 10127×1015=10145. 10360,783/10145=10360,641.</ref> As [[Charles Kittel|Kittel]] and [[Herbert Kroemer|Kroemer]] put it, "The probability of ''Hamlet'' is therefore zero in any operational sense of an event...", and the statement that the monkeys must eventually succeed "gives a misleading conclusion about very, very large numbers." This is from their textbook on [[thermodynamics]], the field whose statistical foundations motivated the first known expositions of typing monkeys.<ref name="KK">{{cite book | author=[[Charles Kittel|Kittel, Charles]] and [[Herbert Kroemer]] | title=Thermal Physics (2nd ed.) | publisher=W. H. Freeman Company | year=1980 | isbn=0-7167-1088-9 |page=53}}</ref>

In fact there is less than a one in a trillion chance of success that such a universe full of monkeys could type any particular document a mere 79 characters long.<ref group="note">As explained at http://www.nutters.org/docs/more-monkeys, the problem can be approximated further. 10145/log10(64)=78.9 characters.</ref>

===Almost surely===
{{See also|Almost surely}}

The probability that an infinite randomly generated string of text will contain a particular finite substring is 1. However, this does not mean the substring's absence is "impossible", despite the absence having a prior probability of 0. For example, the immortal monkey ''could'' randomly type G as its first letter, G as its second, and G as every single letter thereafter, producing an infinite string of Gs; at no point must the monkey be "compelled" to type anything else. (To assume otherwise implies the [[gambler's fallacy]].) However long a randomly generated finite string is, there is a small but nonzero chance that it will turn out to consist of the same character repeated throughout; this chance approaches zero as the string's length approaches infinity. There is nothing special about such a monotonous sequence except that it is easy to describe; the same fact applies to any nameable specific sequence, such as "RGRGRG" repeated forever, or "a-b-aa-bb-aaa-bbb-...", or "Three, Six, Nine, Twelve…".

If the hypothetical monkey has a typewriter with 90 equally likely keys that include numerals and punctuation, then the first typed keys might be "3.14" (the first three [[digits of pi]]) with a probability of (1/90)4, which is 1/65,610,000. Equally probable is any other string of four characters allowed by the typewriter, such as "GGGG", "mATh", or "q%8e". The probability that 100 randomly typed keys will consist of the first 99 digits of pi (including the separator key), or any other ''particular'' sequence of that length, is much lower: (1/90)100. If the monkey's allotted length of text is infinite, the chance of typing only the digits of pi is 0, which is just as ''possible'' as typing nothing but Gs (also probability 0).

The same applies to the event of typing a particular version of ''Hamlet'' followed by endless copies of itself; or ''Hamlet'' immediately followed by all the digits of pi; these specific strings are [[infinite set|equally infinite]] in length, they are not prohibited by the terms of the thought problem, and they each have a prior probability of 0. In fact, ''any'' particular infinite sequence the immortal monkey types will have ''had'' a prior probability of 0, even though the monkey must type something.

This is an extension of the principle that a finite string of random text has a lower and lower probability of ''being'' a particular string the longer it is (though all specific strings are equally unlikely). This probability approaches 0 as the string approaches infinity. Thus, the probability of the monkey typing an endlessly long string, such as all of the digits of pi in order, on a 90-key keyboard is (1/90)∞ which equals (1/∞) which is essentially 0. At the same time, the probability that the sequence ''contains'' a particular subsequence (such as the word MONKEY, or the 12th through 999th digits of pi, or a version of the King James Bible) increases as the total string increases. This probability approaches 1 as the total string approaches infinity, and thus the original theorem is correct.

=== Correspondence between strings and numbers ===
In a simplification of the thought experiment, the monkey could have a typewriter with just two keys: 1 and 0. The infinitely long string thusly produced would correspond to the [[Binary numeral system|binary]] digits of a particular [[real number]] between 0 and 1. A countably infinite set of possible strings end in infinite repetitions, which means the corresponding real number is [[rational number|rational]]. Examples include the strings corresponding to one-third (010101…), five-sixths (11010101…) and five-eighths (1100000…). Only a subset of such strings (albeit a countably infinite subset) contains the entirety of ''Hamlet'' (if the text is translated from [[ASCII]] to binary).

Meanwhile, there is an ''[[uncountably]]'' infinite set of strings which do not end in such repetition; these correspond to the [[irrational numbers]]. These can be sorted into two uncountably infinite subsets: those which contain ''Hamlet'' and those which do not. However, the "largest" subset of all the real numbers are those which not only contain ''Hamlet'', but which contain every other possible string of any length, and with equal distribution of such strings. These irrational numbers are called [[normal number|normal]]. Because almost all numbers are normal, almost all possible strings contain all possible finite substrings. Hence, the probability of the monkey typing a normal number is 1. The same principles apply regardless of the number of keys from which the monkey can choose; a 90-key keyboard can be seen as a generator of numbers written in base 90.

