Reduction of the structure group - Revision history

204.102.224.82: The man claim being made here about cofibers is false. I am removing it.

2014-09-25T04:35:32Z

The man claim being made here about cofibers is false. I am removing it.

← Older revision		Revision as of 06:35, 25 September 2014
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	~~{{about\|Theodorus the mathematician from Cyrene\|the atheist also from Cyrene\|Theodorus the Atheist}}~~		29 years old Gastroenterologist Takacs from Carignan, enjoys to spend time digital art, diet and baking. During the previous year has completed a visit to Monastery of Geghard and the Upper Azat Valley.
	'''Theodorus of [[Cyrene, Libya\|Cyrene]]''' ({{lang-el\|Θεόδωρος ὁ Κυρηναῖος}}) was a [[Ancient Greece\|Greek]] [[mathematician]] of the 5th century BC. The only first-hand accounts of him that survive are in three of [[Plato]]'s dialogues: the ''[[Theaetetus (dialogue)\|Theaetetus]]'', the ''[[Sophist (dialogue)\|Sophist]]'', and the ''[[Statesman (dialogue)\|Statesman]]''. In the former dialogue, he posits a mathematical theorem now known as the [[Spiral of Theodorus]].

	~~==Life==~~
	~~Little is known of Theodorus' biography beyond what can be inferred~~ from Plato's dialogues. He was born in the northern African colony of Cyrene, and apparently taught both there and in Athens.<ref name=nails>[[Debra Nails\|Nails, Debra]]. ''The People of Plato: A Prosopography of Plato and Other Socratics''. Indianapolis: Hackett Publishing, 2002, pp. 281-2.</ref> He complains of old age in the ''Theaetetus'', whose dramatic date of 399 BC suggests his period of flourishing to have occurred in the mid-5th century. The text also associates him with the [[sophist]] [[Protagoras]], ~~with whom he claims to have studied before turning~~ to geometry.<ref>c.f. Plato, ''Theaetetus'', 189a</ref> A dubious tradition repeated among ancient biographers like [[Diogenes Laërtius]]<ref>Diogenes Laërtius 3.6</ref> held that Plato later studied with him at Cyrene.<ref name=nails />

	~~==Work in mathematics==~~
	~~Theodorus' work is known through a sole theorem~~, ~~which is delivered in the literary context of the ''Theaetetus''~~ and ~~has been argued alternately to be historically accurate or fictional~~.~~<ref name=nails /> In the text, his student [[Theaetetus (mathematician)\|Theaetetus]] attributes to him the theorem that the square roots of the non-square numbers up to 17 are irrational:~~
	~~<blockquote>~~
	~~Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with~~ the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet and at that he stopped.<ref>{{cite book \|title=Cratylus, Theaetetus, Sophist, Statesman \|author=Plato \|authorlink= Plato \|page=174d \|url= http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0172%3Atext%3DTheaet.%3Apage%3D147\|accessdate= August 5, 2010}}</ref>
	~~</blockquote>~~
	~~(The square containing ''two'' square units is not mentioned, perhaps because the incommensurability of its side with the unit was already known.)~~
	~~Theodorus's method of proof is not known. It is not even known whether, in the quoted passage,~~
	"up to" (μέχρι) means that seventeen is included. If seventeen is excluded, then Theodorus's proof may have relied merely on considering whether numbers are even or odd. Indeed, Hardy and Wright<ref>{{cite book \|title=''An Introduction to the Theory of Numbers'' \|last1=Hardy \|first1=G. H. \|author1-link= G. H. Hardy \|last2=Wright \|first2=E. M. \|author2-link= E. M. Wright \|year=1979 \|publisher= Oxford\|isbn=0-19-853171-0 \|pages=42–44}}</ref>
	~~<!--~~
	~~<ref>{{cite journal\|title=Theodorus' Irrationality Proofs\|author=James R. Choike\|journal=''The Two-Year College Mathematics Journal''\|year=1980}}</ref>~~
	~~-->~~
	~~and Knorr<ref>{{cite book\|title= ''The Evolution of the Euclidean Elements'' \|first= Wilbur \|last= Knorr \|authorlink= Wilbur Knorr \|~~year= 1975 \|publisher = D. Reidel \|isbn= 90-277-0509-7}}</ref> suggest proofs that rely ultimately on the following theorem: If <math>x^2=ny^2</math> is soluble in integers, and <math>n</math> is odd, then <math>n</math> must be [[Modular arithmetic\|congruent]] to 1 ''modulo'' 8 (since <math>x</math> and <math>y</math> can be assumed odd, so their squares are congruent to 1 ''modulo'' 8).

