Compressed sensing - Revision history

en>EdJohnston: Add semiprotection template

2015-01-13T02:39:21Z

Add semiprotection template

← Older revision		Revision as of 04:39, 13 January 2015
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	~~Hi there,~~ I ~~am Alyson Pomerleau and I think it seems quite great when you say it~~. ~~Her family lives in Ohio~~ but her ~~spouse wants them~~ to ~~transfer~~. I ~~am presently~~ a ~~travel agent. It's not a common thing~~ but ~~what she likes doing is to perform domino but she doesn't have the time recently~~.<br><br>My ~~site~~ - [http://~~myfusionprofits~~.com/~~groups~~/find-out~~-about-self-improvement-tips-~~here~~/ phone psychic~~]		I'm Yoshiko Oquendo. Kansas is exactly where her house is but she needs to move because of her family. The thing I adore most flower arranging and now I have time to consider on new issues. After being out of my job for years I became a production and distribution officer but I plan on changing it.<br><br>My blog - extended car warranty ([http://grtfl.com/index.php?do=/profile-220/info/ find out here])

en>Dough34: fix capitalization, remove redundant links

2014-02-16T22:00:36Z

fix capitalization, remove redundant links

← Older revision		Revision as of 00:00, 17 February 2014
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	A [[two-dimensional]] '''equable shape''' (or perfect shape) is one whose [[area]] is numerically equal to its [[perimeter]].<ref>{{cite book \|title=Challenges in Geometry: For Mathematical Olympians Past and Present		Hi there, I am Alyson Pomerleau and I think it seems quite great when you say it. Her family lives in Ohio but her spouse wants them to transfer. I am presently a travel agent. It's not a common thing but what she likes doing is to perform domino but she doesn't have the time recently.<br><br>My site - [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ phone psychic]
	~~\|first=Christopher J.\|last=Bradley\|publisher=Oxford University Press\|year=2005\|isbn=0-19-856692-1\|page=15}}.</ref> For example, a [[right triangle\|right angled triangle]] with sides 5~~, 12 and ~~13 has area and perimeter both equal to 30 [[Units of measurement\|units]].~~

	~~==Scaling and units==~~
	~~An area cannot be equal to a length except relative to a particular unit of measurement. For example, if shape has an area of 5 square yards and a perimeter of 5 yards, then~~ it ~~has an area of {{convert\|45\|sqft\|m2}} and a perimeter of 15 feet (since 3 feet = 1 yard and hence 9 square feet = 1 square yard)~~. Moreover, contrary to what the name implies, changing the size while leaving the shape intact changes an "equable shape" into a non-equable shape. However its common use as [[GCSE]] coursework has led to ~~its being an accepted concept~~. ~~For any shape, there is a [[Similarity (geometry)\|similar]] equable shape: if a shape ''S'' has perimeter ''p'' and area ''A'', then [[Scaling (geometry)\|scaling]] ''S'' by~~ a ~~factor of ''p/A'' leads to an equable shape~~. ~~Alternatively, one may find equable shapes by setting up and solving an equation in which the area equals the perimeter. In the case of the square, for instance, this equation is~~

	~~:<math>\displaystyle x^2 = 4x.</math>~~

	~~Solving this yields that~~ '~~'x'' = 4, so~~ a ~~4 × 4 square is equable.~~

	~~==Tangential polygons==~~
	~~A [[tangential polygon]]~~ is ~~a polygon in which the sides are all tangent to a common circle. Every tangential polygon may be triangulated by drawing edges from the circle's center~~ to ~~the polygon~~'~~s vertices, forming a collection of triangles that all~~ have height equal to the circle's radius; it follows from this decomposition that the total area of a tangential polygon equals half the perimeter times the radius. Thus, a tangential polygon is equable if and only if its [[inradius]] is two. All triangles are tangential, so in particular the ~~equable triangles are exactly the triangles with inradius two.<ref>{{citation~~
	~~\| last = Kilmer \| first = Jean E.~~
	~~\| issue = 1~~
	~~\| journal = The Mathematics Teacher~~
	~~\| jstor = 27965678~~
	~~\| pages = 65–70~~
	~~\| title = Triangles of Equal Area and Perimeter and Inscribed Circles~~
	~~\| volume = 81}}~~.<~~/ref~~><~~ref~~>~~{{citation\|url=http://jwilson.coe.uga.edu/emt725/Perfect/PerTri.html\|title=Perfect triangles\|first=Jim\|last=WIlson\|publisher=University of Georgia}}. See also Wilson's list of~~ [http://~~jwilson~~.~~coe.uga.edu~~/~~emt725~~/~~Perfect/sol.html solutions].</ref>~~

