Spirograph
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Spirograph is a
geometric Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
drawing device that produces mathematical
roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
curves of the variety technically known as
hypotrochoid In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle. The parametric equations f ...
s and
epitrochoid In geometry, an epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle. The parametric ...
s. The well-known toy version was developed by British engineer
Denys Fisher Denys Fisher (11 May 1918 – 17 September 2002) was an English engineer who invented the spirograph toy and created the company Denys Fisher Toys. He left Leeds University to join the family firmKingfisher (Lubrication) Ltd In 1960 he left the ...
and first sold in 1965. The name has been a registered
trademark A trademark (also written trade mark or trade-mark) is a type of intellectual property consisting of a recognizable sign, design, or expression that identifies products or services from a particular source and distinguishes them from others ...
of
Hasbro Hasbro, Inc. (; a syllabic abbreviation of its original name, Hassenfeld Brothers) is an American multinational conglomerate holding company incorporated and headquartered in Pawtucket, Rhode Island. Hasbro owns the trademarks and products of ...
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, by
Kahootz Toys Kahootz Toys was a toy company based in Ann Arbor, Michigan, best known for the relaunch of the classic toy Spirograph. Kahootz founded in January 2012 by Doug Cass, Colleen Loughman, Joe Yassay, and Brent Oeschger after their previous company, Gi ...
.


History

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. When working in Vienna between 1845 and 1848, Desvignes constructed a version of the machine that would help prevent banknote forgeries, as any of the nearly endless variations of roulette patterns that it could produce were extremely difficult to reverse engineer. The mathematician
Bruno Abakanowicz Bruno Abdank-Abakanowicz (6 October 1852 – 29 August 1900) was a Polish mathematician, inventor, and electrical engineer. Life Abakanowicz was born in 1852 in the Russian Empire (now Lithuania). After graduating from the Riga Technical Univ ...
invented a new Spirograph device between 1881 and 1900. It was used for calculating an area delimited by curves. Drawing toys based on gears have been around since at least 1908, when The Marvelous Wondergraph was advertised in the
Sears catalog Sears, Roebuck and Co. ( ), commonly known as Sears, is an American chain of department stores founded in 1892 by Richard Warren Sears and Alvah Curtis Roebuck and reincorporated in 1906 by Richard Sears and Julius Rosenwald, with what began a ...
. An article describing how to make a Wondergraph drawing machine appeared in the ''Boys Mechanic'' publication in 1913. The definitive Spirograph toy was developed by the British engineer
Denys Fisher Denys Fisher (11 May 1918 – 17 September 2002) was an English engineer who invented the spirograph toy and created the company Denys Fisher Toys. He left Leeds University to join the family firmKingfisher (Lubrication) Ltd In 1960 he left the ...
between 1962 and 1964 by creating drawing machines with
Meccano Meccano is a brand of scale model, model construction system created in 1898 by Frank Hornby in Liverpool, England. The system consists of reusable metal strips, plates, Structural steel#Common structural shapes, angle girders, wheels, axles and ...
pieces. Fisher exhibited his spirograph at the 1965
Nuremberg International Toy Fair The Nuremberg International Toy Fair ( German: ''Spielwarenmesse''), held annually since 1949, is the largest international trade fair for toys and games. Only trade visitors associated with the toy business, journalists and invited guests are ...
. It was subsequently produced by his company. US distribution rights were acquired by
Kenner Kenner Products, known simply as Kenner, was an American toy company founded in 1946. Throughout its history, the Kenner brand produced several highly recognizable toys and merchandise lines including action figures like the original series of ' ...
, 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. 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.


Operation

The original US-released Spirograph consisted of two differently sized plastic rings (or
stator The stator is the stationary part of a rotary system, found in electric generators, electric motors, sirens, mud motors or biological rotors. Energy flows through a stator to or from the rotating component of the system. In an electric mot ...
s), 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 A ballpoint pen, also known as a biro (British English), ball pen (Hong Kong, Indian and Philippine English), or dot pen ( Nepali) is a pen that dispenses ink (usually in paste form) over a metal ball at its point, i.e. over a "ball point". ...
—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).