	A possibility suggested earlier by [[Hieronymus Georg Zeuthen\|Zeuthen]]<ref name=heath>{{cite book \|title=''A History of Greek Mathematics'' \|first=Thomas\|last=Heath \|authorlink=T. L. Heath \|publisher= Dover \|year=1981 \|isbn=0-486-24073-8 \|volume= 1 \|page=206}}</ref> is that Theodorus applied the so-called [[Euclidean algorithm]], formulated in Proposition X.2 of the [[Euclid's Elements\|''Elements'']] as a test for incommensurability. In modern terms, the theorem is that a real number with an ''infinite'' [[continued fraction]] expansion is irrational. Irrational square roots have [[Periodic continued fraction\|periodic expansions]]. The period of the square root of 19 has length 6, which is greater than the period of the square root of any smaller number. The period of √17 has length one (so does √18; but the irrationality of √18 [[Logical consequence\|follows from]] that of √2).

	~~The so-called Spiral of Theodorus is composed of contiguous [[right triangle]]s with [[hypotenuse]] lengths equal √2, √3, √4, …, √17; additional triangles cause the diagram to overlap.~~
	~~[[Philip J. Davis]] [[interpolation\|interpolated]] the vertices of the spiral to get~~ a ~~continuous curve. He discusses the history of attempts to determine Theodorus' method in his book ''Spirals: From Theodorus to Chaos'', and makes brief references~~ to ~~the matter in his fictional ''Thomas Gray'' series.~~

	~~That Theaetetus established a more general theory of irrationals, whereby square roots~~ of ~~non-square numbers are irrational, is suggested in the eponymous Platonic dialogue as well as commentary on,~~ and ~~[[scholia]] to,~~ the ~~''Elements''~~.~~<ref>Heath 209</ref>~~
	~~<!--~~
	Theaetetus made the generalization that the side of any square, represented by a [[Nth_root#Working_with_surds\|surd]], was incommensurable with the linear unit.<ref>{{cite book \| title = ''A Short History of Greek Mathematics'' \| author = James Gow \| publisher = University press \| year = 1884 \| url = http://books.google.com/?id=9d8DAAAAMAAJ&pg=PA85&dq=Theodorus%27+Irrationality+Proofs }}</ref>
	~~-->~~

	~~==See also==~~
	*[[Chronology of ancient Greek mathematicians]]
	*[[List of speakers in Plato's dialogues]]
	*[[Quadratic irrational]]
	*[[Wilbur Knorr]]

	~~==References==~~
	~~{{reflist\|2}}~~

	~~{{Greek mathematics}}~~

	~~{{DEFAULTSORT:Theodorus Of Cyrene}}~~
	~~[[Category:Ancient Greek mathematicians]]~~
	~~[[Category:5th-century BC Greek people]]~~
	~~[[Category:Cyrenean Greeks]]~~

63.252.64.200: added link to 'almost symplectic manifold'

2013-12-02T20:15:53Z

added link to 'almost symplectic manifold'