	~~==Integer dimensions==~~
	Combining restrictions that a shape be equable and that its dimensions be integers is significantly more restrictive than either restriction on its own. For instance, there are infinitely many [[Pythagorean triple]]s describing integer-~~sided [[right triangle]]s, and there are infinitely many equable right triangles with non~~-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10).<ref name="bicycle">{{citation\|title=Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries\|volume=18\|series=Dolciani Mathematical Expositions\|first1=Joseph D. E.\|last1=Konhauser\|first2=Dan\|last2=Velleman\|first3=Stan\|last3=Wagon\|author3-link=Stan Wagon\|publisher=Cambridge University Press\|year=1997\|isbn=9780883853252\|url=http://books.google.com/books?id=ElSi5V5uS2MC&pg=PA29\|page=29\|contribution=95. When does the perimeter equal the area?}}</ref>

	More generally, the problem of finding all equable triangles with integer sides (that is, equable [[Heronian triangle]]s) was considered by B. Yates in 1858.<ref>{{citation\|first=B.\|last=Yates\|title=Quest 2019\|journal=The Lady's and Gentleman's Diary\|year=1858\|page=83}}.</ref><ref>{{citation\|title=[[History of the Theory of Numbers]], Volume Il: Diophantine Analysis\|first=Leonard Eugene\|last=Dickson\|authorlink=Leonard Eugene Dickson\|publisher=Courier Dover Publications\|year=2005\|isbn=9780486442334\|page=195}}.</ref> As [[William Allen Whitworth\|W. A. Whitworth]] and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).<ref>{{harvtxt\|Dickson\|2005}}, p. 199.</~~ref><ref>{{citation~~
	~~\| last = Markowitz \| first = L.~~
	~~\| issue = 3~~
	~~\| journal = The Mathematics Teacher~~
	~~\| pages = 222–223~~
	~~\| title = Area = Perimeter~~
	~~\| volume = 74~~
	~~\| year = 1981~~
	~~\| zbl = 1982d.06561}}.</ref>~~

	The only equable [[rectangle]]s with integer sides are the 4 × 4 square and the 3 × 6 rectangle.<ref name="bicycle"/> An integer rectangle is a special type of [[polyomino]], and more generally there exist polyominoes with equal area and perimeter for any [[even number\|even]] integer area greater than or equal to 16. For smaller areas, the perimeter of a polyomino must exceed its area.<ref>{{citation\|title=Geometry Labs\|first=Henri\|last=Picciotto\|year=1999\|publisher=MathEducationPage.org\|page=208\|url=http://books.google.com/books?id=7gTMKr7TT6gC&pg=PA208}}.</ref>

	~~== Equable solids ==~~
	~~In [[Three-dimensional space\|three dimensions]], a shape is equable when its [[surface area]] is numerically equal to its [[volume]].~~

	~~As with equable shapes in two dimensions, you may find an equable solid, in which the volume is numerically equal to the surface area, by scaling any solid by an appropriate factor.~~

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Geometric shapes]~~]

en>Monkbot: /* Photography */Fix CS1 deprecated date parameter errors

2014-02-04T04:32:42Z

Photography: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 06:32, 4 February 2014
Line 1:		Line 1:
	~~Friends contact him Royal Cummins~~. ~~The preferred pastime for him~~ and ~~his kids~~ is to ~~drive~~ and ~~now he~~ is ~~attempting~~ to ~~make money with~~ it. ~~I am~~ a ~~production~~ and ~~distribution officer~~. ~~Her husband~~ and ~~her chose to reside in Alabama~~.<br><br>~~My site auto warranty~~, [http://~~dotnetnuke~~.~~joinernetworks~~.com/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/61/~~UserId~~/~~5304~~/~~Default~~.~~aspx Ongoing~~],		A [[two-dimensional]] '''equable shape''' (or perfect shape) is one whose [[area]] is numerically equal to its [[perimeter]].<ref>{{cite book \|title=Challenges in Geometry: For Mathematical Olympians Past and Present
			\|first=Christopher J.\|last=Bradley\|publisher=Oxford University Press\|year=2005\|isbn=0-19-856692-1\|page=15}}.</ref> For example, a [[right triangle\|right angled triangle]] with sides 5, 12 and 13 has area and perimeter both equal to 30 [[Units of measurement\|units]].

