Spirograph

Spirograph set (early 1980s UK version)
Inventor(s)	Denys Fisher
Company	Hasbro
Country	United Kingdom
Availability	1965–present
Materials	Plastic
Official website

Spirograph is ageometric drawing device that produces mathematicalroulette curves of the variety technically known ashypotrochoids andepitrochoids. The well-known toy version was developed by British engineerDenys Fisher and first sold in 1965.

The name has been a registeredtrademark ofHasbro Inc. since 1998 following purchase of the company that had acquired the Denys Fisher company. The Spirograph brand was relaunched worldwide in 2013, with its original product configurations, byKahootz Toys.

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 mathematicianBruno Abakanowicz invented a new Spirograph device between 1881 and 1900. It was used for calculating an area delimited by curves.^[3]

Drawing toys based on gears have been around since at least 1908, whenThe Marvelous Wondergraph was advertised in theSears catalog.^[4][5] An article describing how to make a Wondergraph drawing machine appeared in theBoys Mechanic publication in 1913.^[6]

The definitive Spirograph toy was developed by the British engineerDenys Fisher between 1962 and 1964 by creating drawing machines withMeccano pieces. Fisher exhibited his spirograph at the 1965Nuremberg International Toy Fair. It was subsequently produced by his company. US distribution rights were acquired byKenner, 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]

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]

The original US-released Spirograph consisted of two differently sized plastic rings (orstators), 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 (orrotors)—each having holes for aballpoint 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.

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).

• 
  Animation of a Spirograph
• 
  Several Spirograph designs drawn with a Spirograph set using several different-colored pens
• 
  Closeup of a Spirograph wheel

Consider a fixed outer circle${\displaystyle C_o}$ of radius${\displaystyle R}$ centered at the origin. A smaller inner circle${\displaystyle C_i}$ of radius is rolling inside${\displaystyle C_o}$ and is continuously tangent to it.${\displaystyle C_i}$ will be assumed never to slip on${\displaystyle C_o}$ (in a real Spirograph, teeth on both circles prevent such slippage). Now assume that a point${\displaystyle A}$ lying somewhere inside${\displaystyle C_i}$ is located a distance from${\displaystyle C_i}$'s center. This point${\displaystyle A}$ 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${\displaystyle A}$ was on the${\displaystyle X}$ axis. In order to find the trajectory created by a Spirograph, follow point${\displaystyle A}$ as the inner circle is set in motion.

Now mark two points${\displaystyle T}$ on${\displaystyle C_o}$ and${\displaystyle B}$ on${\displaystyle C_i}$. The point${\displaystyle T}$ always indicates the location where the two circles are tangent. Point${\displaystyle B}$, however, will travel on${\displaystyle C_i}$, and its initial location coincides with${\displaystyle T}$. After setting${\displaystyle C_i}$ in motion counterclockwise around${\displaystyle C_o}$,${\displaystyle C_i}$ has a clockwise rotation with respect to its center. The distance that point${\displaystyle B}$ traverses on${\displaystyle C_i}$ is the same as that traversed by the tangent point${\displaystyle T}$ on${\displaystyle C_o}$, due to the absence of slipping.

Now define the new (relative) system of coordinates${\displaystyle (X^{\prime },Y^{\prime })}$ with its origin at the center of${\displaystyle C_i}$ and its axes parallel to${\displaystyle X}$ and${\displaystyle Y}$. Let the parameter${\displaystyle t}$ be the angle by which the tangent point${\displaystyle T}$ rotates on${\displaystyle C_o}$, and${\displaystyle t^{\prime }}$ be the angle by which${\displaystyle C_i}$ rotates (i.e. by which${\displaystyle B}$ travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by${\displaystyle B}$ and${\displaystyle T}$ along their respective circles must be the same, therefore

${\displaystyle tR=(t-t^{\prime })r,}$

or equivalently,

${\displaystyle t^{\prime }=-\frac{R-r}{r}t.}$

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 () accommodates this convention.

Let${\displaystyle (x_c,y_c)}$ be the coordinates of the center of${\displaystyle C_i}$ in the absolute system of coordinates. Then${\displaystyle R-r}$ represents the radius of the trajectory of the center of${\displaystyle C_i}$, which (again in the absolute system) undergoes circular motion thus:

As defined above,${\displaystyle t^{\prime }}$ is the angle of rotation in the new relative system. Because point${\displaystyle A}$ obeys the usual law of circular motion, its coordinates in the new relative coordinate system${\displaystyle (x^{\prime },y^{\prime })}$ are

In order to obtain the trajectory of${\displaystyle A}$ in the absolute (old) system of coordinates, add these two motions:

where${\displaystyle \rho }$ is defined above.

Now, use the relation between${\displaystyle t}$ and${\displaystyle t^{\prime }}$ as derived above to obtain equations describing the trajectory of point${\displaystyle A}$ in terms of a single parameter${\displaystyle t}$:

(using the fact that function${\displaystyle \sin }$ isodd).

It is convenient to represent the equation above in terms of the radius${\displaystyle R}$ of${\displaystyle C_o}$ and dimensionlessparameters describing the structure of the Spirograph. Namely, let

${\displaystyle l=\frac{\rho }{r}}$

and

${\displaystyle k=\frac{r}{R}.}$

The parameter${\displaystyle 0\le l\le 1}$ represents how far the point${\displaystyle A}$ is located from the center of${\displaystyle C_i}$. At the same time,${\displaystyle 0\le k\le 1}$ represents how big the inner circle${\displaystyle C_i}$ is with respect to the outer one${\displaystyle C_o}$.

It is now observed that

${\displaystyle \frac{\rho }{R}=lk,}$

and therefore the trajectory equations take the form

Parameter${\displaystyle R}$ is a scaling parameter and does not affect the structure of the Spirograph. Different values of${\displaystyle R}$ would yieldsimilar Spirograph drawings.

The two extreme cases${\displaystyle k=0}$ and${\displaystyle k=1}$ result in degenerate trajectories of the Spirograph. In the first extreme case, when${\displaystyle k=0}$, we have a simple circle of radius${\displaystyle R}$, corresponding to the case where${\displaystyle C_i}$ has been shrunk into a point. (Division by${\displaystyle k=0}$ in the formula is not a problem, since both${\displaystyle \sin }$ and${\displaystyle \cos }$ are bounded functions.)

The other extreme case${\displaystyle k=1}$ corresponds to the inner circle${\displaystyle C_i}$'s radius${\displaystyle r}$ matching the radius${\displaystyle R}$ of the outer circle${\displaystyle C_o}$, i.e.${\displaystyle r=R}$. In this case the trajectory is a single point. Intuitively,${\displaystyle C_i}$ is too large to roll inside the same-sized${\displaystyle C_o}$ without slipping.

If${\displaystyle l=1}$, then the point${\displaystyle A}$ is on the circumference of${\displaystyle C_i}$. In this case the trajectories are calledhypocycloids and the equations above reduce to those for a hypocycloid.

• Official website
• Voevudko, A. E. (12 March 2018)."Gearographic Curves".Code Project.

• Art and craft toys
• Curves
• Products introduced in 1965
• Hasbro products
• 1970s toys
• Ballpoint pen art

• Articles with short description
• Short description is different from Wikidata
• Use dmy dates from October 2020
• Official website not in Wikidata