← Older revision		Revision as of 22:15, 2 December 2013
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	~~Hello~~ and ~~welcome~~. ~~My name~~ is ~~Irwin~~ and ~~I totally dig that~~ name. ~~For many years I~~'~~ve been operating as a payroll clerk~~. ~~To play baseball is~~ the ~~hobby~~ he ~~will by no means stop performing~~. ~~Years~~ in the ~~past we moved~~ to ~~North Dakota~~.<br><br>~~my webpage; [~~http://~~privatefriendship~~.~~com~~/~~uncategorized~~/~~curing~~-~~your~~-~~candida~~-~~albicans~~-~~how~~-to-~~accomplish~~-it-~~easily~~/ ~~privatefriendship~~.com]		{{about\|Theodorus the mathematician from Cyrene\|the atheist also from Cyrene\|Theodorus the Atheist}}
			'''Theodorus of [[Cyrene, Libya\|Cyrene]]''' ({{lang-el\|Θεόδωρος ὁ Κυρηναῖος}}) was a [[Ancient Greece\|Greek]] [[mathematician]] of the 5th century BC. The only first-hand accounts of him that survive are in three of [[Plato]]'s dialogues: the ''[[Theaetetus (dialogue)\|Theaetetus]]'', the ''[[Sophist (dialogue)\|Sophist]]'', and the ''[[Statesman (dialogue)\|Statesman]]''. In the former dialogue, he posits a mathematical theorem now known as the [[Spiral of Theodorus]].

			==Life==
			Little is known of Theodorus' biography beyond what can be inferred from Plato's dialogues. He was born in the northern African colony of Cyrene, and apparently taught both there and in Athens.<ref name=nails>[[Debra Nails\|Nails, Debra]]. ''The People of Plato: A Prosopography of Plato and Other Socratics''. Indianapolis: Hackett Publishing, 2002, pp. 281-2.</ref> He complains of old age in the ''Theaetetus'', whose dramatic date of 399 BC suggests his period of flourishing to have occurred in the mid-5th century. The text also associates him with the [[sophist]] [[Protagoras]], with whom he claims to have studied before turning to geometry.<ref>c.f. Plato, ''Theaetetus'', 189a</ref> A dubious tradition repeated among ancient biographers like [[Diogenes Laërtius]]<ref>Diogenes Laërtius 3.6</ref> held that Plato later studied with him at Cyrene.<ref name=nails />

			==Work in mathematics==
			Theodorus' work is known through a sole theorem, which is delivered in the literary context of the ''Theaetetus'' and has been argued alternately to be historically accurate or fictional.<ref name=nails /> In the text, his student [[Theaetetus (mathematician)\|Theaetetus]] attributes to him the theorem that the square roots of the non-square numbers up to 17 are irrational:
			<blockquote>
			Theodorus here was drawing some figures for us in illustration of roots, showing that squares containing three square feet and five square feet are not commensurable in length with the unit of the foot, and so, selecting each one in its turn up to the square containing seventeen square feet and at that he stopped.<ref>{{cite book \|title=Cratylus, Theaetetus, Sophist, Statesman \|author=Plato \|authorlink= Plato \|page=174d \|url= http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0172%3Atext%3DTheaet.%3Apage%3D147\|accessdate= August 5, 2010}}</ref>
			</blockquote>
			(The square containing ''two'' square units is not mentioned, perhaps because the incommensurability of its side with the unit was already known.)
			Theodorus's method of proof is not known. It is not even known whether, in the quoted passage,
			"up to" (μέχρι) means that seventeen is included. If seventeen is excluded, then Theodorus's proof may have relied merely on considering whether numbers are even or odd. Indeed, Hardy and Wright<ref>{{cite book \|title=''An Introduction to the Theory of Numbers'' \|last1=Hardy \|first1=G. H. \|author1-link= G. H. Hardy \|last2=Wright \|first2=E. M. \|author2-link= E. M. Wright \|year=1979 \|publisher= Oxford\|isbn=0-19-853171-0 \|pages=42–44}}</ref>
			<!--
			<ref>{{cite journal\|title=Theodorus' Irrationality Proofs\|author=James R. Choike\|journal=''The Two-Year College Mathematics Journal''\|year=1980}}</ref>
			-->
			and Knorr<ref>{{cite book\|title= ''The Evolution of the Euclidean Elements'' \|first= Wilbur \|last= Knorr \|authorlink= Wilbur Knorr \|year= 1975 \|publisher = D. Reidel \|isbn= 90-277-0509-7}}</ref> suggest proofs that rely ultimately on the following theorem: If <math>x^2=ny^2</math> is soluble in integers, and <math>n</math> is odd, then <math>n</math> must be [[Modular arithmetic\|congruent]] to 1 ''modulo'' 8 (since <math>x</math> and <math>y</math> can be assumed odd, so their squares are congruent to 1 ''modulo'' 8).