			==Scaling and units==
			An area cannot be equal to a length except relative to a particular unit of measurement. For example, if shape has an area of 5 square yards and a perimeter of 5 yards, then it has an area of {{convert\|45\|sqft\|m2}} and a perimeter of 15 feet (since 3 feet = 1 yard and hence 9 square feet = 1 square yard). Moreover, contrary to what the name implies, changing the size while leaving the shape intact changes an "equable shape" into a non-equable shape. However its common use as [[GCSE]] coursework has led to its being an accepted concept. For any shape, there is a [[Similarity (geometry)\|similar]] equable shape: if a shape ''S'' has perimeter ''p'' and area ''A'', then [[Scaling (geometry)\|scaling]] ''S'' by a factor of ''p/A'' leads to an equable shape. Alternatively, one may find equable shapes by setting up and solving an equation in which the area equals the perimeter. In the case of the square, for instance, this equation is

			:<math>\displaystyle x^2 = 4x.</math>

			Solving this yields that ''x'' = 4, so a 4 × 4 square is equable.

			==Tangential polygons==
			A [[tangential polygon]] is a polygon in which the sides are all tangent to a common circle. Every tangential polygon may be triangulated by drawing edges from the circle's center to the polygon's vertices, forming a collection of triangles that all have height equal to the circle's radius; it follows from this decomposition that the total area of a tangential polygon equals half the perimeter times the radius. Thus, a tangential polygon is equable if and only if its [[inradius]] is two. All triangles are tangential, so in particular the equable triangles are exactly the triangles with inradius two.<ref>{{citation
			\| last = Kilmer \| first = Jean E.
			\| issue = 1
			\| journal = The Mathematics Teacher
			\| jstor = 27965678
			\| pages = 65–70
			\| title = Triangles of Equal Area and Perimeter and Inscribed Circles
			\| volume = 81}}.</ref><ref>{{citation\|url=http://jwilson.coe.uga.edu/emt725/Perfect/PerTri.html\|title=Perfect triangles\|first=Jim\|last=WIlson\|publisher=University of Georgia}}. See also Wilson's list of [http://jwilson.coe.uga.edu/emt725/Perfect/sol.html solutions].</ref>

			==Integer dimensions==
			Combining restrictions that a shape be equable and that its dimensions be integers is significantly more restrictive than either restriction on its own. For instance, there are infinitely many [[Pythagorean triple]]s describing integer-sided [[right triangle]]s, and there are infinitely many equable right triangles with non-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10).<ref name="bicycle">{{citation\|title=Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries\|volume=18\|series=Dolciani Mathematical Expositions\|first1=Joseph D. E.\|last1=Konhauser\|first2=Dan\|last2=Velleman\|first3=Stan\|last3=Wagon\|author3-link=Stan Wagon\|publisher=Cambridge University Press\|year=1997\|isbn=9780883853252\|url=http://books.google.com/books?id=ElSi5V5uS2MC&pg=PA29\|page=29\|contribution=95. When does the perimeter equal the area?}}</ref>

			More generally, the problem of finding all equable triangles with integer sides (that is, equable [[Heronian triangle]]s) was considered by B. Yates in 1858.<ref>{{citation\|first=B.\|last=Yates\|title=Quest 2019\|journal=The Lady's and Gentleman's Diary\|year=1858\|page=83}}.</ref><ref>{{citation\|title=[[History of the Theory of Numbers]], Volume Il: Diophantine Analysis\|first=Leonard Eugene\|last=Dickson\|authorlink=Leonard Eugene Dickson\|publisher=Courier Dover Publications\|year=2005\|isbn=9780486442334\|page=195}}.</ref> As [[William Allen Whitworth\|W. A. Whitworth]] and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).<ref>{{harvtxt\|Dickson\|2005}}, p. 199.</ref><ref>{{citation
			\| last = Markowitz \| first = L.
			\| issue = 3
			\| journal = The Mathematics Teacher
			\| pages = 222–223
			\| title = Area = Perimeter
			\| volume = 74
			\| year = 1981
			\| zbl = 1982d.06561}}.</ref>

