Beginners often slip the gears, especially when using the holes near the edge of the larger wheels, resulting in broken or irregular lines. Experienced users may learn 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). Shaped wheels: Shaped wheels come in a wide variety of shapes, including bar, quad, triangle, and oval. Like the round wheels, shaped versions also have multiple holes to vary the design.

The definitive Spirograph toy was developed by the British engineer Denys Fisher between 1962 and 1964 by creating drawing machines with Meccano pieces. Fisher exhibited his spirograph at the 1965 Nuremberg International Toy Fair. It was subsequently produced by his company. US distribution rights were acquired by Kenner, Inc., which introduced it to the United States market in 1966 and promoted it as a creative children's toy. Kenner later introduced Spirotot, Magnetic Spirograph, Spiroman, and various refill sets. [7] x ( t ) = R [ ( 1 − k ) cos ⁡ t + l k cos ⁡ 1 − k k t ] , y ( t ) = R [ ( 1 − k ) sin ⁡ t − l k sin ⁡ 1 − k k t ] . {\displaystyle {\begin{aligned}x(t)&=R\left[(1-k)\cos t+lk\cos {\frac {1-k}{k}}t\right],\\y(t)&=R\left[(1-k)\sin t-lk\sin {\frac {1-k}{k}}t\right].\\\end{aligned}}}Drawing toys based on gears have been around since at least 1908, when The Marvelous Wondergraph was advertised in the Sears catalog. [4] [5] An article describing how to make a Wondergraph drawing machine appeared in the Boys Mechanic publication in 1913. [6] The other extreme case k = 1 {\displaystyle k=1} corresponds to the inner circle C i {\displaystyle C_{i}} 's radius r {\displaystyle r} matching the radius R {\displaystyle R} of the outer circle C o {\displaystyle C_{o}} , i.e. r = R {\displaystyle r=R} . In this case the trajectory is a single point. Intuitively, C i {\displaystyle C_{i}} is too large to roll inside the same-sized C o {\displaystyle C_{o}} without slipping. Wheels create the magic. Toothed edges and strategically placed holes provide multiple design options with each wheel. Spirograph sets come with anywhere from six to 25 wheels with the following options.

Most Spirograph sets have plastic wheels, but there are a few out there with metal wheels. Of course, metal is more durable than plastic, but metal is heavier to carry, and sets with metal wheels usually have fewer wheels for the price. The parameter 0 ≤ l ≤ 1 {\displaystyle 0\leq l\leq 1} represents how far the point A {\displaystyle A} is located from the center of C i {\displaystyle C_{i}} . At the same time, 0 ≤ k ≤ 1 {\displaystyle 0\leq k\leq 1} represents how big the inner circle C i {\displaystyle C_{i}} is with respect to the outer one C o {\displaystyle C_{o}} . Now define the new (relative) system of coordinates ( X ′ , Y ′ ) {\displaystyle (X',Y')} with its origin at the center of C i {\displaystyle C_{i}} and its axes parallel to X {\displaystyle X} and Y {\displaystyle Y} . Let the parameter t {\displaystyle t} be the angle by which the tangent point T {\displaystyle T} rotates on C o {\displaystyle C_{o}} , and t ′ {\displaystyle t'} be the angle by which C i {\displaystyle C_{i}} rotates (i.e. by which B {\displaystyle B} travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by B {\displaystyle B} and T {\displaystyle T} along their respective circles must be the same, thereforex = x c + x ′ = ( R − r ) cos ⁡ t + ρ cos ⁡ t ′ , y = y c + y ′ = ( R − r ) sin ⁡ t + ρ sin ⁡ t ′ , {\displaystyle {\begin{aligned}x&=x_{c}+x'=(R-r)\cos t+\rho \cos t',\\y&=y_{c}+y'=(R-r)\sin t+\rho \sin t',\\\end{aligned}}} In 1827, Greek-born English architect and engineer Peter Hubert Desvignes developed and advertised a "Speiragraph", a device to create elaborate spiral drawings. A man named J. Jopling soon claimed to have previously invented similar methods. [1] When working in Vienna between 1845 and 1848, Desvignes constructed a version of the machine that would help prevent banknote forgeries, [2] as any of the nearly endless variations of roulette patterns that it could produce were extremely difficult to reverse engineer. The mathematician Bruno Abakanowicz invented a new Spirograph device between 1881 and 1900. It was used for calculating an area delimited by curves. [3]