Mathematical basis

Consider a fixed outer circle C_o of radius R centered at the origin. A smaller inner circle C_i of radius r < R is rolling inside C_o and is continuously tangent to it. C_i will be assumed never to slip on C_o (in a real Spirograph, teeth on both circles prevent such slippage). Now assume that a point A lying somewhere inside C_i is located a distance \rho from C_i's center. This point 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 A was on the X axis. In order to find the trajectory created by a Spirograph, follow point A as the inner circle is set in motion. Now mark two points T on C_o and B on C_i. The point T always indicates the location where the two circles are tangent. Point B, however, will travel on C_i, and its initial location coincides with T. After setting C_i in motion counterclockwise around C_o, C_i has a clockwise rotation with respect to its center. The distance that point B traverses on C_i is the same as that traversed by the tangent point T on C_o, due to the absence of slipping. Now define the new (relative) system of coordinates (X', Y') with its origin at the center of C_i and its axes parallel to X and Y. Let the parameter t be the angle by which the tangent point T rotates on C_o, and t' be the angle by which C_i rotates (i.e. by which B travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by B and T along their respective circles must be the same, therefore : tR = (t - t')r, or equivalently, : t' = -\frac 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 (t' < 0) accommodates this convention. Let (x_c, y_c) be the coordinates of the center of C_i in the absolute system of coordinates. Then R - r represents the radius of the trajectory of the center of C_i, which (again in the absolute system) undergoes circular motion thus: : \begin x_c &= (R - r)\cos t,\\ y_c &= (R - r)\sin t. \end As defined above, t' is the angle of rotation in the new relative system. Because point A obeys the usual law of circular motion, its coordinates in the new relative coordinate system (x', y') are : \begin x' &= \rho\cos t',\\ y' &= \rho\sin t'. \end In order to obtain the trajectory of A in the absolute (old) system of coordinates, add these two motions: : \begin x &= x_c + x' = (R - r)\cos t + \rho\cos t',\\ y &= y_c + y' = (R - r)\sin t + \rho\sin t',\\ \end where \rho is defined above. Now, use the relation between t and t' as derived above to obtain equations describing the trajectory of point A in terms of a single parameter t: : \begin x &= x_c + x' = (R - r)\cos t + \rho\cos \fract,\\ y &= y_c + y' = (R - r)\sin t - \rho\sin \fract\\ \end (using the fact that function \sin is
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
). It is convenient to represent the equation above in terms of the radius R of C_o and dimensionless parameters describing the structure of the Spirograph. Namely, let : l = \frac and : k = \frac. The parameter 0 \le l \le 1 represents how far the point A is located from the center of C_i. At the same time, 0 \le k \le 1 represents how big the inner circle C_i is with respect to the outer one C_o. It is now observed that : \frac = lk, and therefore the trajectory equations take the form : \begin x(t) &= R\left 1 - k)\cos t + lk\cos \fract\right\\ y(t) &= R\left 1 - k)\sin t - lk\sin \fract\right\\ \end Parameter R is a scaling parameter and does not affect the structure of the Spirograph. Different values of R would yield similar Spirograph drawings. The two extreme cases k = 0 and k = 1 result in degenerate trajectories of the Spirograph. In the first extreme case, when k = 0, we have a simple circle of radius R, corresponding to the case where C_i has been shrunk into a point. (Division by k = 0 in the formula is not a problem, since both \sin and \cos are bounded functions.) The other extreme case k = 1 corresponds to the inner circle C_i's radius r matching the radius R of the outer circle C_o, i.e. r = R. In this case the trajectory is a single point. Intuitively, C_i is too large to roll inside the same-sized C_o without slipping. If l = 1, then the point A is on the circumference of C_i. In this case the trajectories are called
hypocycloid 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. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid crea ...
s and the equations above reduce to those for a hypocycloid.


See also

*
Cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spi ...
*
Apsidal precession In celestial mechanics, apsidal precession (or apsidal advance) is the precession (gradual rotation) of the line connecting the apsides (line of apsides) of an astronomical body's orbit. The apsides are the orbital points closest (periapsi ...
*
Cyclograph A cyclograph (also known as an arcograph) is an instrument for drawing arcs of large diameter circles whose centres are inconveniently or inaccessibly located, one version of which was invented by Scottish architect and mathematician Peter Nichols ...
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Geometric lathe A geometric lathe was used for making ornamental patterns on the plates used in printing bank notes and postage stamps. It is sometimes called a guilloché lathe. It was developed early in the nineteenth century when efforts were introduced to com ...
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Guilloché Guilloché (; or guilloche) is a decorative technique in which a very precise, intricate and repetitive pattern is mechanically engraved into an underlying material via engine turning, which uses a machine of the same name, also called a ros ...
*
Harmonograph A 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. The devices, which began to appear in the mid-19th century ...
*
Hypotrochoid In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle. The parametric equations f ...
*
Lissajous curve A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dire ...
*
List of periodic functions This is a list of some well-known periodic functions. The constant function , where is independent of , is periodic with any period, but lacks a ''fundamental period''. A definition is given for some of the following functions, though each funct ...
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Pantograph A pantograph (, from their original use for copying writing) is a mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line dr ...
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Pinion A pinion is a round gear—usually the smaller of two meshed gears—used in several applications, including drivetrain and rack and pinion systems. Applications Drivetrain Drivetrains usually feature a gear known as the pinion, which may ...
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Rose (mathematics) In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied ...
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Rosetta (orbit) A Rosetta orbit is a complex type of orbit. In astronomy, a Rosetta orbit occurs when there is a periastron shift during each orbital cycle. A retrograde Newtonian shift can occur when the central mass is extended rather than a point gravitati ...
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Spirograph Nebula IC 418, also known as Spirograph Nebula, is a planetary nebula located in the constellation of Lepus (constellation), Lepus about 3,600 Light-year, ly away. It spans 0.3 light-years across. The name derives from the intricate pattern of the nebul ...
, a
planetary nebula A planetary nebula (PN, plural PNe) is a type of emission nebula consisting of an expanding, glowing shell of ionized gas ejected from red giant stars late in their lives. The term "planetary nebula" is a misnomer because they are unrelated to ...
that displays delicate, spirograph-like filigree. *
Tusi couple The Tusi couple is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and fort ...


References


External links

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