==History==

===Statistical mechanics===
In one of the forms in which probabilists now know this theorem, with its "dactylographic" [i.e., typewriting] monkeys ({{lang-fr|singes dactylographes}}; the French word ''singe'' covers both the monkeys and the apes), appeared in [[Émile Borel]]'s 1913 article "''Mécanique Statistique et Irréversibilité''" (''[[Statistical mechanics]] and irreversibility''),<ref name=":0">{{cite journal | author=Émile Borel | title=Mécanique Statistique et Irréversibilité | journal=J. Phys. 5e série | volume=3 | year=1913 | pages=189–196}}</ref> and in his book "Le Hasard" in 1914. His "monkeys" are not actual monkeys; rather, they are a metaphor for an imaginary way to produce a large, [[random sequence]] of letters. Borel said that if a million monkeys typed ten hours a day, it was extremely unlikely that their output would exactly equal all the books of the richest libraries of the world; and yet, in comparison, it was even more unlikely that the laws of statistical mechanics would ever be violated, even briefly.

The physicist [[Arthur Eddington]] drew on Borel's image further in ''The Nature of the Physical World'' (1928), writing:

{{quote|If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly more favourable than the chance of the molecules returning to one half of the vessel.<ref name="Arthur1928">{{cite book | author=Arthur Eddington | title=The Nature of the Physical World: The [[Gifford Lectures]] | publisher=Macmillan | location=New York | year=1928 | page=72 | isbn=0-8414-3885-4}}</ref><ref name='Arthur1927'>{{cite web | url = http://www.giffordlectures.org/Browse.asp?PubID=TPNOPW&Volume=0&Issue=0&ArticleID=6 | title = Chapter IV: The Running-Down of the Universe | accessdate = 2012-01-22 | last = Eddington | first = Arthur | work = The Nature of the Physical World 1926–1927: The [[Gifford Lectures]]}}</ref>}}

These images invite the reader to consider the incredible improbability of a large but finite number of monkeys working for a large but finite amount of time producing a significant work, and compare this with the even greater improbability of certain physical events. Any physical process that is even less likely than such monkeys' success is effectively impossible, and it may safely be said that such a process will never happen.<ref name="KK" />

===Origins and "The Total Library"===
In a 1939 essay entitled "The Total Library", Argentine writer [[Jorge Luis Borges]] traced the infinite-monkey concept back to [[Aristotle]]'s ''Metaphysics.'' Explaining the views of [[Leucippus]], who held that the world arose through the random combination of [[atom]]s, Aristotle notes that the atoms themselves are homogeneous and their possible arrangements only differ in shape, position and ordering. In ''[[On Generation and Corruption]]'', the Greek philosopher compares this to the way that a tragedy and a comedy consist of the same "atoms", ''i.e.'', alphabetic characters.<ref>Aristotle, ''Περὶ γενέσεως καὶ φθορᾶς'' (''On Generation and Corruption''), 315b14.</ref> Three centuries later, [[Cicero]]'s ''De natura deorum'' (''On the Nature of the Gods'') argued against the atomist worldview:

{{quote|He who believes this may as well believe that if a great quantity of the one-and-twenty letters, composed either of gold or any other matter, were thrown upon the ground, they would fall into such order as legibly to form the ''Annals'' of Ennius. I doubt whether fortune could make a single verse of them.<ref>Marcus Tullius Cicero, ''De natura deorum'', 2.37. Translation from ''Cicero's Tusculan Disputations; Also, Treatises On The Nature Of The Gods, And On The Commonwealth'', C. D. Yonge, principal translator, New York, Harper & Brothers Publishers, Franklin Square. (1877). [http://www.gutenberg.org/etext/14988 Downloadable text].</ref>}}

Borges follows the history of this argument through [[Blaise Pascal]] and [[Jonathan Swift]],<ref>The English translation of "The Total Library" lists the title of Swift's essay as "Trivial Essay on the Faculties of the Soul." The appropriate reference is, instead: Swift, Jonathan, Temple Scott et al. "A Tritical Essay upon the Faculties of the Mind." The Prose Works of Jonathan Swift, Volume 1. London: G. Bell, 1897, pp. 291-296. [http://books.google.com/books?id=FctEAAAAYAAJ&printsec=frontcover&dq=The+Prose+Works+of+Jonathan+Swift&hl=en&ei=JdyDTb-yM8u3tweNmcy8BA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC4Q6AEwAA#v=onepage&q&f=false Google Books]</ref> then observes that in his own time, the vocabulary had changed. By 1939, the idiom was "that a half-dozen monkeys provided with typewriters would, in a few eternities, produce all the books in the British Museum." (To which Borges adds, "Strictly speaking, one immortal monkey would suffice.") Borges then imagines the contents of the Total Library which this enterprise would produce if carried to its fullest extreme:

{{quote|Everything would be in its blind volumes. Everything: the detailed history of the future, [[Aeschylus]]' ''The Egyptians'', the exact number of times that the waters of the Ganges have reflected the flight of a falcon, the secret and true nature of Rome, the encyclopedia Novalis would have constructed, my dreams and half-dreams at dawn on August 14, 1934, the proof of [[Pierre Fermat]]'s [[Fermat's last theorem|theorem]], the unwritten chapters of ''[[Edwin Drood]]'', those same chapters translated into the language spoken by the [[Garamantes]], the paradoxes Berkeley invented concerning Time but didn't publish, Urizen's books of iron, the premature epiphanies of [[Stephen Dedalus]], which would be meaningless before a cycle of a thousand years, the Gnostic [[Gospel of Basilides]], the song the sirens sang, the complete catalog of the Library, the proof of the inaccuracy of that catalog. Everything: but for every sensible line or accurate fact there would be millions of meaningless cacophonies, verbal farragoes, and babblings. Everything: but all the generations of mankind could pass before the dizzying shelves—shelves that obliterate the day and on which chaos lies—ever reward them with a tolerable page.<ref>[[Jorge Luis Borges|Borges, Jorge Luis]]. "''La biblioteca total''" (The Total Library), ''Sur'' No. 59, August 1939. Trans. by [[Eliot Weinberger]]. In ''Selected Non-Fictions'' (Penguin: 1999), ISBN 0-670-84947-2.</ref>}}

Borges' total library concept was the main theme of his widely read 1941 short story "[[The Library of Babel]]", which describes an unimaginably vast library consisting of interlocking hexagonal chambers, together containing every possible volume that could be composed from the letters of the alphabet and some punctuation characters.