If l = 1 {\displaystyle l=1} , then the point A {\displaystyle A} is on the circumference of C i {\displaystyle C_{i}} . In this case the trajectories are called hypocycloids and the equations above reduce to those for a hypocycloid. Now, use the relation between t {\displaystyle t} and t ′ {\displaystyle t'} as derived above to obtain equations describing the trajectory of point A {\displaystyle A} in terms of a single parameter t {\displaystyle t} :Each Spirograph wheel has several holes into which you place a pencil or pen to create a design. Some sets have larger holes than others, and newer sets tend to have larger holes than older sets. Larger holes allow for the use of a greater variety of writing instruments. While small holes limit instrument choice, they hold the pen or pencil tighter, which can make drawing easier. Patterns can be made using both hands, though it may take some practice. If you really want to be creative, try drawing a single picture with the help of another person. You can get several wheels going around the same plate to make something truly unique. Spirograph set size and portability The two extreme cases k = 0 {\displaystyle k=0} and k = 1 {\displaystyle k=1} result in degenerate trajectories of the Spirograph. In the first extreme case, when k = 0 {\displaystyle k=0} , we have a simple circle of radius R {\displaystyle R} , corresponding to the case where C i {\displaystyle C_{i}} has been shrunk into a point. (Division by k = 0 {\displaystyle k=0} in the formula is not a problem, since both sin {\displaystyle \sin } and cos {\displaystyle \cos } are bounded functions.) Let ( x c , y c ) {\displaystyle (x_{c},y_{c})} be the coordinates of the center of C i {\displaystyle C_{i}} in the absolute system of coordinates. Then R − r {\displaystyle R-r} represents the radius of the trajectory of the center of C i {\displaystyle C_{i}} , which (again in the absolute system) undergoes circular motion thus: x = x c + x ′ = ( R − r ) cos ⁡ t + ρ cos ⁡ R − r r t , y = y c + y ′ = ( R − r ) sin ⁡ t − ρ sin ⁡ R − r r t {\displaystyle {\begin{aligned}x&=x_{c}+x'=(R-r)\cos t+\rho \cos {\frac {R-r}{r}}t,\\y&=y_{c}+y'=(R-r)\sin t-\rho \sin {\frac {R-r}{r}}t\\\end{aligned}}}

x c = ( R − r ) cos ⁡ t , y c = ( R − r ) sin ⁡ t . {\displaystyle {\begin{aligned}x_{c}&=(R-r)\cos t,\\y_{c}&=(R-r)\sin t.\end{aligned}}} To create designs, wheels are placed either within or along the outside of the plate or ring. Plates and rings have teeth on the outside and inside edge. Consequently, wheels can be used on either side. Plates and rings are held in place using Spiro-putty, magnets, or pins. In 2013 the Spirograph brand was re-launched worldwide, with the original gears and wheels, by Kahootz Toys. The modern products use removable putty in place of pins to hold the stationary pieces in place. The Spirograph was Toy of the Year in 1967, and Toy of the Year finalist, in two categories, in 2014. Kahootz Toys was acquired by PlayMonster LLC in 2019. [8] Operation edit Animation of a Spirograph Several Spirograph designs drawn with a Spirograph set using several different-colored pens Closeup of a Spirograph wheelor purse and store everything you need, though you’d need to replace the paper often. These are by far the most portable sets, but even large Spirograph sets are designed with portability in mind. They come with a carrying case in which to store wheels, pens, and paper so you can make art anywhere. Spirograph set features Round wheels: These basic wheels are probably the type with which you are familiar. They may have five to 35 holes. Each hole will create a slightly different pattern using the same wheel. As defined above, t ′ {\displaystyle t'} is the angle of rotation in the new relative system. Because point A {\displaystyle A} obeys the usual law of circular motion, its coordinates in the new relative coordinate system ( x ′ , y ′ ) {\displaystyle (x',y')} are It is convenient to represent the equation above in terms of the radius R {\displaystyle R} of C o {\displaystyle C_{o}} and dimensionless The original US-released Spirograph consisted of two differently sized plastic rings (or stators), with gear teeth on both the inside and outside of their circumferences. Once either of these rings were held in place (either by pins, with an adhesive, or by hand) any of several provided gearwheels (or rotors)—each having holes for a ballpoint pen—could be spun around the ring to draw geometric shapes. Later, the Super-Spirograph introduced additional shapes such as rings, triangles, and straight bars. All edges of each piece have teeth to engage any other piece; smaller gears fit inside the larger rings, but they also can rotate along the rings' outside edge or even around each other. Gears can be combined in many different arrangements. Sets often included variously colored pens, which could enhance a design by switching colors, as seen in the examples shown here.