Spirograph
Encyclopedia
Spirograph is a geometric drawing toy that produces mathematical curves of the variety technically known as hypotrochoid
s and epitrochoid
s. The term has also been used to describe a variety of software applications that display similar curves, and applied to the class of curves that can be produced with the drawing equipment (so in this sense it may be regarded as a synonym of hypotrochoid). The name is a registered trademark
of Hasbro
, Inc.
between 1881 and 1900 for calculating an area delimited by curves. The Spirograph itself was developed by the British engineer Denys Fisher
, who exhibited it in 1965 at the Nuremberg International Toy Fair
. It was subsequently produced by his company. Distribution rights were acquired by Kenner
, Inc., which introduced it to the United States market in 1966, promoting it as a creative children's toy.
In 1968, Kenner introduced Spirotot, a less complex version of Spirograph, for preschool-age children, too young for Spirograph.
A Spirograph consists of a set of plastic gear
s and other shapes such as rings, triangles, or straight bars. There are 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.
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 pinned to the paper and cardboard. Another plastic piece—called the 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 actual 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. 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.
of radius 
centered at the origin. A smaller circle 
of radius 
is rolling inside 
and it is tangent to 
. The inner circle cannot slip since teeth are present in a real Spirograph. Now assume that a point 
that corresponds to hole in the inner circle of the Spirograph is located at the distance 
from the center of 
. Without loss of generality it can be assumed that at the initial moment the point 
was on the 
-axis. In order to find the trajectory created by a Spirograph, follow 
as the inner circle is set in motion.
Now mark two points
on 
and 
on 
. The point 
indicates where two circles are tangent all the time. Point 
however will travel on 
and its initial location coincides with 
. After setting 
in motion counterclockwise, there is a clockwise rotation with respect to its center. The distances that point 
traverses on the small circle is the same as the distance that the tangent point 
travels on the large circle due to absence of any slipping effects.
Now the new (relative) system of coordinates
with its origin at the center of 
and its axes parallel to 
and 
is obserbable. If the parameter 
is defined as the angle by which the tangent point 
rotates on 
and 
is the angle by which 
rotates (i.e. by which 
travels) in the relative system of coordinates, then the distances traveled by 
and 
along their respective circles must be the same (no slipping). Therefore
or equivalently
It is common to assume that a counterclockwise motion results in a positive change of angle and a clockwise one will correspond to a negative change of angle. A minus sign in the above formula. (
)to accommodate this convention.
Let
be the coordinates of the center of 
in the absolute system of coordinates. Then 
represents the radius of the trajectory of the center of the inner circle, and
The coordinates of
in the new system are 
and they obey the regular law of circular motion (the angle of rotation in the relative system is 
):
In order to obtain the trajectory of
in the absolute (old) system of coordinates, add these two motions:
where
is defined above.
Now, use the relation between
and 
as discussed above to obtain equations describing the trajectory of point 
in terms of one parameter 
:
(using the fact that function
is odd)
It is convenient to represent the equation above in terms the radius
of the largest circle and dimensionless
parameters describing the structure of the Spirograph. Namely, let
and
The parameter
represents how far the point 
is located from the center of the inner circle. At the same time, 
represents how big the inner circle is with respect to the large one.
It is now observed that
and therefore the trajectory equations take form of
Parameter
is a scaling parameter and will not affect the structure of the Spirograph. It is interesting to note that two extreme cases of 
and 
will result in degenerate trajectories of the Spirograph. Namely when 
we will have a simple circle of radius 
. And indeed this case corresponds to the case when the inner circle is shrunk into a point. (Division by 
in the formula is not a problem since both 
and 
are bounded functions).
The other extreme case
corresponds to the inner circle matching the large circle. In this case the trajectory is a single point since the inner circle is too large to roll without slipping.
If
then it is the case when the point 
is on the circumference of the inner circle. In this case the trajectories are called hypocycloid
s and the equations will match the one describing a hypocycloid.
Hypotrochoid
A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle....
s and epitrochoid
Epitrochoid
An epitrochoid is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle....
s. The term has also been used to describe a variety of software applications that display similar curves, and applied to the class of curves that can be produced with the drawing equipment (so in this sense it may be regarded as a synonym of hypotrochoid). The name is a registered trademark
Trademark
A trademark, trade mark, or trade-mark is a distinctive sign or indicator used by an individual, business organization, or other legal entity to identify that the products or services to consumers with which the trademark appears originate from a unique source, and to distinguish its products or...
of Hasbro
Hasbro
Hasbro is a multinational toy and boardgame company from the United States of America. It is one of the largest toy makers in the world. The corporate headquarters is located in Pawtucket, Rhode Island, United States...
, Inc.
History
Drawing toys based on gears have been around since at least 1908, when "The Marvelous Wondergraph" was advertised in the Sears catalog. The Boys Mechanic publication of 1913 had an article describing how to make a Wondergraph drawing machine. An instrument called a spirograph was invented by the mathematician Bruno AbakanowiczBruno Abakanowicz
Bruno Abdank-Abakanowicz was a mathematician, inventor and electrical engineer.- Life and Nationality :Abakanowicz was born in 1852 in Vilkmergė, Lithuania, then part of the Russian Empire. After graduating from the Riga Technical University, Abakanowicz passed his habilitation and began an...
between 1881 and 1900 for calculating an area delimited by curves. The Spirograph itself was developed by the British engineer Denys Fisher
Denys Fisher
Denys Fisher was an English engineer who invented the spirograph toy....