==Real monkeys==
In 2003, lecturers and students from the [[University of Plymouth]] MediaLab Arts course used a £2,000 grant from the [[Arts Council]] to study the literary output of real monkeys. They left a computer keyboard in the enclosure of six [[Celebes Crested Macaque]]s in [[Paignton Zoo]] in [[Devon]] in [[England]] for a month, with a radio link to broadcast the results on a website.<ref name="BBC News">{{cite news | title= No words to describe monkeys' play | date= 2003-05-09 | publisher= BBC News | url= http://news.bbc.co.uk/2/hi/3013959.stm | accessdate = 2009-07-25}} {{Dead link|date=March 2013}}</ref>

Not only did the monkeys produce nothing but five pages<ref>{{cite web | title = Notes Towards the Complete Works of Shakespeare | url= http://www.vivaria.net/experiments/notes/publication/NOTES_EN.pdf | format = PDF | publisher = vivaria.net | year= 2002 | accessdate = 2006-06-13}}{{dead link|date=September 2013}}</ref> consisting largely of the letter [[S]], the lead male began by bashing the keyboard with a stone, and the monkeys continued by urinating and defecating on it. Phillips said that the artist-funded project was primarily performance art, and they had learned "an awful lot" from it. He concluded that monkeys "are not random generators. They're more complex than that. ... They were quite interested in the screen, and they saw that when they typed a letter, something happened. There was a level of intention there."<ref name="BBC News" /><ref name="Associated2003">{{cite news |url=http://www.wired.com/news/culture/0,1284,58790,00.html |title=Monkeys Don't Write Shakespeare |agency=Associated Press |publisher=Wired News |date= 2003-05-09| accessdate = 2007-03-02}}</ref>

==Applications and criticisms==

===Evolution===
[[Image:Thomas Henry Huxley - Project Gutenberg eText 16935.jpg|thumb|upright|[[Thomas Huxley]] is sometimes misattributed with proposing a variant of the theory in his debates with [[Samuel Wilberforce]].]]
In his 1931 book ''The Mysterious Universe'', Eddington's rival [[James Hopwood Jeans|James Jeans]] attributed the monkey parable to a "Huxley", presumably meaning [[Thomas Henry Huxley]]. This attribution is incorrect.<ref name="Padmanabhan2005">{{cite journal |first=Thanu |last=Padmanabhan |title=The dark side of astronomy |journal=Nature |volume=435 |pages=20–21 |year=2005 |doi=10.1038/435020a |issue=7038}} {{cite book |author=Platt, Suzy; Library of Congress Congressional Research Service |title=Respectfully quoted: a dictionary of quotations |year=1993 |publisher=Barnes & Noble |isbn=0-88029-768-9|pages=388–389}}</ref> Today, it is sometimes further reported that Huxley applied the example in a [[1860 Oxford evolution debate|now-legendary debate]] over [[Charles Darwin]]'s ''[[On the Origin of Species]]'' with the Anglican Bishop of Oxford, [[Samuel Wilberforce]], held at a meeting of the [[British Association for the Advancement of Science]] at Oxford on June 30, 1860. This story suffers not only from a lack of evidence, but the fact that in 1860 the typewriter itself had yet to emerge.<ref name="Rescher2006">{{cite book |first=Nicholas |last=Rescher |title=Studies in the Philosophy of Science |year=2006 |publisher=ontos verlag |isbn=3-938793-20-1 |page=103}}</ref>

Despite the original mix-up, monkey-and-typewriter arguments are now common in arguments over evolution. For example, Doug Powell argues as a [[Christian apologist]] that even if a monkey accidentally types the letters of ''Hamlet'', it has failed to produce ''Hamlet'' because it lacked the intention to communicate. His parallel implication is that natural laws could not produce the information content in [[DNA]].<ref name="Powell2006">{{cite book |last=Powell |first=Doug |title=Holman Quicksource Guide to Christian Apologetics |year=2006 |publisher=Broadman & Holman | isbn = 0-8054-9460-X |pages=60, 63}}</ref> A more common argument is represented by Reverend [[John F. MacArthur]], who claims that the genetic mutations necessary to produce a tapeworm from an amoeba are as unlikely as a monkey typing Hamlet's soliloquy, and hence the odds against the evolution of all life are impossible to overcome.<ref name="MacArthur2003">{{cite book |first=John |last=MacArthur |title=Think Biblically!: Recovering a Christian Worldview |year=2003 |publisher=Crossway Books |isbn=1-58134-412-0 |pages=78–79}}</ref>

