Spirograph

Spirograph

Inventor Denys Fisher

Company Hasbro Kahootz Toys

(http://www.originalspirograph.com)

Country United Kingdom

Availability 1965present

Materials Plastic

Official website

(http://www.originalspirograph.com)

SpirographFrom Wikipedia, the free encyclopedia

Spirograph is a geometric drawing toy that

produces mathematical roulette curves of the

variety technically known as hypotrochoids and

epitrochoids. It was developed by British engineer

Denys Fisher and first sold in 1965. The name has

been a registered trademark of Hasbro, Inc., since

it bought the Denys Fisher company. The

Spirograph brand was relaunched with original

product configurations in 2013 by Kahootz Toys.

Contents

1 History

2 Operation

3 Mathematical basis

4 See also

5 References

6 External links

History

The mathematician Bruno Abakanowicz invented the spirograph between 1881 and 1900. It was

used for calculating an area delimited by curves.[1] Drawing toys based on gears have been around

since at least 1908, when The Marvelous Wondergraph was advertised in the Sears catalog.[2][3]

An article describing how to make a Wondergraph drawing machine appeared in the Boys Mechanic

publication in 1913.[4] 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, Inc., which introduced it to the United

States market in 1966 and promoted it as a creative children's toy.

In 2013, Kahootz Toys (http://www.originalspirograph.com) relaunched the Spirograph brand with

products that returned to the use of the original gears and wheels. The modern products use

removable putty in place of pins to hold the stationery pieces in place on the paper. The Spirograph

was a 2014 Toy of the Year finalist in 2 categories, almost 50 years after the toy was named Toy of

the Year in 1967.

Operation

Page 1 of 5Spirograph - Wikipedia, the free encyclopedia

18/08/2014http://en.wikipedia.org/wiki/Spirograph

Several Spirograph designs drawn

with a Spirograph set

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 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 Super-Spirograph

consisted of a set of plastic gears 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.

To use it, a sheet of paper is placed on a heavy cardboard backing, and one of the plastic

piecesknown as a statoris secured via pins or reusable adhesive to the paper and cardboard.

Another plastic piececalled the rotoris 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.

Mathematical basis

Consider a fixed outer circle of radius

centered at the origin. A smaller inner circle

of radius is rolling inside and is

continuously tangent to it. will be assumed

never to slip on (in a real Spirograph, teeth

on both circles prevent such slippage). Now

assume that a point lying somewhere inside

is located a distance from 's center.

This point 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 was on the -axis. In order

to find the trajectory created by a Spirograph,

follow point as the inner circle is set in

motion.

Now mark two points on and on .

The point always indicates the location where

the two circles are tangent. Point however will travel on and its initial location coincides with



. After setting in motion counterclockwise around , has a clockwise rotation with respect

to its center. The distance that point traverses on is the same as that traversed by the tangent

point on , due to the absence of slipping.

Now define the new (relative) system of coordinates with its origin at the center of and

its axes parallel to and . Let the parameter be the angle by which the tangent point rotates

on and be the angle by which rotates (i.e. by which travels) in the relative system of

coordinates. Because there is no slipping, the distances traveled by and along their respective

circles must be the same, therefore

or equivalently

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 be the coordinates of the center of in the absolute system of coordinates. Then

represents the radius of the trajectory of the center of , which (again in the absolute

system) undergoes circular motion thus:

As defined above, is the angle of rotation in the new relative system. Because point obeys the

usual law of circular motion, its coordinates in the new relative coordinate system obey:

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 derived above to obtain equations describing the trajectory

of point in terms of a single parameter :

(using the fact that function is odd).



It is convenient to represent the equation above in terms of the radius of 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 . At the

same time, represents how big the inner circle is with respect to the outer one .

It is now observed that

and therefore the trajectory equations take the form

Parameter is a scaling parameter and does not affect the structure of the Spirograph. Different

values of would yield similar Spirograph drawings.

It is interesting to note that the two extreme cases and result in degenerate trajectories

of the Spirograph. In the first extreme case when we have a simple circle of radius ,

corresponding to the case where has been 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 's radius matching the radius

of the outer circle , ie . In this case the trajectory is a single point. Intuitively, is too

large to roll inside the same-sized without slipping.

If then the point is on the circumference of . In this case the trajectories are called

hypocycloids and the equations above reduce to those for a hypocycloid.

See also

Cyclograph

Guilloch

Harmonograph

List of periodic functions

Pantograph

Spirograph Nebula, a planetary nebula that displays delicate, spirograph-like filigree.



References

1. ^ Goldstein, Cathrine; Gray, Jeremy; Ritter, Jim (1996). L'Europe mathmatique: histoires, mythes,

identits (http://books.google.com/books?id=Ri46VxE7Pc0C&pg=PA293). Editions MSH. p. 293.

Retrieved 17 July 2011.

2. ^ Kaveney, Wendy. "CONTENTdm Collection : Compound Object

Viewer" (http://digitallibrary.imcpl.org/cdm4/document.php?

CISOROOT=/tcm&CISOPTR=787&REC=4). digitallibrary.imcpl.org. Retrieved 17 July 2011.

3. ^ Linderman, Jim. "ArtSlant - Spirograph? No, MAGIC

PATTERN!" (http://www.artslant.com/chi/articles/show/16968). artslant.com. Retrieved 17 July 2011.

4. ^ "From The Boy Mechanic (1913) - A

Wondergraph" (http://www.marcdatabase.com/~lemur/lemur.com/library-of-antiquarian-

technology/philosophical-instruments/boy-mechanic-1913/index.html#introduction). marcdatabase.com.

2004. Retrieved 17 July 2011.

External links

Official Website (http://www.originalspirograph.com)

HTML5 Interactive Spirograph Creator

(http://www.artbylogic.com/spirographart/spirograph.htm)

A Spirograph software for MS Windows (http://www.mathiversity.com/spirograph/)

Retrieved from "http://en.wikipedia.org/w/index.php?title=Spirograph&oldid=619674860"

Categories: Art and craft toys Curves Products introduced in 1965 Hasbro products

1970s toys

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