, who exhibited it in 1965 at the Nuremberg International Toy Fair
Nuremberg International Toy Fair
Spielwarenmesse International Toy Fair Nürnberg is the largest international trade fair for toys and games. Only trade visitors associated with the toy business, journalists and invited guests are admitted...
. It was subsequently produced by his company. Distribution rights were acquired by Kenner
Kenner
Kenner Products was a toy company founded in 1947 by three brothers, Albert, Phillip, and Joseph L. Steiner, in Cincinnati, Ohio, United States, and was named after the street where the original corporate offices were located, which is just north of Cincinnati's Union Terminal.Kenner introduced its...
, Inc., which introduced it to the United States market in 1966, promoting it as a creative children's toy.
In 1968, Kenner introduced Spirotot, a less complex version of Spirograph, for preschool-age children, too young for Spirograph.
Operation
Gear
A gear is a rotating machine part having cut teeth, or cogs, which mesh with another toothed part in order to transmit torque. Two or more gears working in tandem are called a transmission and can produce a mechanical advantage through a gear ratio and thus may be considered a simple machine....
s and other shapes such as rings, triangles, or straight bars. There are 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.
To use it, a sheet of paper is placed on a heavy cardboard backing, and one of the plastic pieces—known as a stator
Stator
The stator is the stationary part of a rotor system, found in an electric generator, electric motor and biological rotors.Depending on the configuration of a spinning electromotive device the stator may act as the field magnet, interacting with the armature to create motion, or it may act as the...
—is pinned to the paper and cardboard. Another plastic piece—called the 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 actual 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. 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.
Mathematical basis
Consider a fixed circle of radius centered at the origin. A smaller circle of radius is rolling inside and it is tangent to . The inner circle cannot slip since teeth are present in a real Spirograph. Now assume that a point that corresponds to hole in the inner circle of the Spirograph is located at the distance from the center of . Without loss of generality it can be assumed that at the initial moment the point was on the -axis. In order to find the trajectory created by a Spirograph, follow as the inner circle is set in motion.Now mark two points
on and on . The point indicates where two circles are tangent all the time. Point however will travel on and its initial location coincides with . After setting in motion counterclockwise, there is a clockwise rotation with respect to its center. The distances that point traverses on the small circle is the same as the distance that the tangent point travels on the large circle due to absence of any slipping effects.Now the new (relative) system of coordinates
with its origin at the center of and its axes parallel to and is obserbable. If the parameter is defined as the angle by which the tangent point rotates on and is the angle by which rotates (i.e. by which travels) in the relative system of coordinates, then the distances traveled by and along their respective circles must be the same (no slipping). Thereforeor equivalently
It is common to assume that a counterclockwise motion results in a positive change of angle and a clockwise one will correspond to a negative change of angle. A minus sign in the above formula. (
)to accommodate this convention.Let
be the coordinates of the center of in the absolute system of coordinates. Then represents the radius of the trajectory of the center of the inner circle, andThe coordinates of
in the new system are and they obey the regular law of circular motion (the angle of rotation in the relative system is ):In order to obtain the trajectory of
in the absolute (old) system of coordinates, add these two motions:where
is defined above.Now, use the relation between
and as discussed above to obtain equations describing the trajectory of point in terms of one parameter :(using the fact that function
is odd)It is convenient to represent the equation above in terms the radius
of the largest circle and dimensionlessparameters describing the structure of the Spirograph. Namely, let
and
The parameter
represents how far the point is located from the center of the inner circle. At the same time, represents how big the inner circle is with respect to the large one.It is now observed that
and therefore the trajectory equations take form of
Parameter
is a scaling parameter and will not affect the structure of the Spirograph. It is interesting to note that two extreme cases of and will result in degenerate trajectories of the Spirograph. Namely when we will have a simple circle of radius . And indeed this case corresponds to the case when the inner circle is shrunk into a point. (Division by in the formula is not a problem since both and are bounded functions).The other extreme case
corresponds to the inner circle matching the large circle. In this case the trajectory is a single point since the inner circle is too large to roll without slipping.If
then it is the case when the point is on the circumference of the inner circle. In this case the trajectories are called hypocycloidHypocycloid
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle...
s and the equations will match the one describing a hypocycloid.
See also
- GuillochéGuillochéGuilloché is a decorative engraving technique in which a very precise intricate repetitive pattern or design is mechanically engraved into an underlying material with fine detail...
- HarmonographHarmonographA harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. The drawings created typically are Lissajous curves, or related drawings of greater complexity...
- Spirograph Nebula, a planetary nebulaPlanetary nebulaA planetary nebula is an emission nebula consisting of an expanding glowing shell of ionized gas ejected during the asymptotic giant branch phase of certain types of stars late in their life...
that displays delicate, spirograph-like filigree.