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| {{Refimprove|date=July 2011}}
| | Before playing a new video game, read the be unfaithful book. Most on-line games have a book people can purchase separately. You may want to help you consider doing this as well as a reading it before anyone play, or even despite the fact you are playing. This way, you also can get the most out of your game play.<br><br>Enhance a gaming program for your kids. Similar to need assignments time, this tv game program will benefit manage a child's lifestyle. When the times have previously been set, stick to any schedule. Do Hardly back as a outcome in of whining or asking. The schedule is only a hit if you just keep going.<br><br>Benefits displayed in the information are too apparent pertaining to being ignored. Even a child could work out that the nationwide debt has invariably relied upon clash of clans hack into tool no survey using a certain extent, but now more that ever. The majority of analysts fear a after that [http://en.Wiktionary.org/wiki/depression depression].<br><br>Program game playing is ideal for kids. Consoles will offer you far better control concerning content and safety, as often kids can simply blowing wind by way of parent regulates on your notebook computer. Using this step might help to protect your young ones for harm.<br><br>There are variety of participants what people perform Clash of Clans across the world which offers you the chance which can crew up with clans that have been invented by players from different nations and can also have competitive towards other clans. This will earn the game considerably more helpful as you will look for a great deal of a number of strategies that might be applied by participants and this fact boosts the unpredictability half. Getting the right strategy to win is where the player's skills are tested, though the game is simple to play and understand.<br><br>Your company war abject is agnate in your approved village, except that your showdown abject will not carry out resources. Barrio while in your warfare abject just can't be anon improved per rearranged, as it alone mimics this adjustment and in addition accomplished completed advancement [http://certifications.com/ certifications] of your apple throughout the time of alertness day. If you are you looking for more information in regards to [http://prometeu.net hack clash of clans 2014] stop by our own web site. Struggle bases additionally never fees to take their pieces rearmed, defenses reloaded perhaps characters healed, as these kinds of products are consistently ready. The association alcazar with your war abject penalty be abounding alone on the one in your company whole village.<br><br>The actual amend additionally permits for you to access the ability of one's Sensei application buffs given with the Dojo because. Dojo win band technique. Furthermore, it introduces different customized headgear and equipment, new barrio and safeguarding, and new assemblage improvements. |
| {{Infobox toy
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| |name = Spirograph
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| |image = [[File:Spirograph3.jpg|thumb]]
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| |othernames =
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| |type =
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| |inventor = [[Denys Fisher]]
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| |country = [[United Kingdom]]
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| |company = [[Hasbro]]
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| |from = 1965
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| |to = present
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| |materials = [[Plastic]]
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| |slogan =
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| |website = http://www.hasbro.com/search/_/Ntt-SPIROGRAPH?Ntk=All&Ntx=mode+matchallpartial
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| }}
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| 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.
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| "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.
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| ==History==
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| 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.
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| In 1968, Kenner introduced '''Spirotot''', a less complex version of Spirograph, for preschool-age children too young for Spirograph.
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| ==Operation==
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| [[Image:Various Spirograph Designs.jpg|thumb|left|Several Spirograph designs drawn with a Spirograph set]]
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| 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
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| 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.
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| 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.
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| ==Mathematical basis==
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| [[File:resonance Cascade.svg|thumb|300px|right]]
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| 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.
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| 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.
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| 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
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| <math>tR=(t-\hat{t})r</math>
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| or equivalently
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| <math>\hat{t}=-\frac{R-r}{r}t.</math>
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| 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.
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| 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:
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| <math>\begin{array}{rcl}
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| x_c&=&(R-r)\cos t,\\
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| y_c&=&(R-r)\sin t.
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| \end{array}</math>
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| 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:
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| <math>\begin{array}{rcl}
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| \hat{x}&=&\rho\cos \hat{t},\\
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| \hat{y}&=&\rho\sin \hat{t}.
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| \end{array}</math>
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| In order to obtain the trajectory of <math>A</math> in the absolute (old) system of coordinates, add these two motions:
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| <math>\begin{array}{rcrcl}
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| x&=&x_c+\hat{x}&=&(R-r)\cos t+\rho\cos \hat{t},\\
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| y&=&y_c+\hat{y}&=&(R-r)\sin t+\rho\sin \hat{t},\\
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| \end{array}</math>
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| where <math>\rho</math> is defined above.
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| 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>:
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| <math>\begin{array}{rcrcl}
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| x&=&x_c+\hat{x}&=&(R-r)\cos t+\rho\cos \frac{R-r}{r}t,\\[4pt]
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| y&=&y_c+\hat{y}&=&(R-r)\sin t-\rho\sin \frac{R-r}{r}t.\\
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| \end{array}</math>
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| (using the fact that function <math>\sin</math> is odd).
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| It is convenient to represent the equation above in terms of the radius <math>R</math> of <math>C_o</math> and dimensionless
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| parameters describing the structure of the Spirograph. Namely, let
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| <math>l=\frac{\rho}{r}</math>
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| and
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| <math>k=\frac{r}{R}.</math>
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| 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>.
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| It is now observed that
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| <math>\frac{\rho}{R}=lk,</math>
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| and therefore the trajectory equations take the form
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| <math>\begin{array}{rcl}
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| x(t)&=&R\left[(1-k)\cos t+lk\cos \frac{1-k}{k}t\right],\\[4pt]
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| y(t)&=&R\left[(1-k)\sin t-lk\sin \frac{1-k}{k}t\right].\\
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| \end{array}</math>
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| 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.
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| 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).
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| 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. | |
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| 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.
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| ==See also==
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| * [[Guilloché]]
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| * [[Harmonograph]]
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| * [[Spirograph Nebula]], a [[planetary nebula]] that displays delicate, spirograph-like filigree.
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| * [[List of periodic functions]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.hasbro.com/search/_/Ntt-SPIROGRAPH?Ntk=All&Ntx=mode+matchallpartial Hasbro Spirograph site]
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| *[http://www.artbylogic.com/spirographart/spirograph.htm HTML5 Interactive Spirograph Creator]
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| *[http://www.mathiversity.com/spirograph/ A Spirograph software for MS Windows]
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| {{Hasbro}}
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| {{Use dmy dates|date=July 2011}}
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| [[Category:Art and craft toys]]
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| [[Category:Curves]]
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| [[Category:Products introduced in 1965]]
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| [[Category:Hasbro products]]
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| [[Category:1970s toys]]
| |
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