[[Evolutionary biology|Evolutionary biologist]] [[Richard Dawkins]] employs the typing monkey concept in his book ''[[The Blind Watchmaker]]'' to demonstrate the ability of [[natural selection]] to produce biological [[complexity]] out of random [[mutation]]s. In a simulation experiment Dawkins has his [[weasel program]] produce the Hamlet phrase ''METHINKS IT IS LIKE A WEASEL'', starting from a randomly typed parent, by "breeding" subsequent generations and always choosing the closest match from progeny that are copies of the parent, with random mutations. The chance of the target phrase appearing in a single step is extremely small, yet Dawkins showed that it could be produced rapidly (in about 40 generations) using cumulative selection of phrases. The random choices furnish raw material, while cumulative selection imparts information. As Dawkins acknowledges, however, the weasel program is an imperfect analogy for evolution, as "offspring" phrases were selected "according to the criterion of resemblance to a ''distant ideal'' target." In contrast, Dawkins affirms, evolution has no long-term plans and does not progress toward some distant goal (such as humans). The weasel program is instead meant to illustrate the difference between [[non-random]] cumulative selection, and [[random]] single-step selection.<ref name="Dawkins1996">{{cite book |last=Dawkins |first=Richard |year=1996 |title=The Blind Watchmaker |publisher=W.W. Norton & Co. |isbn=0-393-31570-3 |pages=46–50}}</ref> In terms of the typing monkey analogy, this means that ''Romeo and Juliet'' could be produced relatively quickly if placed under the constraints of a nonrandom, Darwinian-type selection, by freezing in place any letters that happened to match the target text, and making that the template for the next generation of typing monkeys.

A different avenue for exploring the analogy between evolution and an unconstrained monkey lies in the problem that the monkey types only one letter at a time, independently of the other letters. Hugh Petrie argues that a more sophisticated setup is required, in his case not for biological evolution but the evolution of ideas:

{{quote|In order to get the proper analogy, we would have to equip the monkey with a more complex typewriter. It would have to include whole Elizabethan sentences and thoughts. It would have to include Elizabethan beliefs about human action patterns and the causes, Elizabethan morality and science, and linguistic patterns for expressing these. It would probably even have to include an account of the sorts of experiences which shaped Shakespeare's belief structure as a particular example of an Elizabethan. Then, perhaps, we might allow the monkey to play with such a typewriter and produce variants, but the impossibility of obtaining a Shakespearean play is no longer obvious. What is varied really does encapsulate a great deal of already-achieved knowledge.<ref name="Blachowicz1998">As quoted in {{cite book |first=James |last=Blachowicz |title=Of Two Minds: Nature of Inquiry |year=1998 |publisher=SUNY Press |isbn=0-7914-3641-1 |page=109}}</ref>}}

[[James W. Valentine]], while admitting that the classic monkey's task is impossible, finds that there is a worthwhile analogy between written English and the [[metazoa]]n genome in this other sense: both have "combinatorial, hierarchical structures" that greatly constrain the immense number of combinations at the alphabet level.<ref name="Valentine2004">{{cite book |first=James |last=Valentine |title=On the Origin of Phyla |year=2004 |publisher=University of Chicago Press |isbn=0-226-84548-6 |pages=77–80}}</ref>

===Literary theory===
[[R. G. Collingwood]] argued in 1938 that art cannot be produced by accident, and wrote as a sarcastic aside to his critics,

{{quote|...some ... have denied this proposition, pointing out that if a monkey played with a typewriter ... he would produce ... the complete text of Shakespeare. Any reader who has nothing to do can amuse himself by calculating how long it would take for the probability to be worth betting on. But the interest of the suggestion lies in the revelation of the mental state of a person who can identify the 'works' of Shakespeare with the series of letters printed on the pages of a book...<ref name="Sclafani1975">p.126 of ''The Principles of Art'', as summarized and quoted by {{cite journal |first=Richard J. |last=Sclafani |title=The logical primitiveness of the concept of a work of art |journal=British Journal of Aesthetics |year=1975 |volume=15 |issue=1 |doi=10.1093/bjaesthetics/15.1.14 |page=14}}</ref>}}

[[Nelson Goodman]] took the contrary position, illustrating his point along with Catherine Elgin by the example of Borges' "[[Pierre Menard, Author of the Quixote]]",

{{quote|What Menard wrote is simply another inscription of the text. Any of us can do the same, as can printing presses and photocopiers. Indeed, we are told, if infinitely many monkeys ... one would eventually produce a replica of the text. That replica, we maintain, would be as much an instance of the work, ''Don Quixote'', as Cervantes' manuscript, Menard's manuscript, and each copy of the book that ever has been or will be printed.<ref name="John2004">{{cite book |author=John, Eileen and Dominic Lopes, editors |title=The Philosophy of Literature: Contemporary and Classic Readings: An Anthology |year=2004 |publisher=Blackwell |isbn=1-4051-1208-5 |page=96}}</ref>}}

In another writing, Goodman elaborates, "That the monkey may be supposed to have produced his copy randomly makes no difference. It is the same text, and it is open to all the same interpretations...." [[Gérard Genette]] dismisses Goodman's argument as [[begging the question]].<ref name="Genette1997">{{cite book |first=Gérard |last= Genette |title=The Work of Art: Immanence and Transcendence |year=1997 |publisher=Cornell UP |isbn=0-8014-8272-0}}</ref>

