Parasitic drag - Revision history

128.163.7.131: /* Skin friction */

2014-12-15T17:00:57Z

Skin friction

← Older revision		Revision as of 19:00, 15 December 2014
Line 1:		Line 1:
	~~[[File:Missing Square Animation.gif\|thumb\|right\|300px\|Missing square puzzle animation]]~~		I'm Stewart and I live in a seaside city in northern Netherlands, Capelle Aan Den Ijssel. I'm 27 and I'm will soon finish my study at Journalism.<br><br>Here is my homepage: [http://Www.filipinoadobo.com/testsite/how-to-get-free-fifa-15-coins-2 fifa 15 Coin generator]
	~~The '''missing square puzzle''~~' is an [[optical illusion]] used in [[mathematics]] classes to help students reason about geometrical figures. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled [[triangle]], but one has a 1×1 hole in it.

	~~==Solution==~~
	~~[[File:Missing square puzzle.svg\|thumb\|right\|200px\|The missing square shown in the lower triangle, where both triangles are in a perfect grid]]~~
	The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, because what appears to be the [[hypotenuse]] is bent. In other words, the "hypotenuse" does not maintain a consistent [[slope]], even though it may appear that way to the human eye. A true 13×5 triangle cannot be created from the given component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area. The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be <math>\textstyle{S=\frac{13 \times 5}{2}=32.5}</math> units. However, the blue triangle has a ratio of 5:2 (=2.5:1), while the red triangle has the ratio 8:3 (≈2.667:1), so the apparent combined [[hypotenuse]] in each figure is actually bent. So with the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square. The amount of bending is approximately 1/28th of a unit (1.245364267°), which is difficult to see on the diagram of this puzzle. Note the grid point where the red and blue triangles in the lower image meet (5 squares to the right and two units up from the lower left corner of the combined figure), and compare it to the same point on the other figure; the edge is slightly under the mark in the upper image, but goes through it in the lower. Overlaying the hypotenuses from both figures results in a ~~very thin [[parallelogram]] with an area of exactly one grid square—the same area "missing" from the second figure.~~

	~~===Principle===~~
	~~According to [[Martin Gardner]],<ref>~~
	~~{{cite book~~
	~~\|last= Martin~~
	~~\|first= Gardner~~
	~~\|title= Mathematics Magic and Mystery~~
	~~\|year= 1956~~
	~~\|publisher= Dover~~
	~~\|pages= 139–150~~
	~~\|isbn= 9780486203355~~
	~~}}</ref> this particular puzzle was invented by a [[New York City]] amateur magician, [[Paul Curry]],~~ in ~~1953. However~~, ~~the principle of a dissection paradox has been known since the start of the 16th century~~. The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive [[Fibonacci numbers]]. Many other geometric [[dissection puzzle]]s are based on a few simple properties of the Fibonacci sequence.<ref>{{cite web \|publisher=Math World \|last=Weisstein \|first=Eric \|title=Cassini'~~s Identity \|url=http://mathworld.wolfram.com/CassinisIdentity.html}}</ref>~~

	~~[[File:Missing square puzzle dimensions.png\|thumb\|right\|200px\|Missing square puzzle dimensions]]~~

	~~==Similar puzzles==~~
	~~[[File:Loyd64-65-dis b.svg\|thumb\|right\|200px\|[[Sam Loyd]]~~'~~s paradoxical dissection]]~~
	[[Sam Loyd]]'s paradoxical dissection. In the "larger" rearrangement, the gaps between the figures have a combined unit square more area than their square gaps counterparts, creating an illusion that the figures there take up more space than those in the square figure. In the "smaller" rearrangement, each quadrilateral needs to overlap the triangle by an area of half a unit for its top/bottom edge to align with a grid line.

	~~[[File:Missing square edit~~.~~gif\|thumb\|left\|150px\|A variant of Mitsunobu Matsuyama's "Paradox"]]~~
	Mitsunobu Matsuyama's "Paradox" uses four congruent [[quadrilateral]]s and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If ''a'' is the side of the large square and ''θ'' is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec<~~sup~~>2<~~/sup~~>~~''θ'' − 1. For ''θ'' = 5°, this~~ is ~~approximately 1.00765, which corresponds to a difference of about 0.8%.~~