			A possibility suggested earlier by [[Hieronymus Georg Zeuthen\|Zeuthen]]<ref name=heath>{{cite book \|title=''A History of Greek Mathematics'' \|first=Thomas\|last=Heath \|authorlink=T. L. Heath \|publisher= Dover \|year=1981 \|isbn=0-486-24073-8 \|volume= 1 \|page=206}}</ref> is that Theodorus applied the so-called [[Euclidean algorithm]], formulated in Proposition X.2 of the [[Euclid's Elements\|''Elements'']] as a test for incommensurability. In modern terms, the theorem is that a real number with an ''infinite'' [[continued fraction]] expansion is irrational. Irrational square roots have [[Periodic continued fraction\|periodic expansions]]. The period of the square root of 19 has length 6, which is greater than the period of the square root of any smaller number. The period of √17 has length one (so does √18; but the irrationality of √18 [[Logical consequence\|follows from]] that of √2).

			The so-called Spiral of Theodorus is composed of contiguous [[right triangle]]s with [[hypotenuse]] lengths equal √2, √3, √4, …, √17; additional triangles cause the diagram to overlap.
			[[Philip J. Davis]] [[interpolation\|interpolated]] the vertices of the spiral to get a continuous curve. He discusses the history of attempts to determine Theodorus' method in his book ''Spirals: From Theodorus to Chaos'', and makes brief references to the matter in his fictional ''Thomas Gray'' series.

			That Theaetetus established a more general theory of irrationals, whereby square roots of non-square numbers are irrational, is suggested in the eponymous Platonic dialogue as well as commentary on, and [[scholia]] to, the ''Elements''.<ref>Heath 209</ref>
			<!--
			Theaetetus made the generalization that the side of any square, represented by a [[Nth_root#Working_with_surds\|surd]], was incommensurable with the linear unit.<ref>{{cite book \| title = ''A Short History of Greek Mathematics'' \| author = James Gow \| publisher = University press \| year = 1884 \| url = http://books.google.com/?id=9d8DAAAAMAAJ&pg=PA85&dq=Theodorus%27+Irrationality+Proofs }}</ref>
			-->

			==See also==
			*[[Chronology of ancient Greek mathematicians]]
			*[[List of speakers in Plato's dialogues]]
			*[[Quadratic irrational]]
			*[[Wilbur Knorr]]

			==References==
			{{reflist\|2}}

			{{Greek mathematics}}

			{{DEFAULTSORT:Theodorus Of Cyrene}}
			[[Category:Ancient Greek mathematicians]]
			[[Category:5th-century BC Greek people]]
			[[Category:Cyrenean Greeks]]

en>AnomieBOT: Dating maintenance tags: {{Merge}}

2012-08-10T19:55:03Z

Dating maintenance tags: {{Merge}}

New page

Hello and welcome. My name is Irwin and I totally dig that name. For many years I've been operating as a payroll clerk. To play baseball is the hobby he will by no means stop performing. Years in the past we moved to North Dakota.<br><br>my webpage; [http://privatefriendship.com/uncategorized/curing-your-candida-albicans-how-to-accomplish-it-easily/ privatefriendship.com]