			The only equable [[rectangle]]s with integer sides are the 4 × 4 square and the 3 × 6 rectangle.<ref name="bicycle"/> An integer rectangle is a special type of [[polyomino]], and more generally there exist polyominoes with equal area and perimeter for any [[even number\|even]] integer area greater than or equal to 16. For smaller areas, the perimeter of a polyomino must exceed its area.<ref>{{citation\|title=Geometry Labs\|first=Henri\|last=Picciotto\|year=1999\|publisher=MathEducationPage.org\|page=208\|url=http://books.google.com/books?id=7gTMKr7TT6gC&pg=PA208}}.</ref>

			== Equable solids ==
			In [[Three-dimensional space\|three dimensions]], a shape is equable when its [[surface area]] is numerically equal to its [[volume]].

			As with equable shapes in two dimensions, you may find an equable solid, in which the volume is numerically equal to the surface area, by scaling any solid by an appropriate factor.

			==References==
			{{reflist}}

			[[Category:Geometric shapes]]

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

2012-08-31T06:33:03Z

Dating maintenance tags: {{Cn}}

New page

Friends contact him Royal Cummins. The preferred pastime for him and his kids is to drive and now he is attempting to make money with it. I am a production and distribution officer. Her husband and her chose to reside in Alabama.<br><br>My site auto warranty, [http://dotnetnuke.joinernetworks.com/ActivityFeed/MyProfile/tabid/61/UserId/5304/Default.aspx Ongoing],

← Older revision		Revision as of 00:00, 17 February 2014
Line 1:		Line 1:
	A [[two-dimensional]] '''equable shape''' (or perfect shape) is one whose [[area]] is numerically equal to its [[perimeter]].<ref>{{cite book \|title=Challenges in Geometry: For Mathematical Olympians Past and Present		Hi there, I am Alyson Pomerleau and I think it seems quite great when you say it. Her family lives in Ohio but her spouse wants them to transfer. I am presently a travel agent. It's not a common thing but what she likes doing is to perform domino but she doesn't have the time recently.<br><br>My site - [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ phone psychic]
	~~\|first=Christopher J.\|last=Bradley\|publisher=Oxford University Press\|year=2005\|isbn=0-19-856692-1\|page=15}}.</ref> For example, a [[right triangle\|right angled triangle]] with sides 5~~, 12 and ~~13 has area and perimeter both equal to 30 [[Units of measurement\|units]].~~

	~~==Scaling and units==~~
	~~An area cannot be equal to a length except relative to a particular unit of measurement. For example, if shape has an area of 5 square yards and a perimeter of 5 yards, then~~ it ~~has an area of {{convert\|45\|sqft\|m2}} and a perimeter of 15 feet (since 3 feet = 1 yard and hence 9 square feet = 1 square yard)~~. Moreover, contrary to what the name implies, changing the size while leaving the shape intact changes an "equable shape" into a non-equable shape. However its common use as [[GCSE]] coursework has led to ~~its being an accepted concept~~. ~~For any shape, there is a [[Similarity (geometry)\|similar]] equable shape: if a shape ''S'' has perimeter ''p'' and area ''A'', then [[Scaling (geometry)\|scaling]] ''S'' by~~ a ~~factor of ''p/A'' leads to an equable shape~~. ~~Alternatively, one may find equable shapes by setting up and solving an equation in which the area equals the perimeter. In the case of the square, for instance, this equation is~~