	~~{{-}}~~

	~~==References==~~
	~~{{Reflist}}~~

	~~==External links==~~
	~~{{Commons category\|Missing square puzzle}}~~
	*A printable [http://www.archimedes-lab.org/workshop13skulls.html Missing Square variant] with a video demonstration.
	*[http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml Curry's Paradox: How Is It Possible?] at [[cut-the-knot]]
	*[http:~~//www.archimedes-lab.org/page3b.html Triangles and Paradoxes] at [[archimedes-lab.org]]~~
	*[http://~~www~~.~~marktaw~~.com/~~blog~~/~~TheTriangleProblem.html The Triangle Problem or What's Wrong with the Obvious Truth]~~
	*[http://www.mathematik.uni-~~bielefeld.de/~sillke/PUZZLES/jigsaw~~-~~paradox.html Jigsaw Paradox]~~
	*[http://www.slideshare.net/sualeh/the-~~eleven~~-~~holes~~-~~puzzle The Eleven Holes Puzzle]~~
	*[http://www.excelhero.com/blog/2010/09/excel-~~optical~~-~~illusions-week-30.html Very nice animated Excel workbook of the Missing Square Puzzle]~~
	*A video explaining [http://www.youtube.com/watch?v=eFw0878Ig-A&feature=related Curry's Paradox and Area] by James Stanton

	~~{{DEFAULTSORT:Missing Square Puzzle}}~~
	~~[[Category:Optical illusions]]~~
	~~[[Category:Fibonacci numbers]]~~
	~~[[Category:Elementary mathematics]]~~
	~~[[Category:Mathematics paradoxes]]~~
	~~[[Category:Puzzles]]~~
	~~[[Category:Geometric dissection]~~]

195.229.181.253 at 08:45, 18 December 2013

2013-12-18T08:45:34Z

← Older revision		Revision as of 10:45, 18 December 2013
Line 1:		Line 1:
	~~Sony Vegas Pro 12 Overview - displaying newer possibilities within the movie croping and editing software application~~. ~~I also offer a walkthrough~~ of ~~my Vimeo Give Settings~~.~~<br><br>Sony is~~ a ~~well known design~~, ~~however its competent online video changing software package is significantly less prevalent as distinct competent online video editors equivalent to Adobe Elite~~ [~~http~~:~~//Www~~.~~Guardian.Co.uk/search?q=Seasoned Seasoned~~] ~~professional and Avid Media channels Composer. It's a terrific software program without~~ a ~~doubt~~, ~~nevertheless can't contest with most of~~ the other ~~(dearer) movie publishers~~, ~~since it doesn't possess~~ the ~~whole qualified options that some foes have. Sony Vegas Pro 12 -~~ [~~http://sonyvegaspro12.info/ http://sonyvegaspro12.info~~], even ~~facilitates 4K Incredibly High definition~~. ~~Supporting a real amount~~ of ~~decision enables you to convey more place for fine detail~~ and ~~flexibility and its suitable to~~ the ~~video recording results/compositions to function correctly as it has superior quality production.~~<br><br>~~Sony's tech help crew is incredibly slow-moving to react~~. ~~We regularly discover~~ the ~~different on-line panels as being a~~ a ~~lot more practical method~~ of ~~get explanations on small changes and fixes that can help proficiency~~. ~~We stumbled upon a scenario~~ the ~~place~~ the ~~explorer tab~~ in ~~Sony Vegas Pro 12 no longer viewable~~ the ~~contents into~~ the ~~folders~~. ~~It really stopped doing work~~. ~~We didn't make any alterations~~ to ~~your personal preferences or perhaps~~ the ~~style~~. ~~Beyond doubt any time you not have~~ the ~~capability to browse for documents using~~ the ~~Explorer windowpane~~ in ~~Sony Vegas it turns into incredibly aggravating~~ to ~~apply requiring you to toggle inside~~ and ~~outside~~ of ~~internet directories.<br><br>To start with start looking~~ the ~~software interface resembles a lot of~~ the ~~diverse no-linear publishers with~~ the ~~review timelines~~, ~~windows 7 and instruments~~. ~~Almost all~~ of the ~~solutions labored the same as they have~~ to and was ~~reasonably easy to have began experiencing~~ by ~~no means touched~~ the ~~more modern type~~ of ~~Vegas Skilled~~. ~~In case you wanted this information along with you wish to receive particulars regarding sony vegas pro 12 repair download~~. ~~From understanding Sony Vegas Pro 12 -~~ [http://~~sonyvegaspro12~~.~~info~~/ ~~http~~:~~//sonyvegaspro12~~.~~info~~], ~~Sony’s internet site~~, ~~I was aware~~ that ~~Sony Vegas had numerous newest alternatives on~~ top of ~~up-to-date models~~ and that ~~i wished to make certain I found myself capable~~ of ~~take~~ a ~~look at lots of them - so as~~ a ~~substitute~~ of ~~studying~~ the ~~guidance tutorial I looked to You tube guides~~ and ~~Sony’s Webinars and was quickly announced nearly quickness on with~~ the ~~program.~~<br><br>~~Find out extra relating~~ to the ~~most current features Sony Vegas Professional 12 should give in this particular introduction and examination video of Sony~~'s ~~skilled low~~-~~linear improving computer software~~. ~~Vegas Professional 12 is really an included, onward~~-~~looking~~ at ~~generation setting up suitable for a whole new period of very creative experts~~. ~~Blending a nicely~~-~~well~~-~~known path~~-~~established timeline with many different of loving work~~-~~flow developments, Vegas Professional 12 is really a exhilarating, contemporary method to submit~~-~~manufacturing~~. Assisting even more formats, supplemental workflows, plus much more imaginative control than ever before sooner than, Vegas Pro 12 makes it simple to generate your information and categorical your artistic perception. ~~Sound Forge, Acidity, and Vegas application have described electronic digital written content development for any era~~ of ~~artisticspecialists~~ and ~~new comers, and fanatics.~~		[[File:Missing Square Animation.gif\|thumb\|right\|300px\|Missing square puzzle animation]]
			The '''missing square puzzle''' is an [[optical illusion]] used in [[mathematics]] classes to help students reason about geometrical figures. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled [[triangle]], but one has a 1×1 hole in it.