For [[Jorge J. E. Gracia]], the question of the identity of texts leads to a different question, that of author. If a monkey is capable of typing ''Hamlet'', despite having no intention of meaning and therefore disqualifying itself as an author, then it appears that texts do not require authors. Possible solutions include saying that whoever finds the text and identifies it as ''Hamlet'' is the author; or that Shakespeare is the author, the monkey his agent, and the finder merely a user of the text. These solutions have their own difficulties, in that the text appears to have a meaning separate from the other agents: what if the monkey operates before Shakespeare is born, or if Shakespeare is never born, or if no one ever finds the monkey's typescript?<ref name="Gracia1996">{{cite book |last=Gracia |first=Jorge |title=Texts: Ontological Status, Identity, Author, Audience |year=1996 |publisher=SUNY Press |isbn=0-7914-2901-6 |pages=1–2, 122–125}}</ref>

===Random document generation===
The theorem concerns a [[thought experiment]] which cannot be fully carried out in practice, since it is predicted to require prohibitive amounts of time and resources. Nonetheless, it has inspired efforts in finite random text generation.

One computer program run by Dan Oliver of Scottsdale, Arizona, according to an article in ''[[The New Yorker]]'', came up with a result on August 4, 2004: After the group had worked for 42,162,500,000 billion billion monkey-years, one of the "monkeys" typed, "<tt>VALENTINE. Cease toIdor:eFLP0FRjWK78aXzVOwm)-‘;8.t</tt>" The first 19 letters of this sequence can be found in "The Two Gentlemen of Verona". Other teams have reproduced 18 characters from "Timon of Athens", 17 from "Troilus and Cressida", and 16 from "Richard II".<ref name="ja">[http://www.newyorker.com/arts/critics/books/2007/04/09/070409crbo_books_acocella?currentPage=all Newyorker.com] Acocella, Joan "The Typing Life: How writers used to write" ''[[The New Yorker]]'' April 9, 2007, a review of ''The Iron Whim: A Fragmented History of Typewriting'' (Cornell) 2007, by Darren Wershler-Henry</ref>

A website entitled ''The Monkey Shakespeare Simulator'', launched on July 1, 2003, contained a [[Java applet]] that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. For example, it produced this partial line from ''[[Henry IV, Part 2]]'', reporting that it took "2,737,850 million billion billion billion monkey-years" to reach 24 matching characters:
:<tt>RUMOUR. Open your ears; 9r"5j5&?OWTY Z0d</tt>...''

Due to processing power limitations, the program uses a probabilistic model (by using a [[random number generator]] or RNG) instead of actually generating random text and comparing it to Shakespeare. When the simulator "detects a match" (that is, the RNG generates a certain value or a value within a certain range), the simulator simulates the match by generating matched text.

More sophisticated methods are used in practice for [[natural language generation]]. If instead of simply generating random characters one restricts the generator to a meaningful vocabulary and conservatively following grammar rules, like using a [[context-free grammar]], then a random document generated this way can even fool some humans (at least on a cursory reading) as shown in the experiments with [[SCIgen]], [[snarXiv]], and the [[Postmodernism Generator]].

===Testing of random number generators===
{{main|Diehard tests}}
Questions about the statistics describing how often an ideal monkey is [[expected value|expected]] to type certain strings translate into [[Randomness tests|practical tests for random number generators]]; these range from the simple to the "quite sophisticated". Computer science professors [[George Marsaglia]] and [[Arif Zaman]] report that they used to call one such category of tests "overlapping m-[[tuple]] tests" in lecture, since they concern overlapping m-tuples of successive elements in a random sequence. But they found that calling them "monkey tests" helped to motivate the idea with students. They published a report on the class of tests and their results for various RNGs in 1993.<ref name="Marsaglia1993">{{Cite journal | title = Monkey tests for random number generators | journal = Computers & mathematics with applications | issn = 0898-1221 | year = 1993 | volume = 26 | pages = 1–10 | publisher = Elsevier, Oxford | author = Marsaglia G. and Zaman A. | doi = 10.1016/0898-1221(93)90001-C | postscript = [http://stat.fsu.edu/pub/diehard/cdrom/pscript/monkey.ps PostScript version] | issue = 9}}</ref>

==Popular culture==
{{Main|Infinite monkey theorem in popular culture}}
The infinite monkey theorem and its associated imagery is considered a popular and proverbial illustration of the mathematics of probability, widely known to the general public because of its transmission through popular culture rather than through formal education.<ref group="note">Examples of the theorem being referred to as proverbial include: Why Creativity Is Not like the Proverbial Typing Monkey. Jonathan W. Schooler, Sonya Dougal, ''Psychological Inquiry'', Vol. 10, No. 4 (1999); and ''The Case of the Midwife Toad'' ([[Arthur Koestler]], New York, 1972, page 30): ''"Neo-Darwinism does indeed carry the nineteenth-century brand of materialism to its extreme limits—to the proverbial monkey at the typewriter, hitting by pure chance on the proper keys to produce a Shakespeare sonnet."'' The latter is sourced from [http://www.angelfire.com/in/hypnosonic/Parable_of_the_Monkeys.html Parable of the Monkeys], a collection of historical references to the theorem in various formats.</ref>

In his 1978 radio play, ''[[The Hitchhiker's Guide to the Galaxy]]'', [[Douglas Adams]] invoked the theorem to illustrate the power of the ‘Infinite Improbability Drive’ that powered a spaceship. From Episode 2: ''"[[Ford Prefect (character)|Ford]], there’s an infinite number of monkeys outside who want to talk to us about this script for [[Hamlet (play)|Hamlet]] they’ve worked out."''