	~~:<math>\displaystyle x^2 = 4x.</math>~~

	~~Solving this yields that~~ '~~'x'' = 4, so~~ a ~~4 × 4 square is equable.~~

	~~==Tangential polygons==~~
	~~A [[tangential polygon]]~~ is ~~a polygon in which the sides are all tangent to a common circle. Every tangential polygon may be triangulated by drawing edges from the circle's center~~ to ~~the polygon~~'~~s vertices, forming a collection of triangles that all~~ have height equal to the circle's radius; it follows from this decomposition that the total area of a tangential polygon equals half the perimeter times the radius. Thus, a tangential polygon is equable if and only if its [[inradius]] is two. All triangles are tangential, so in particular the ~~equable triangles are exactly the triangles with inradius two.<ref>{{citation~~
	~~\| last = Kilmer \| first = Jean E.~~
	~~\| issue = 1~~
	~~\| journal = The Mathematics Teacher~~
	~~\| jstor = 27965678~~
	~~\| pages = 65–70~~
	~~\| title = Triangles of Equal Area and Perimeter and Inscribed Circles~~
	~~\| volume = 81}}~~.<~~/ref~~><~~ref~~>~~{{citation\|url=http://jwilson.coe.uga.edu/emt725/Perfect/PerTri.html\|title=Perfect triangles\|first=Jim\|last=WIlson\|publisher=University of Georgia}}. See also Wilson's list of~~ [http://~~jwilson~~.~~coe.uga.edu~~/~~emt725~~/~~Perfect/sol.html solutions].</ref>~~

	~~==Integer dimensions==~~
	Combining restrictions that a shape be equable and that its dimensions be integers is significantly more restrictive than either restriction on its own. For instance, there are infinitely many [[Pythagorean triple]]s describing integer-~~sided [[right triangle]]s, and there are infinitely many equable right triangles with non~~-integer sides; however, there are only two equable integer right triangles, with side lengths (5,12,13) and (6,8,10).<ref name="bicycle">{{citation\|title=Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries\|volume=18\|series=Dolciani Mathematical Expositions\|first1=Joseph D. E.\|last1=Konhauser\|first2=Dan\|last2=Velleman\|first3=Stan\|last3=Wagon\|author3-link=Stan Wagon\|publisher=Cambridge University Press\|year=1997\|isbn=9780883853252\|url=http://books.google.com/books?id=ElSi5V5uS2MC&pg=PA29\|page=29\|contribution=95. When does the perimeter equal the area?}}</ref>

	More generally, the problem of finding all equable triangles with integer sides (that is, equable [[Heronian triangle]]s) was considered by B. Yates in 1858.<ref>{{citation\|first=B.\|last=Yates\|title=Quest 2019\|journal=The Lady's and Gentleman's Diary\|year=1858\|page=83}}.</ref><ref>{{citation\|title=[[History of the Theory of Numbers]], Volume Il: Diophantine Analysis\|first=Leonard Eugene\|last=Dickson\|authorlink=Leonard Eugene Dickson\|publisher=Courier Dover Publications\|year=2005\|isbn=9780486442334\|page=195}}.</ref> As [[William Allen Whitworth\|W. A. Whitworth]] and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already listed, with sides (6,25,29), (7,15,20), and (9,10,17).<ref>{{harvtxt\|Dickson\|2005}}, p. 199.</~~ref><ref>{{citation~~
	~~\| last = Markowitz \| first = L.~~
	~~\| issue = 3~~
	~~\| journal = The Mathematics Teacher~~
	~~\| pages = 222–223~~
	~~\| title = Area = Perimeter~~
	~~\| volume = 74~~
	~~\| year = 1981~~
	~~\| zbl = 1982d.06561}}.</ref>~~

	The only equable [[rectangle]]s with integer sides are the 4 × 4 square and the 3 × 6 rectangle.<ref name="bicycle"/> An integer rectangle is a special type of [[polyomino]], and more generally there exist polyominoes with equal area and perimeter for any [[even number\|even]] integer area greater than or equal to 16. For smaller areas, the perimeter of a polyomino must exceed its area.<ref>{{citation\|title=Geometry Labs\|first=Henri\|last=Picciotto\|year=1999\|publisher=MathEducationPage.org\|page=208\|url=http://books.google.com/books?id=7gTMKr7TT6gC&pg=PA208}}.</ref>

	~~== Equable solids ==~~
	~~In [[Three-dimensional space\|three dimensions]], a shape is equable when its [[surface area]] is numerically equal to its [[volume]].~~

	~~As with equable shapes in two dimensions, you may find an equable solid, in which the volume is numerically equal to the surface area, by scaling any solid by an appropriate factor.~~

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Geometric shapes]~~]