			==Solution==
			[[File:Missing square puzzle.svg\|thumb\|right\|200px\|The missing square shown in the lower triangle, where both triangles are in a perfect grid]]
			The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, because what appears to be the [[hypotenuse]] is bent. In other words, the "hypotenuse" does not maintain a consistent [[slope]], even though it may appear that way to the human eye. A true 13×5 triangle cannot be created from the given component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area. The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be <math>\textstyle{S=\frac{13 \times 5}{2}=32.5}</math> units. However, the blue triangle has a ratio of 5:2 (=2.5:1), while the red triangle has the ratio 8:3 (≈2.667:1), so the apparent combined [[hypotenuse]] in each figure is actually bent. So with the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square. The amount of bending is approximately 1/28th of a unit (1.245364267°), which is difficult to see on the diagram of this puzzle. Note the grid point where the red and blue triangles in the lower image meet (5 squares to the right and two units up from the lower left corner of the combined figure), and compare it to the same point on the other figure; the edge is slightly under the mark in the upper image, but goes through it in the lower. Overlaying the hypotenuses from both figures results in a very thin [[parallelogram]] with an area of exactly one grid square—the same area "missing" from the second figure.

			===Principle===
			According to [[Martin Gardner]],<ref>
			{{cite book
			\|last= Martin
			\|first= Gardner
			\|title= Mathematics Magic and Mystery
			\|year= 1956
			\|publisher= Dover
			\|pages= 139–150
			\|isbn= 9780486203355
			}}</ref> this particular puzzle was invented by a [[New York City]] amateur magician, [[Paul Curry]], in 1953. However, the principle of a dissection paradox has been known since the start of the 16th century. The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive [[Fibonacci numbers]]. Many other geometric [[dissection puzzle]]s are based on a few simple properties of the Fibonacci sequence.<ref>{{cite web \|publisher=Math World \|last=Weisstein \|first=Eric \|title=Cassini's Identity \|url=http://mathworld.wolfram.com/CassinisIdentity.html}}</ref>

			[[File:Missing square puzzle dimensions.png\|thumb\|right\|200px\|Missing square puzzle dimensions]]

			==Similar puzzles==
			[[File:Loyd64-65-dis b.svg\|thumb\|right\|200px\|[[Sam Loyd]]'s paradoxical dissection]]
			[[Sam Loyd]]'s paradoxical dissection. In the "larger" rearrangement, the gaps between the figures have a combined unit square more area than their square gaps counterparts, creating an illusion that the figures there take up more space than those in the square figure. In the "smaller" rearrangement, each quadrilateral needs to overlap the triangle by an area of half a unit for its top/bottom edge to align with a grid line.

			[[File:Missing square edit.gif\|thumb\|left\|150px\|A variant of Mitsunobu Matsuyama's "Paradox"]]
			Mitsunobu Matsuyama's "Paradox" uses four congruent [[quadrilateral]]s and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If ''a'' is the side of the large square and ''θ'' is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec<sup>2</sup>''θ'' − 1. For ''θ'' = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.