A quotation attributed <ref>{{cite web|author=Robert Wilensky, speech at a 1996 conference |url=http://www.quotationspage.com/quote/27695.html |title=Quote Details: Robert Wilensky: We've heard that a... |publisher=The Quotations Page |date= |accessdate=2012-01-18}}{{Verify credibility|date=August 2012}}</ref> to a 1996 speech by Robert Wilensky stated, ''"We’ve heard that a million monkeys at a million keyboards could produce the complete works of Shakespeare; now, thanks to the Internet, we know that is not true."''

The enduring, widespread popularity of the theorem was noted in the introduction to a 2001 paper, ''"Monkeys, Typewriters and Networks: The Internet in the Light of the Theory of Accidental Excellence"'' (Hoffmann & Hofmann, 2001).<ref>[http://skylla.wz-berlin.de/pdf/2002/ii02-101.pdf Monkeys, Typewriters and Networks], Ute Hoffmann & Jeanette Hofmann, Wissenschaftszentrum Berlin für Sozialforschung gGmbH (WZB), 2001.</ref> In 2002, an article in the [[Washington Post]] said, ''"Plenty of people have had fun with the famous notion that an infinite number of monkeys with an infinite number of typewriters and an infinite amount of time could eventually write the works of Shakespeare."''
<ref>[http://www.washingtonpost.com/ac2/wp-dyn/A28521-2002Oct27?language=printer "Hello? This is Bob"], Ken Ringle, ''Washington Post'', 28 October 2002, page C01.</ref> In 2003, the previously mentioned [[Arts Council England|Arts Council]] funded experiment involving real monkeys and a computer keyboard received widespread press coverage.<ref>[http://www.vivaria.net/experiments/notes/documentation/press/ Notes Towards the Complete Works of Shakespeare] – some press clippings.</ref> In 2007, the theorem was listed by ''[[Wired (magazine)|Wired]]'' magazine in a list of eight classic [[thought experiment]]s.<ref>[http://www.wired.com/science/discoveries/magazine/15-06/st_best The Best Thought Experiments: Schrödinger's Cat, Borel’s Monkeys], Greta Lorge, Wired Magazine: Issue 15.06, May 2007.</ref>

==See also==
* [[Normal number]]
* [[Hilbert's paradox of the Grand Hotel]], another thought experiment involving infinity
* ''[[The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos]]'', explains the multiverse in which every possible event will occur an infinite amount of times
* [[The Engine]]
* [[Infinite Monkeys]]

==Notes==
{{Reflist|30em|group="note"}}

==References==
{{Reflist|35em}}

==External links==
* [http://mathforum.org/library/drmath/view/55871.html Ask Dr. Math article], August 1998, Adam Bridge
* [http://www.angelfire.com/in/hypnosonic/Parable_of_the_Monkeys.html The Parable of the Monkeys], a bibliography with quotations
* [http://azureworld.blogspot.com/2007/04/planck-monkeys.html Planck Monkeys], on populating the cosmos with monkey particles
* [http://www.pixelmonkeys.org/ PixelMonkeys.org] - Artist, Matt Kane's application of the Infinite Monkey Theorem on pixels to create images.
* [http://tools.ietf.org/html/rfc2795 RFC 2795] - Humorous RFC on the implementation of the Infinite monkey theorem.

{{DEFAULTSORT:Infinite Monkey Theorem}}
[[Category:Articles containing proofs]]
[[Category:Fictional monkeys]]
[[Category:Infinity]]
[[Category:Literary theory]]
[[Category:Metaphors referring to animals]]
[[Category:Probability theory]]
[[Category:Randomness]]
[[Category:Random text generation]]
[[Category:Thought experiments]]

{{Link GA|de}}
{{Link FA|pl}}
{{Link FA|tr}}

Relativistic Euler equations

2012-11-20T03:53:35Z

5.151.82.75:

{{Refimprove|date=July 2011}}
{{Infobox toy
|name = Spirograph
|image = [[File:Spirograph3.jpg|thumb]]
|othernames =
|type =
|inventor = [[Denys Fisher]]
|country = [[United Kingdom]]
|company = [[Hasbro]]
|from = 1965
|to = present
|materials = [[Plastic]]
|slogan =
|website = http://www.hasbro.com/search/_/Ntt-SPIROGRAPH?Ntk=All&Ntx=mode+matchallpartial
}}

A '''Spirograph''' is a [[geometric]] drawing toy that produces mathematical [[roulette (curve)|roulette]] curves of the variety technically known as [[hypotrochoid]]s and [[epitrochoid]]s. It was developed by British engineer [[Denys Fisher]] and first sold in 1965.

"Spirograph" has also been used to describe a variety of software applications that display similar curves. It has also been applied to the class of curves that can be produced with the drawing equipment, and therefore may be regarded as a synonym of [[hypotrochoid]]. The name has been a registered [[trademark]] of [[Hasbro]], Inc., since it bought the Denys Fisher company.

==History==
The mathematician [[Bruno Abakanowicz]] invented the spirograph between 1881 and 1900. It was used for calculating an area delimited by curves.<ref>{{cite book |url= http://books.google.com/books?id=Ri46VxE7Pc0C&pg=PA293 |title=L'Europe mathématique: histoires, mythes, identités|page=293|first1= Cathérine |last1=Goldstein |first2= Jeremy |last2= Gray |first3=Jim |last3=Ritter |publisher= Editions MSH |year= 1996 |accessdate=17 July 2011}}</ref> Drawing toys based on gears have been around since at least 1908, when '''The Marvelous Wondergraph''' was advertised in the [[Sears]] catalog.<ref>{{cite web |url= http://digitallibrary.imcpl.org/cdm4/document.php?CISOROOT=/tcm&CISOPTR=787&REC=4 |title=CONTENTdm Collection : Compound Object Viewer |last=Kaveney | first=Wendy |work=digitallibrary.imcpl.org |date= |accessdate=17 July 2011}}</ref><ref>{{cite web |url= http://www.artslant.com/chi/articles/show/16968 |title=ArtSlant - Spirograph? No, MAGIC PATTERN! |first=Jim |last=Linderman |work=artslant.com |accessdate=17 July 2011}}</ref> An article describing how to make a Wondergraph drawing machine appeared in the ''Boys Mechanic'' publication in 1913.<ref>{{cite web |url= http://www.marcdatabase.com/~lemur/lemur.com/library-of-antiquarian-technology/philosophical-instruments/boy-mechanic-1913/index.html#introduction |title=From ''The Boy Mechanic'' (1913) - A Wondergraph |work=marcdatabase.com |year=2004 |accessdate=17 July 2011}}</ref> The Spirograph itself was developed by the British engineer [[Denys Fisher]], who exhibited at the 1965 [[Nuremberg International Toy Fair]]. It was subsequently produced by his company. US distribution rights were acquired by [[Kenner Products|Kenner]], Inc., which introduced it to the United States market in 1966 and promoted it as a creative children's toy.

In 1968, Kenner introduced '''Spirotot''', a less complex version of Spirograph, for preschool-age children too young for Spirograph.

==Operation==
[[Image:Various Spirograph Designs.jpg|thumb|left|Several Spirograph designs drawn with a Spirograph set]]
The original US-released Spirograph consisted of two different-sized plastic rings, with gear teeth on both the inside and outside of their circumferences. They were pinned to a [[cardboard (paper product)|cardboard]] backing with pins, and any of several provided gearwheels, which had holes provided for a [[ballpoint pen]] to extend through them to an underlying paper writing surface. It could be spun around to make geometric shapes on the underlying paper medium. Later, the
Super-Spirograph consisted of a set of plastic [[gear]]s and other interlocking shape-segments such as rings, triangles, or straight bars. It has several sizes of gears and shapes, and all edges have teeth to engage any other piece. For instance, smaller gears fit inside the larger rings, but also can engage the outside of the rings in such a fashion that they rotate around the inside or along the outside edge of the rings.

To use it, a sheet of paper is placed on a heavy cardboard backing, and one of the plastic pieces—known as a [[stator]]—is secured via pins or reusable adhesive to the paper and cardboard. Another plastic piece—called the [[wikt:rotor|rotor]]—is placed so that its teeth engage with those of the pinned piece. For example, a ring may be pinned to the paper and a small gear placed inside the ring. The number of arrangements possible by combining different gears is very large. The point of a pen is placed in one of the holes of the rotor. As the rotor is moved, the pen traces out a curve. The pen is used both to draw and to provide locomotive force; some practice is required before the Spirograph can be operated without disengaging the stator and rotor, particularly when using the holes close to the edge of the larger rotors. More intricate and unusual-shaped patterns may be made through the use of both hands, one to draw and one to guide the pieces. It is possible to move several pieces in relation to each other (say, the triangle around the ring, with a circle "climbing" from the ring onto the triangle), but this requires concentration or even additional assistance from other artists.

==Mathematical basis==

[[File:resonance Cascade.svg|thumb|300px|right]]
Consider a fixed outer circle <math>C_o</math> of radius <math>R</math> centered at the origin. A smaller inner circle <math>C_i</math> of radius <math>r<R</math> is rolling inside <math>C_o</math> and is continuously tangent to it. <math>C_i</math> will be assumed never to slip on <math>C_o</math> (in a real Spirograph, teeth on both circles prevent such slippage). Now assume that a point <math>A</math> lying somewhere inside <math>C_i</math> is located a distance <math>\rho<r</math> from <math>C_i</math>'s center. This point <math>A</math> corresponds to the pen-hole in the inner disk of a real Spirograph. Without loss of generality it can be assumed that at the initial moment the point <math>A</math> was on the <math>X</math>-axis. In order to find the trajectory created by a Spirograph, follow point <math>A</math> as the inner circle is set in motion.

Now mark two points <math>T</math> on <math>C_o</math> and <math>B</math> on <math>C_i</math>. The point <math>T</math> always indicates the location where the two circles are tangent. Point <math>B</math> however will travel on <math>C_i</math> and its initial location coincides with <math>T</math>. After setting <math>C_i</math> in motion counterclockwise around <math>C_o</math>, <math>C_i</math> has a clockwise rotation with respect to its center. The distance that point <math>B</math> traverses on <math>C_i</math> is the same as that traversed by the tangent point <math>T</math> on <math>C_o</math>, due to the absence of slipping.