			{{-}}

			==References==
			{{Reflist}}

			==External links==
			{{Commons category\|Missing square puzzle}}
			*A printable [http://www.archimedes-lab.org/workshop13skulls.html Missing Square variant] with a video demonstration.
			*[http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml Curry's Paradox: How Is It Possible?] at [[cut-the-knot]]
			*[http://www.archimedes-lab.org/page3b.html Triangles and Paradoxes] at [[archimedes-lab.org]]
			*[http://www.marktaw.com/blog/TheTriangleProblem.html The Triangle Problem or What's Wrong with the Obvious Truth]
			*[http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/jigsaw-paradox.html Jigsaw Paradox]
			*[http://www.slideshare.net/sualeh/the-eleven-holes-puzzle The Eleven Holes Puzzle]
			*[http://www.excelhero.com/blog/2010/09/excel-optical-illusions-week-30.html Very nice animated Excel workbook of the Missing Square Puzzle]
			*A video explaining [http://www.youtube.com/watch?v=eFw0878Ig-A&feature=related Curry's Paradox and Area] by James Stanton

			{{DEFAULTSORT:Missing Square Puzzle}}
			[[Category:Optical illusions]]
			[[Category:Fibonacci numbers]]
			[[Category:Elementary mathematics]]
			[[Category:Mathematics paradoxes]]
			[[Category:Puzzles]]
			[[Category:Geometric dissection]]

en>Michael Hardy at 04:24, 2 September 2012

2012-09-02T04:24:10Z

New page

Sony Vegas Pro 12 Overview - displaying newer possibilities within the movie croping and editing software application. I also offer a walkthrough of my Vimeo Give Settings. Sony is a well known design, however its competent online video changing software package is significantly less prevalent as distinct competent online video editors equivalent to Adobe Elite [http://Www.Guardian.Co.uk/search?q=Seasoned Seasoned] professional and Avid Media channels Composer. It's a terrific software program without a doubt, nevertheless can't contest with most of the other (dearer) movie publishers, since it doesn't possess the whole qualified options that some foes have. Sony Vegas Pro 12 - [http://sonyvegaspro12.info/ http://sonyvegaspro12.info], even facilitates 4K Incredibly High definition. Supporting a real amount of decision enables you to convey more place for fine detail and flexibility and its suitable to the video recording results/compositions to function correctly as it has superior quality production. Sony's tech help crew is incredibly slow-moving to react. We regularly discover the different on-line panels as being a a lot more practical method of get explanations on small changes and fixes that can help proficiency. We stumbled upon a scenario the place the explorer tab in Sony Vegas Pro 12 no longer viewable the contents into the folders. It really stopped doing work. We didn't make any alterations to your personal preferences or perhaps the style. Beyond doubt any time you not have the capability to browse for documents using the Explorer windowpane in Sony Vegas it turns into incredibly aggravating to apply requiring you to toggle inside and outside of internet directories. To start with start looking the software interface resembles a lot of the diverse no-linear publishers with the review timelines, windows 7 and instruments. Almost all of the solutions labored the same as they have to and was reasonably easy to have began experiencing by no means touched the more modern type of Vegas Skilled. In case you wanted this information along with you wish to receive particulars regarding sony vegas pro 12 repair download. From understanding Sony Vegas Pro 12 - [http://sonyvegaspro12.info/ http://sonyvegaspro12.info], Sony’s internet site, I was aware that Sony Vegas had numerous newest alternatives on top of up-to-date models and that i wished to make certain I found myself capable of take a look at lots of them - so as a substitute of studying the guidance tutorial I looked to You tube guides and Sony’s Webinars and was quickly announced nearly quickness on with the program. Find out extra relating to the most current features Sony Vegas Professional 12 should give in this particular introduction and examination video of Sony's skilled low-linear improving computer software. Vegas Professional 12 is really an included, onward-looking at generation setting up suitable for a whole new period of very creative experts. Blending a nicely-well-known path-established timeline with many different of loving work-flow developments, Vegas Professional 12 is really a exhilarating, contemporary method to submit-manufacturing. Assisting even more formats, supplemental workflows, plus much more imaginative control than ever before sooner than, Vegas Pro 12 makes it simple to generate your information and categorical your artistic perception. Sound Forge, Acidity, and Vegas application have described electronic digital written content development for any era of artisticspecialists and new comers, and fanatics.