Now define the new (relative) system of coordinates <math>(\hat{X},\hat{Y})</math> with its origin at the center of <math>C_i</math> and its axes parallel to <math>X</math> and <math>Y</math>. Let the parameter <math>t</math> be the angle by which the tangent point <math>T </math> rotates on <math>C_o</math> and <math>\hat{t}</math> be the angle by which <math>C_i</math> rotates (i.e. by which <math>B</math> travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by <math>B</math> and <math>T</math> along their respective circles must be the same, therefore

<math>tR=(t-\hat{t})r</math>

or equivalently

<math>\hat{t}=-\frac{R-r}{r}t.</math>

It is common to assume that a counterclockwise motion corresponds to a positive change of angle and a clockwise one to a negative change of angle. A minus sign in the above formula (<math>\hat{t}<0</math>) accommodates this convention.

Let <math>(x_c,y_c)</math> be the coordinates of the center of <math>C_i</math> in the absolute system of coordinates. Then <math>R-r</math> represents the radius of the trajectory of the center of <math>C_i</math>, which (again in the absolute system) undergoes circular motion thus:

<math>\begin{array}{rcl}
x_c&=&(R-r)\cos t,\\
y_c&=&(R-r)\sin t.
\end{array}</math>

As defined above, <math>\hat{t}</math> is the angle of rotation in the new relative system. Because point <math>A</math> obeys the usual law of circular motion, its coordinates in the new relative coordinate system <math>(\hat{x},\hat{y})</math> obey:

<math>\begin{array}{rcl}
\hat{x}&=&\rho\cos \hat{t},\\
\hat{y}&=&\rho\sin \hat{t}.
\end{array}</math>

In order to obtain the trajectory of <math>A</math> in the absolute (old) system of coordinates, add these two motions:

<math>\begin{array}{rcrcl}
x&=&x_c+\hat{x}&=&(R-r)\cos t+\rho\cos \hat{t},\\
y&=&y_c+\hat{y}&=&(R-r)\sin t+\rho\sin \hat{t},\\
\end{array}</math>

where <math>\rho</math> is defined above.

Now, use the relation between <math>t</math> and <math>\hat{t}</math> as derived above to obtain equations describing the trajectory of point <math>A</math> in terms of a single parameter <math>t</math>:

<math>\begin{array}{rcrcl}
x&=&x_c+\hat{x}&=&(R-r)\cos t+\rho\cos \frac{R-r}{r}t,\\[4pt]
y&=&y_c+\hat{y}&=&(R-r)\sin t-\rho\sin \frac{R-r}{r}t.\\
\end{array}</math>

(using the fact that function <math>\sin</math> is odd).

It is convenient to represent the equation above in terms of the radius <math>R</math> of <math>C_o</math> and dimensionless
parameters describing the structure of the Spirograph. Namely, let

<math>l=\frac{\rho}{r}</math>

and

<math>k=\frac{r}{R}.</math>

The parameter <math>0\le l \le 1</math> represents how far the point <math>A</math> is located from the center of <math>C_i</math>. At the same time, <math>0\le k \le 1</math> represents how big the inner circle <math>C_i</math> is with respect to the outer one <math>C_o</math>.

It is now observed that

<math>\frac{\rho}{R}=lk,</math>

and therefore the trajectory equations take the form

<math>\begin{array}{rcl}
x(t)&=&R\left[(1-k)\cos t+lk\cos \frac{1-k}{k}t\right],\\[4pt]
y(t)&=&R\left[(1-k)\sin t-lk\sin \frac{1-k}{k}t\right].\\
\end{array}</math>

Parameter <math>R</math> is a scaling parameter and does not affect the structure of the Spirograph. Different values of <math>R</math> would yield [[similarity (geometry)|similar]] Spirograph drawings.

It is interesting to note that the two extreme cases <math>k=0</math> and <math>k=1</math> result in degenerate trajectories of the Spirograph. In the first extreme case when <math>k=0</math> we have a simple circle of radius <math>R</math>, corresponding to the case where <math>C_i</math> has been shrunk into a point. (Division by <math>k=0</math> in the formula is not a problem since both <math>\sin</math> and <math>\cos</math> are bounded functions).

The other extreme case <math>k=1</math> corresponds to the inner circle <math>C_i</math>'s radius <math>r</math> matching the radius <math>R</math> of the outer circle <math>C_o</math>, ie <math>r=R</math>. In this case the trajectory is a single point. Intuitively, <math>C_i</math> is too large to roll inside the same-sized <math>C_o</math> without slipping.

If <math>l=1</math> then the point <math>A</math> is on the circumference of <math>C_i</math>. In this case the trajectories are called [[hypocycloid]]s and the equations above reduce to those for a hypocycloid.

==See also==

* [[Guilloché]]
* [[Harmonograph]]
* [[Spirograph Nebula]], a [[planetary nebula]] that displays delicate, spirograph-like filigree.
* [[List of periodic functions]]

==References==
{{reflist}}

==External links==
*[http://www.hasbro.com/search/_/Ntt-SPIROGRAPH?Ntk=All&Ntx=mode+matchallpartial Hasbro Spirograph site]
*[http://www.artbylogic.com/spirographart/spirograph.htm HTML5 Interactive Spirograph Creator]
*[http://www.mathiversity.com/spirograph/ A Spirograph software for MS Windows]

{{Hasbro}}
{{Use dmy dates|date=July 2011}}

[[Category:Art and craft toys]]
[[Category:Curves]]
[[Category:Products introduced in 1965]]
[[Category:Hasbro products]]
[[Category:1970s toys]]